From f02a4ab5ab70a8af80415ac35c5b48cd5a93a475 Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 16:44:49 +0200 Subject: [PATCH 1/9] Certify rational unitary FFT and rounded Toeplitz shear --- research/rational-unitary-fft/PROOF.md | 349 +++ research/rational-unitary-fft/README.md | 116 + research/rational-unitary-fft/RESULT.json | 2524 ++++++++++++++++++++ research/rational-unitary-fft/SOURCES.json | 16 + research/rational-unitary-fft/TASK.json | 64 + research/rational-unitary-fft/VERIFY.json | 28 + research/rational-unitary-fft/engine.py | 258 ++ research/rational-unitary-fft/exact.py | 223 ++ research/rational-unitary-fft/verify.py | 318 +++ 9 files changed, 3896 insertions(+) create mode 100644 research/rational-unitary-fft/PROOF.md create mode 100644 research/rational-unitary-fft/README.md create mode 100644 research/rational-unitary-fft/RESULT.json create mode 100644 research/rational-unitary-fft/SOURCES.json create mode 100644 research/rational-unitary-fft/TASK.json create mode 100644 research/rational-unitary-fft/VERIFY.json create mode 100644 research/rational-unitary-fft/engine.py create mode 100644 research/rational-unitary-fft/exact.py create mode 100644 research/rational-unitary-fft/verify.py diff --git a/research/rational-unitary-fft/PROOF.md b/research/rational-unitary-fft/PROOF.md new file mode 100644 index 0000000000..99a55596f6 --- /dev/null +++ b/research/rational-unitary-fft/PROOF.md @@ -0,0 +1,349 @@ +# Certified rational unitary FFT and rounded Toeplitz shear + +Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance. +Apache-2.0. + +I instantiate Chafik Boukhalfa/PR628's rational unit-phase construction +inside Vlado Vojdanovski/PR1673's retained FFT and Toeplitz-shear schedule. +PR3709 already supplies certified inward dyadic twiddles and rounded FFT +execution. I retain those credits. The result here concerns the unitary +child, its composed approximation and its fixed-grid dirty-data contract. + +## 1. Contract and precision allocation + +Let \(n=2^d\ge1\), and let \(T\in\mathbb Q(i)^{n\times n}\) be Toeplitz, +with first column \(c\) and first row \(r\). Define +\[ +g_0=h_0=c_0/2,\quad +g_j=(c_j+r_{n-j})/2,\quad h_j=(c_j-r_{n-j})/2\quad(1\le j2/3\), while the endpoint error is below \(1/6\). +For the true rotated phase \(v_0\), put +\[ +\tau=\Im(v_0)/(1+\Re(v_0)). +\] +The denominator is at least \(3/2\). Compute quotient extrema at the box +corners, round their midpoint to the nearest \((P+4)\)-grid value \(t_0\), +and form +\[ +\sigma=C\frac{1-t_0^2+2it_0}{1+t_0^2}. +\] +Then \(|\sigma|=1\) exactly, \(|t_0|<1\), and +\[ +|\sigma-e^{2\pi ik/n}|\le4E_B+2^{-P-4}. +\] +This follows from the quotient bound \(2E_B\) and the stereographic +derivative bound \(2/(1+t^2)\le2\). + +The executable checks the **maximum endpoint** component-\(\ell^1\) +distance from \(\sigma\) to the entire oracle box. That distance is at +most \(10E_B+2^{-P-3}<\nu\). A minimum distance to the box would not +certify the approximation. + +The parameter denominator is at most \(2^{P+4}\). Unit-phase components +therefore have numerator/denominator height at most \(2P+9\), with +slightly larger allowances in the code. The oracle's offline exact +Machin sums, factorials and phase-description arithmetic use polynomial +time and space in \(P\) and the input description length. A long external +phase numerator is an input cost, not a free oracle argument. + +## 3. Exactly unitary butterflies and FFT approximation + +Use the certified phase \(a+ib\) approximating \(e^{i\pi/4}\), and a +certified twiddle \(w\). Define +\[ +B(a,b,w)= +\begin{pmatrix}a&bw\\b&-aw\end{pmatrix} += +\begin{pmatrix}a&b\\b&-a\end{pmatrix}\operatorname{diag}(1,w). +\] +Since \(a^2+b^2=|w|^2=1\), this matrix is exactly unitary. +Its adjoint is +\(\operatorname{diag}(1,\bar w)\left[\begin{smallmatrix}a&b\\b&-a\end{smallmatrix}\right]\). +It is generally not the same expression with \(w\) merely conjugated. + +The splitter difference from the ideal \(1/\sqrt2\) splitter has norm +\(|(a+ib)-e^{i\pi/4}|\). Hence each butterfly differs from its ideal gate +by at most \(2\nu\). Disjoint gates give the same layer bound. I retain +the source's exact bit reversal and its \(d\) butterfly layers, so +\[ +\boxed{\|\widetilde F-F\|_2\le2d\nu,\qquad +F=\mathrm{DFT}_{-}/\sqrt n.} +\] +We telescope products of unitary layers, so no amplification factor +appears. Every exact FFT prefix and suffix is unitary. Reverse the layers +and adjoint the stored gates to obtain \(\widetilde F^*\); do not generate +a new approximate inverse phase table. + +## 4. Spectral preparation and Toeplitz approximation + +For a kernel \(k\), the ideal circulant eigenvalues are the +**unnormalized** DFT +\[ +\lambda_\ell=\sum_j k_j e^{-2\pi i\ell j/n}. +\] +I approximate each root independently by a certified rational unit, +sum exactly offline, and round each multiplier component once to the +nearest \(S\)-grid value. If \(\|k\|_1\le M_k\), then +\[ +\|\widehat\lambda-\lambda\|_\infty\le M_k\nu+v,\qquad +\|\widehat\lambda\|_\infty\le M_k+v. +\] +The second bound uses exact unit modulus before rounding. It does not +add a spurious phase-error magnitude factor. + +I store the rounded multipliers before online use. The exact offline +sum can have a denominator whose size grows with \(n\); carrying that +sum into data gates would violate the stated coefficient-width bound. + +The actual circulant child applies +\[ +\widetilde C_k=\widetilde F^*\operatorname{diag}(\widehat\lambda)\widetilde F, +\] +so +\[ +\|\widetilde C_k-C_k\|_2 +\le M_k\nu+v+2M_k\|\widetilde F-F\|_2. +\] +Exact unitarity alone does not make \(\widetilde F\) an exact diagonalizer. + +For \(D=\operatorname{diag}(e^{\pi ij/n})\), the source decomposition is +\[ +T=C_g+D^*C_{Dh}D. +\] +I generate each diagonal entry of \(\widetilde D\) independently. Thus +\(\|\widetilde D-D\|_2\le\nu\), without a powers' accumulated error or +an \(nP\)-bit phase representation. The actual schedule is the first +circulant shear, the common twist \(\widetilde D\) on both banks, the +circulant shear prepared from \(\widetilde D h\), and the inverse twist. + +Twisting preserves \(\|h\|_1\). Charge \(M_h\nu\) for changing the +second kernel and \(2M_h\nu\) for its two exterior twists. Combining +these terms with the two circulant estimates gives +\[ +\|\widetilde T-T\|_2 +\le(4d+1)M_g\nu+(4d+4)M_h\nu+2v +\le\boxed{\delta}. +\] +In exact rational arithmetic the complete map is +\((x,y)\mapsto(x,y+\widetilde T x)\). The input matrix may be zero +or singular because the pointwise shear gates remain invertible. + +## 5. Prefix bounds and rounded execution + +Every non-spectral gate stays within one bank and is unitary. +Each of the two spectral layers has orientation +\[ +S(A)=\begin{pmatrix}I&0\\A&I\end{pmatrix}. +\] +For a bank unitary \(U=\operatorname{diag}(U_x,U_y)\), +\[ +US(A)U^*=S(U_yAU_x^*). +\] +Move the bank unitaries to the ends of any contiguous gate segment. +The remaining shear matrix is a sum of at most two unitarily transformed +spectral matrices. Their norms are at most \(M_g+v\) and \(M_h+v\). +Consequently every segment, inverse segment, prefix and suffix has norm +at most +\[ +\boxed{H=1+M_g+M_h+2v.} +\] +Masked partial spectral layers satisfy the same estimate. +I do not multiply a shear bound once per stage. + +Nearest component rounding has error at most \(u/2\); a complex output +has error at most \(u/\sqrt2\). A disjoint layer updates at most \(2n\) +complex wires, so its error vector has norm at most \(\sqrt n\,u\). +The source schedule has at most \(L=4d+8\) layers, including empty +bit-reversal layers. Transporting each local residual through its +remaining exact suffix gives +\[ +\|\widehat z-\widetilde S z_0\|_2 +\le LH\sqrt n\,u\le LHnu. +\] +Every pre-round and rounded state has amplitude at most +\[ +A=H(R+Lnu). +\] +The source-bank error receives no coefficient-approximation term, +because the unrounded rational circuit restores that bank exactly. +Fixed-grid execution can still have nonzero source-restoration error; +the verifier includes a small-grid witness. + +The precision choices in Section 1 close the requested budget: +\[ +4(d+1)M\nu R<2^{-t}/4,\quad 2vR<2^{-t}/4,\quad +LHnu<2^{-t}/2. +\] +Here \(v\le1/32\) and \(H= 1, "positive phase precision") + self.bits, self.nu = bits, Q(1, 1 << bits) + self.working_bits = 2 * (bits + 32) + self.oracle = {"__name__": "_authenticated_phase_oracle"} + exec(compile(source, "", "exec", optimize=0), self.oracle) + need(self.oracle["pi_interval"].cache_info().currsize == 0 + and self.oracle["kernel"].cache_info().currsize == 0, "fresh precision namespace") + self.oracle["BITS"] = self.working_bits + self.oracle["SCALE"] = 1 << self.working_bits + self.values, self.boxes, self.certificates = {}, {}, {} + + @staticmethod + def key(denominator, numerator): + need(type(denominator) is int and denominator > 0 and type(numerator) is int, + "integer phase description") + numerator %= denominator + divisor = gcd(denominator, numerator) + return denominator // divisor, numerator // divisor + + def box(self, denominator, numerator): + key = self.key(denominator, numerator) + if key not in self.boxes: + exact = {(1, 0): C(1), (2, 1): C(-1), (4, 1): C(0, 1), (4, 3): C(0, -1)} + if key in exact: + z = exact[key] + self.boxes[key] = ((z.re, z.re), (z.im, z.im)) + else: + self.boxes[key] = self.oracle["root_interval"](*key) + return self.boxes[key] + + def phase(self, denominator, numerator): + key = self.key(denominator, numerator) + if key in self.values: + return self.values[key] + box = self.box(*key) + if box[0][0] == box[0][1] and box[1][0] == box[1][1]: + sigma, error, parameter = C(box[0][0], box[1][0]), Q(0), None + else: + choices = [] + for center in (C(1), C(-1), C(0, 1), C(0, -1)): + conjugate = center.conjugate() + point = self.oracle["point"] + rotated = self.oracle["cmul"](box, (point(conjugate.re), point(conjugate.im))) + choices.append((rotated[0][0], center, rotated)) + _, center, rotated = max(choices, key=lambda item: item[0]) + need(rotated[0][0] >= Q(1, 2), "certified unit-phase chart") + parameters = [imag / (1 + real) for imag in rotated[1] for real in rotated[0]] + parameter = round_q((min(parameters) + max(parameters)) / 2, self.bits + 4) + need(abs(parameter) < 1, "bounded stereographic parameter") + sigma = center * C((1 - parameter ** 2) / (1 + parameter ** 2), + 2 * parameter / (1 + parameter ** 2)) + # Maximum endpoint distance, not minimum distance to a rectangle. + error = sum(max(abs(endpoint - value) for endpoint in interval) + for interval, value in zip(box, (sigma.re, sigma.im))) + need(sigma.norm2() == 1 and error <= self.nu, "certified exact rational unit phase") + need(max(qheight(sigma.re), qheight(sigma.im)) <= 2 * self.bits + 12, + "unit-phase representation bound") + self.values[key] = sigma + self.certificates[key] = {"denominator": key[0], "numerator": key[1], + "unit": sigma.record(), "error_bound": str(error), + "parameter": None if parameter is None else str(parameter)} + return sigma + + def record(self): + return {"phase_bits": self.bits, "oracle_bits": self.working_bits, + "certificates": [self.certificates[key] for key in sorted(self.certificates)]} + + +def split_kernel(matrix): + n = len(matrix) + need(n > 0 and n & (n - 1) == 0 and matrix.shape == (n, n), "square dyadic tile") + column, row = matrix[:, 0], matrix[0, :] + need(all(matrix[i][j] == (column[i - j] if i >= j else row[j - i]) + for i in range(n) for j in range(n)), "Toeplitz input") + g, h = Vec([0] * n), Vec([0] * n) + g[0] = h[0] = column[0] / 2 + for j in range(1, n): + g[j] = (column[j] + row[n - j]) / 2 + h[j] = (column[j] - row[n - j]) / 2 + return g, h + + +def budget(matrix, radius, target, input_fractional_bits=0): + g, h = split_kernel(matrix) + n, depth = len(matrix), len(matrix).bit_length() - 1 + mg, mh = component_l1(g), component_l1(h) + total = mg + mh + need(type(target) is int and target >= 1 and type(radius) in (int, Q) and radius >= 1, + "target and exact input radius") + need(type(input_fractional_bits) is int and input_fractional_bits >= 0, + "declared incoming fractional precision") + layers = 4 * depth + 8 + phase_bits = target + ceil_log2(16 * (depth + 1) * (total + 1) * (radius + 1)) + multiplier_bits = target + ceil_log2(8 * (radius + 1)) + data_bits = max(input_fractional_bits, target + ceil_log2(2 * layers * n * (total + 2))) + return certificate(n, mg, mh, radius, target, phase_bits, multiplier_bits, data_bits, + input_fractional_bits) + + +def certificate(n, mg, mh, radius, target, phase_bits, multiplier_bits, data_bits, + input_fractional_bits=0): + need(type(n) is int and n > 0 and n & (n - 1) == 0, "dyadic certificate dimension") + need(all(type(x) is int for x in (target, phase_bits, multiplier_bits, data_bits)) + and all(type(x) in (int, Q) for x in (mg, mh, radius)), "exact budget fields") + need(type(input_fractional_bits) is int and 0 <= input_fractional_bits <= data_bits, + "preserve incoming data grid") + need(min(phase_bits, multiplier_bits, data_bits, target) >= 1 and min(mg, mh) >= 0 + and radius >= 1, "positive precisions and valid envelopes") + depth, layers = n.bit_length() - 1, 4 * (n.bit_length() - 1) + 8 + nu, v, u = (Q(1, 1 << b) for b in (phase_bits, multiplier_bits, data_bits)) + total = mg + mh + coefficient_error = 4 * (depth + 1) * total * nu + 2 * v + prefix_bound = 1 + total + 2 * v + rounding_error = layers * prefix_bound * n * u + error = coefficient_error * radius + rounding_error + need(error <= Q(1, 1 << target), "complete target error budget") + need(layers * n * u <= 1, "bounded rounding accumulation") + amplitude = prefix_bound * (radius + layers * n * u) + integer_bits = max(1, ceil_log2(max(Q(1), amplitude))) + 2 + coefficient_height = max(4 * phase_bits + 24, + multiplier_bits + ceil_log2(total + 2) + 2) + temporary_bits = 2 * data_bits + 8 * coefficient_height + 2 * integer_bits + 32 + return dict(n=n, depth=depth, layer_bound=layers, mg=mg, mh=mh, radius=radius, + target_bits=target, phase_bits=phase_bits, multiplier_bits=multiplier_bits, + data_bits=data_bits, input_fractional_bits=input_fractional_bits, + coefficient_error=coefficient_error, prefix_bound=prefix_bound, + rounding_error=rounding_error, total_error=error, amplitude_bound=amplitude, + integer_bits=integer_bits, coefficient_height_bound=coefficient_height, + temporary_bits=temporary_bits) + + +class Adapt(ast.NodeTransformer): + def __init__(self): + self.function = None + self.changes = [] + + def visit_FunctionDef(self, node): + old = self.function + self.function = node.name + result = self.generic_visit(node) + self.function = old + return result + + def visit_Assign(self, node): + if len(node.targets) == 1 and isinstance(node.targets[0], ast.Name): + name = node.targets[0].id + if self.function == "fft" and name == "w": + self.changes.append("fft-twiddle") + node.value = ast.Call(func=ast.Name(id="phase", ctx=ast.Load()), + args=[ast.Name(id="size", ctx=ast.Load()), + ast.UnaryOp(op=ast.USub(), operand=ast.Name(id="j", ctx=ast.Load()))], + keywords=[]) + return node + if self.function == "toeplitz_shear" and name == "gamma": + self.changes.append("independent-twist-coefficients") + return ast.Pass() + if self.function == "toeplitz_shear" and name == "d": + self.changes.append("twist-vector") + node.value = ast.Call(func=ast.Name(id="certified_twist_vector", ctx=ast.Load()), + args=[ast.Name(id="n", ctx=ast.Load())], keywords=[]) + return node + return self.generic_visit(node) + + def visit_BinOp(self, node): + if (self.function == "fft" and isinstance(node.op, ast.Div) + and isinstance(node.right, ast.Call) and isinstance(node.right.func, ast.Attribute) + and node.right.func.attr == "sqrt"): + self.changes.append("unitary-splitter") + return ast.Call(func=ast.Name(id="butterfly", ctx=ast.Load()), + args=[ast.Name(id="w", ctx=ast.Load())], keywords=[]) + return self.generic_visit(node) + + +class Builder: + def __init__(self, phase_oracle, multiplier_bits, circuit_source): + self.phases, self.multiplier_bits = phase_oracle, multiplier_bits + self.spectra, self.twist_vectors, self.bounds = [], [], [] + selected = {"diag", "inverse", "parallel", "fft", "circ_shear", "toeplitz_shear"} + tree = ast.parse(circuit_source) + tree.body = [copy.deepcopy(n) for n in tree.body + if isinstance(n, ast.FunctionDef) and n.name in selected] + need({node.name for node in tree.body} == selected, "complete source function set") + self.original_ast = sha256(ast.dump(tree).encode()).hexdigest() + adapter = Adapt() + tree = ast.fix_missing_locations(adapter.visit(tree)) + need(adapter.changes == ["fft-twiddle", "unitary-splitter", + "independent-twist-coefficients", "twist-vector"], + "exact declared coefficient adaptation") + self.adapted_ast = sha256(ast.dump(tree).encode()).hexdigest() + self.np = SimpleNamespace(array=lambda rows, dtype=None: Mat(rows), + zeros=lambda n, dtype=None: Vec([0] * n), + fft=SimpleNamespace(fft=self.spectrum), + linalg=SimpleNamespace(inv=inverse_gate)) + self.namespace = {"np": self.np, "phase": self.phases.phase, + "butterfly": self.butterfly, "certified_twist_vector": self.twists} + exec(compile(tree, "", "exec", optimize=0), self.namespace) + + def butterfly(self, w): + splitter = self.phases.phase(8, 1) + a, b = splitter.re, splitter.im + return Mat([[a, b * w], [b, -a * w]]) + + def twists(self, n): + result = Vec(self.phases.phase(2 * n, j) for j in range(n)) + self.twist_vectors.append(result) + return result + + def spectrum(self, kernel): + n = len(kernel) + bound = self.bounds[len(self.spectra)] + exact = Vec(sum((kernel[j] * self.phases.phase(n, -i * j) for j in range(n)), C()) + for i in range(n)) + result = Vec(rounded(value, self.multiplier_bits) for value in exact) + v = Q(1, 1 << self.multiplier_bits) + need(all(value.norm2() <= bound ** 2 for value in exact), "unit-root multiplier bound") + need(all(value.norm2() <= (bound + v) ** 2 for value in result), "rounded multiplier bound") + need(all((x - y).norm2() <= v ** 2 for x, y in zip(result, exact)), "multiplier rounding") + self.spectra.append({"kernel": Vec(kernel), "multipliers": result, "bound": bound}) + return result + + def plan(self, matrix): + g, h = split_kernel(matrix) + n = len(matrix) + self.bounds = [component_l1(g), component_l1(h)] + self.spectra, self.twist_vectors = [], [] + result = self.namespace["toeplitz_shear"](list(range(n, 2 * n)), list(range(n)), matrix) + need(len(self.spectra) == 2 and len(self.twist_vectors) == 1, "two circulants and one twist table") + need(self.spectra[0]["kernel"] == g, "first source kernel") + need(self.spectra[1]["kernel"] == self.twist_vectors[0] * h, "twisted second source kernel") + need(all(a.norm2() == b.norm2() for a, b in zip(self.spectra[1]["kernel"], h)), + "twisting preserves kernel magnitudes") + return result + + +def inverse_sqrt_box(n, bits): + scale = 1 << bits + lower = isqrt((1 << (2 * bits)) // n) + if lower * lower * n == 1 << (2 * bits): + return Q(lower, scale), Q(lower, scale) + return Q(lower, scale), Q(lower + 1, scale) diff --git a/research/rational-unitary-fft/exact.py b/research/rational-unitary-fft/exact.py new file mode 100644 index 0000000000..0475e7d803 --- /dev/null +++ b/research/rational-unitary-fft/exact.py @@ -0,0 +1,223 @@ +"""Exact complex rational arithmetic for coefficient and rounded-gate certificates. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +from fractions import Fraction as Q +from hashlib import sha256 + +TRACK = None + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def qheight(value): + return max(abs(value.numerator).bit_length(), value.denominator.bit_length()) + + +class C: + __slots__ = ("re", "im") + + def __init__(self, re=0, im=0): + need(type(re) in (int, Q) and type(im) in (int, Q), "exact rational components") + self.re, self.im = Q(re), Q(im) + if TRACK is not None: + TRACK["max_observed_height"] = max(TRACK["max_observed_height"], + qheight(self.re), qheight(self.im)) + + @staticmethod + def wrap(value): + return value if isinstance(value, C) else C(value) + + def __add__(self, other): + other = C.wrap(other) + return C(self.re + other.re, self.im + other.im) + + __radd__ = __add__ + + def __neg__(self): + return C(-self.re, -self.im) + + def __sub__(self, other): + return self + -C.wrap(other) + + def __rsub__(self, other): + return C.wrap(other) + -self + + def __mul__(self, other): + other = C.wrap(other) + return C(self.re * other.re - self.im * other.im, + self.re * other.im + self.im * other.re) + + __rmul__ = __mul__ + + def conjugate(self): + return C(self.re, -self.im) + + def norm2(self): + return self.re * self.re + self.im * self.im + + def __truediv__(self, other): + other = C.wrap(other) + denominator = other.norm2() + need(denominator > 0, "nonzero divisor") + product = self * other.conjugate() + return C(product.re / denominator, product.im / denominator) + + def __rtruediv__(self, other): + return C.wrap(other) / self + + def __eq__(self, other): + if not isinstance(other, C) and type(other) not in (int, Q): + return False + other = C.wrap(other) + return self.re == other.re and self.im == other.im + + def record(self): + return [str(self.re), str(self.im)] + + +class Vec(list): + def __init__(self, values): + super().__init__(C.wrap(x) for x in values) + + def __mul__(self, other): + need(len(self) == len(other), "elementwise vector dimensions") + return Vec(x * y for x, y in zip(self, other)) + + +class Mat(list): + def __init__(self, rows): + super().__init__([C.wrap(x) for x in row] for row in rows) + + @property + def shape(self): + return (len(self), len(list.__getitem__(self, 0)) if self else 0) + + def __getitem__(self, key): + if not isinstance(key, tuple): + return list.__getitem__(self, key) + row, column = key + if isinstance(row, slice): + rows = list.__getitem__(self, row) + return Mat([r[column] for r in rows]) if isinstance(column, slice) else Vec(r[column] for r in rows) + selected = list.__getitem__(self, row) + return Vec(selected[column]) if isinstance(column, slice) else selected[column] + + +def eye(n): + return Mat([[int(i == j) for j in range(n)] for i in range(n)]) + + +def adjoint(matrix): + return Mat([[matrix[j][i].conjugate() for j in range(len(matrix))] + for i in range(len(matrix[0]))]) + + +def multiply(left, right): + return Mat([[sum((x * right[k][j] for k, x in enumerate(row)), C()) + for j in range(len(right[0]))] for row in left]) + + +def unitary(matrix): + return multiply(adjoint(matrix), matrix) == eye(len(matrix)) + + +def inverse_gate(matrix): + if unitary(matrix): + return adjoint(matrix) + if matrix.shape == (2, 2) and matrix[0][0] == matrix[1][1] == 1 and matrix[0][1] == 0: + return Mat([[1, 0], [-matrix[1][0], 1]]) + raise ValueError("unsupported nonunitary inverse gate") + + +def round_q(value, bits): + scale = 1 << bits + shifted = value * scale + Q(1, 2) + return Q(shifted.numerator // shifted.denominator, scale) + + +def rounded(value, bits): + return C(round_q(value.re, bits), round_q(value.im, bits)) + + +def norm2(vector): + return sum((x.norm2() for x in vector), Q(0)) + + +def component_l1(vector): + return sum((abs(x.re) + abs(x.im) for x in vector), Q(0)) + + +def ceil_log2(value): + value = Q(value) + need(value > 0, "positive logarithm bound") + result = value.numerator.bit_length() - value.denominator.bit_length() + power = Q(1 << result) if result >= 0 else Q(1, 1 << -result) + while power < value: + result += 1 + power *= 2 + while power / 2 >= value: + result -= 1 + power /= 2 + return result + + +def upper(value, bits=160): + scale = 1 << bits + scaled = value * scale + return Q(-((-scaled.numerator) // scaled.denominator), scale) + + +def vector_hash(vector): + digest = sha256() + for entry in vector: + for value in (entry.re, entry.im): + for integer in (value.numerator, value.denominator): + raw = abs(integer).to_bytes(max(1, (abs(integer).bit_length() + 7) // 8), "big") + digest.update(bytes([integer < 0])) + digest.update(len(raw).to_bytes(8, "big")) + digest.update(raw) + return digest.hexdigest() + + +def apply(plan, vector, bits=None): + global TRACK + state = Vec(vector) + old_track = TRACK + stats = {"max_observed_height": 0, "max_grid_numerator_bits": 0, + "max_reduced_fractional_bits": 0, "rounded_outputs": 0} + if bits is not None: + need(bits >= 1, "positive data precision") + for entry in state: + for value in (entry.re, entry.im): + need((value * (1 << bits)).denominator == 1, "input already on data grid") + TRACK = stats + try: + for layer in plan: + used = set() + for ids, gate in layer: + need(len(ids) == len(gate) in (1, 2), "one- or two-wire gate") + need(not used.intersection(ids) and len(set(ids)) == len(ids), "disjoint layer") + need(all(0 <= i < len(state) for i in ids), "wire bounds") + used.update(ids) + before = [state[i] for i in ids] + outputs = [sum((a * x for a, x in zip(row, before)), C()) for row in gate] + for i, value in zip(ids, outputs): + if bits is not None: + value = rounded(value, bits) + stats["rounded_outputs"] += 1 + for part in (value.re, value.im): + grid_integer = part * (1 << bits) + need(grid_integer.denominator == 1, "rounded output grid") + stats["max_grid_numerator_bits"] = max(stats["max_grid_numerator_bits"], + abs(grid_integer.numerator).bit_length()) + stats["max_reduced_fractional_bits"] = max(stats["max_reduced_fractional_bits"], + part.denominator.bit_length() - 1) + state[i] = value + finally: + TRACK = old_track + return state, stats diff --git a/research/rational-unitary-fft/verify.py b/research/rational-unitary-fft/verify.py new file mode 100644 index 0000000000..60d02a6f0d --- /dev/null +++ b/research/rational-unitary-fft/verify.py @@ -0,0 +1,318 @@ +"""Exact gate certificates and fixed-grid replay of the retained FFT/shear child. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +sys.path.insert(0, str(HERE)) +from exact import (C, Mat, Vec, adjoint, apply, component_l1, eye, inverse_gate, + need, norm2, qheight, round_q, unitary, upper, vector_hash) +from engine import Builder, Phases, budget, certificate, inverse_sqrt_box + +ORACLE = "research/cubic-cyclic-inverse/vendor/interval_kernel.py" +CIRCUIT = "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_circuits.py" + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(",", ":")).encode() + + +def rejected(operation, label): + try: + operation() + except (ValueError, AssertionError): + return label + raise ValueError("control unexpectedly passed: " + label) + + +def authenticate(): + rows = [r for r in json.loads((HERE / "SOURCES.json").read_text())["files"] if r["usage"] == "input"] + raw = {} + for row in rows: + content = (ROOT / row["path"]).read_bytes() + need(sha256(content).hexdigest() == row["sha256"], "source SHA256: " + row["path"]) + rejected(lambda: need(sha256(content + b"\0").hexdigest() == row["sha256"], "changed source"), + "source corruption") + raw[row["path"]] = content + return raw, [{"path": r["path"], "sha256": r["sha256"]} for r in rows] + + +def operator(plan, n): + columns = [apply(plan, Vec(int(i == j) for i in range(n)))[0] for j in range(n)] + return Mat([[columns[j][i] for j in range(n)] for i in range(n)]) + + +def mv(matrix, vector): + return Vec(sum((a * x for a, x in zip(row, vector)), C()) for row in matrix) + + +def plan_hash(plan): + return sha256(canonical([[[list(ids), [[x.record() for x in row] for row in gate]] + for ids, gate in layer] for layer in plan])).hexdigest() + + +def fft_check(builder, n, phases): + plan = builder.namespace["fft"](list(range(n))) + before = len(phases.certificates) + inverse = builder.namespace["inverse"](plan) + need(len(phases.certificates) == before, "inverse reuses stored coefficients") + for layer in plan: + used = set() + for ids, gate in layer: + need(not used.intersection(ids) and unitary(gate), "disjoint exactly unitary FFT gate") + used.update(ids) + actual = operator(plan, n) + for j in range(n): + basis = Vec(int(i == j) for i in range(n)) + image = Vec(actual[i][j] for i in range(n)) + need(apply(inverse, image)[0] == basis, "exact FFT adjoint round trip") + scale_box = inverse_sqrt_box(n, phases.working_bits) + need(scale_box[0] ** 2 * n <= 1 <= scale_box[1] ** 2 * n, "Fourier normalization enclosure") + squared_error = Q(0) + for i in range(n): + for j in range(n): + box = phases.box(n, -i * j) + for interval, coordinate in zip(box, (actual[i][j].re, actual[i][j].im)): + candidates = [x * y for x in interval for y in scale_box] + squared_error += max(abs(coordinate - min(candidates)), + abs(coordinate - max(candidates))) ** 2 + epsilon = 2 * (n.bit_length() - 1) * phases.nu + need(squared_error <= n * epsilon ** 2, "independent dense Fourier enclosure") + return {"n": n, "layers": len(plan), "gates": sum(len(layer) for layer in plan), + "operator_sha256": vector_hash([x for row in actual for x in row]), + "fourier_frobenius_squared_upper": str(upper(squared_error)), + "operator_error_bound": str(epsilon)} + + +def structure(plan, n, certificate_row): + height = 0 + shear_layers = 0 + unitary_gates = 0 + for layer in plan: + used = set() + layer_shears = [] + for ids, gate in layer: + need(not used.intersection(ids) and len(set(ids)) == len(ids), "disjoint source layer") + need(all(0 <= i < 2 * n for i in ids), "owned source wires") + used.update(ids) + for row in gate: + for entry in row: + height = max(height, qheight(entry.re), qheight(entry.im)) + crossing = len(ids) == 2 and ids[0] < n <= ids[1] + if crossing: + need(gate[0][0] == gate[1][1] == 1 and gate[0][1] == 0, "source-oriented pointwise shear") + layer_shears.append(gate[1][0]) + else: + need(all(i < n for i in ids) or all(i >= n for i in ids), "unitary bank ownership") + need(unitary(gate), "exactly unitary bank gate") + unitary_gates += 1 + if layer_shears: + need(len(layer_shears) == n and len(used) == 2 * n, "complete spectral layer") + bound = (certificate_row["mg"] if shear_layers == 0 else certificate_row["mh"]) + bound += Q(1, 1 << certificate_row["multiplier_bits"]) + need(all(value.norm2() <= bound ** 2 for value in layer_shears), "charged spectral norm") + shear_layers += 1 + need(shear_layers == 2 and len(plan) <= certificate_row["layer_bound"], "complete child schedule") + need(height <= certificate_row["coefficient_height_bound"], "bounded online coefficient encoding") + return {"unitary_gates": unitary_gates, "spectral_layers": shear_layers, + "max_coefficient_height": height} + + +def ideal_endpoint(matrix, vector): + n = len(matrix) + source, target = vector[:n], vector[n:] + image = mv(matrix, source) + return Vec(source + [a + b for a, b in zip(target, image)]) + + +def one_case(matrix, target, raw, phase_pool, fine_grid=False): + n = len(matrix) + base = Vec(C(Q((7 * i + 3) % 17 - 8, 8), Q((5 * i + 1) % 13 - 6, 8)) + for i in range(2 * n)) + source = Vec(base[:n]) + cancelled = Vec(source + [-x for x in mv(matrix, source)]) + large = Vec(C((1 if i % 2 == 0 else -1) * (1 << 20), + (1 if i % 3 else -1) * (1 << 19)) for i in range(2 * n)) + vectors = [base, Vec(-x for x in base), cancelled, large] + if fine_grid: + vectors.append(Vec([C(Q(1, 1 << 80))] + [C() for _ in range(2 * n - 1)])) + radius = max(Q(1), *(component_l1(x) for x in vectors)) + components = [part for vector in vectors for value in vector for part in (value.re, value.im)] + need(all(value.denominator & (value.denominator - 1) == 0 for value in components), + "dyadic incoming fixtures") + incoming_bits = max(value.denominator.bit_length() - 1 for value in components) + bound = budget(matrix, radius, target, incoming_bits) + p = bound["phase_bits"] + if p not in phase_pool: + phase_pool[p] = Phases(p, raw[ORACLE]) + phases = phase_pool[p] + builder = Builder(phases, bound["multiplier_bits"], raw[CIRCUIT]) + fft = fft_check(builder, n, phases) + plan = builder.plan(matrix) + structural = structure(plan, n, bound) + actual = operator(plan, 2 * n) + for i in range(2 * n): + for j in range(2 * n): + if i < n or j >= n: + need(actual[i][j] == int(i == j), "exact unrounded dirty-source and target-identity blocks") + coefficient_squared = sum(((actual[n + i][j] - matrix[i][j]).norm2() + for i in range(n) for j in range(n)), Q(0)) + need(coefficient_squared <= n * bound["coefficient_error"] ** 2, + "independent exact Toeplitz reference") + micro = Vec([C(Q(2, 1 << bound["data_bits"]))] + [C() for _ in range(2 * n - 1)]) + vectors.append(micro) + declared_bits = max(part.denominator.bit_length() - 1 for vector in vectors + for value in vector for part in (value.re, value.im)) + final_bound = budget(matrix, radius, target, declared_bits) + need(all(final_bound[key] == value for key, value in bound.items() + if key != "input_fractional_bits"), "micro fixture preserves execution precision") + bound = final_bound + states = [] + for vector in vectors: + need(norm2(vector) <= radius ** 2, "input radius") + rounded_out, stats = apply(plan, vector, bound["data_bits"]) + rational_out = mv(actual, vector) + wanted = ideal_endpoint(matrix, vector) + coefficient_error = norm2(Vec(a - b for a, b in zip(rational_out, wanted))) + rounding_error = norm2(Vec(a - b for a, b in zip(rounded_out, rational_out))) + total_error = norm2(Vec(a - b for a, b in zip(rounded_out, wanted))) + source_error = norm2(Vec(a - b for a, b in zip(rounded_out[:n], vector[:n]))) + need(coefficient_error <= (bound["coefficient_error"] * radius) ** 2, "coefficient forward error") + need(rounding_error <= bound["rounding_error"] ** 2, "actual rounded gate error") + need(total_error <= bound["total_error"] ** 2, "complete rounded child error") + need(source_error <= bound["rounding_error"] ** 2, "rounded source restoration allowance") + need(stats["max_observed_height"] <= bound["temporary_bits"], "observed temporary envelope") + need(stats["max_reduced_fractional_bits"] <= bound["data_bits"] + and stats["max_grid_numerator_bits"] + 1 <= bound["data_bits"] + bound["integer_bits"], + "fixed data record width") + states.append({"input_sha256": vector_hash(vector), "output_sha256": vector_hash(rounded_out), + "coefficient_error_squared_upper": str(upper(coefficient_error)), + "rounding_error_squared_upper": str(upper(rounding_error)), + "total_error_squared_upper": str(upper(total_error)), + "source_error_squared_upper": str(upper(source_error)), + "source_restoration_exact": source_error == 0, "statistics": stats}) + record_bound = {k: (str(v) if isinstance(v, Q) else v) for k, v in bound.items()} + return {"input_sha256": vector_hash([x for row in matrix for x in row]), "n": n, + "budget": record_bound, "fft": fft, "structure": structural, + "source_ast_sha256": builder.original_ast, "adapted_ast_sha256": builder.adapted_ast, + "plan_sha256": plan_hash(plan), "layers": len(plan), + "gates": sum(len(layer) for layer in plan), + "toeplitz_frobenius_squared_upper": str(upper(coefficient_squared)), + "states": states}, builder, bound + + +def controls(raw): + n = 4 + matrix = Mat([[C(Q(2 + i - j, 16), Q(1 if i >= j else -1, 32)) + for j in range(n)] for i in range(n)]) + bound = budget(matrix, Q(8), 20) + def build(): + return Builder(Phases(bound["phase_bits"], raw[ORACLE]), bound["multiplier_bits"], raw[CIRCUIT]) + def fails_target(plan): + actual = operator(plan, 2 * n) + wanted = Mat([[int(i == j) if i < n or j >= n else matrix[i - n][j] + for j in range(2 * n)] for i in range(2 * n)]) + error = sum(((a - b).norm2() for ar, br in zip(actual, wanted) for a, b in zip(ar, br)), Q(0)) + need(error > n * bound["coefficient_error"] ** 2, "mutant violates exact-operator certificate") + controls = [] + builder = build() + builder.namespace["inverse"] = lambda p: [[(ids, inverse_gate(gate)) for ids, gate in layer] + for layer in p] + fails_target(builder.plan(matrix)) + controls.append("wrong-inverse-layer-order") + builder = build() + plan = builder.plan(matrix) + swap = Mat([[0, 1], [1, 0]]) + fails_target([[(ids, gate) for ids, gate in layer if gate != swap] for layer in plan]) + controls.append("missing-bit-reversal") + wrong_scale = [] + for layer in plan: + changed = [] + for ids, gate in layer: + if len(ids) == 2 and ids[0] < n <= ids[1]: + gate = Mat([[1, 0], [gate[1][0] / 2, 1]]) + changed.append((ids, gate)) + wrong_scale.append(changed) + fails_target(wrong_scale) + controls.append("normalized-instead-of-unnormalized-multipliers") + builder = build() + builder.np.linalg.inv = lambda gate: gate if len(gate) == 1 else inverse_gate(gate) + fails_target(builder.plan(matrix)) + controls.append("missing-inverse-twist") + q = round_q(Q(707, 1000), 8) + bad_gate = Mat([[q, q], [q, -q]]) + controls.append(rejected(lambda: need(unitary(bad_gate), "unitarity"), "independent-butterfly-rounding")) + controls.append(rejected(lambda: certificate(n, bound["mg"], bound["mh"], Q(8), 20, + bound["phase_bits"], bound["multiplier_bits"], 4), + "underfunded-data-precision")) + controls.append(rejected(lambda: certificate(n, bound["mg"], bound["mh"], Q(8), 20, + 4, bound["multiplier_bits"], bound["data_bits"]), + "underfunded-phase-precision")) + controls.append(rejected(lambda: certificate(n, bound["mg"], bound["mh"], Q(8), 20, + bound["phase_bits"], bound["multiplier_bits"], + bound["data_bits"], bound["data_bits"] + 1), + "discarded-incoming-grid")) + low, high = Phases(8, raw[ORACLE]), Phases(24, raw[ORACLE]) + stale = low.phase(7, 1) + high.phase(7, 1) + box = high.box(7, 1) + distance = sum(max(abs(x - y) for x in interval) + for interval, y in zip(box, (stale.re, stale.im))) + need(distance > high.nu and low.oracle["pi_interval"] is not high.oracle["pi_interval"], + "stale lower-precision phase rejects") + controls.append("stale-phase-at-higher-precision") + controls.append(rejected(lambda: C(0.5), "floating coefficient")) + need(canonical({"n": 1}) != canonical({"n": True}), "typed receipt") + return controls + ["source-corruption", "typed-receipt"] + + +def run(): + raw, sources = authenticate() + phase_pool, records = {}, [] + for n in (1, 2, 4, 8, 16): + for complex_case in (False, True): + column = Vec([C(Q(1, 2))] + [C(Q((-1) ** i, 16 * n), + Q(i, 32 * n) if complex_case else 0) for i in range(1, n)]) + row = Vec([C(Q(1, 2))] + [C(Q(1, 8 * n), + Q((-1) ** j, 16 * n) if complex_case else 0) for j in range(1, n)]) + matrix = Mat([[column[i - j] if i >= j else row[j - i] for j in range(n)] for i in range(n)]) + record, _, _ = one_case(matrix, 12 if not complex_case else 20, raw, phase_pool) + records.append(record) + for value in (C(), C(Q(1, 8))): + matrix = Mat([[value for _ in range(4)] for _ in range(4)]) + record, _, _ = one_case(matrix, 20, raw, phase_pool, fine_grid=(value == 0)) + records.append(record) + need(any(not state["source_restoration_exact"] for row in records for state in row["states"]), + "nonzero rounded source error witness") + return {"schema_version": 1, "scope": "actual rational FFT/Toeplitz gate schedule and fixed-grid data", + "sources": sources, "cases": records, + "phases": [phase_pool[key].record() for key in sorted(phase_pool)], + "controls": controls(raw), "physical_tape_execution": False} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-result", action="store_true") + args = parser.parse_args() + result = run() + path = HERE / "RESULT.json" + if args.write_result: + path.write_text(json.dumps(result, indent=2) + "\n") + else: + need(canonical(result) == canonical(json.loads(path.read_text())), "complete typed receipt") + print(json.dumps({"status": "PASS_RATIONAL_UNITARY_FFT", "cases": len(result["cases"]), + "rounded_states": sum(len(row["states"]) for row in result["cases"]), + "gates": sum(row["gates"] for row in result["cases"]), + "result_sha256": sha256(canonical(result)).hexdigest()}, sort_keys=True)) + + +if __name__ == "__main__": + main() From 203d04313c0cb2b4a4663a5a2d4d407e5a5b6d20 Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 17:26:01 +0200 Subject: [PATCH 2/9] Certify rounded parity recursion and stored inverse --- research/rational-unitary-fft/README.md | 57 +- research/rational-unitary-fft/TASK.json | 14 +- .../rational-unitary-fft/TRIANGLE-PROOF.md | 256 ++ .../rational-unitary-fft/TRIANGLE-RESULT.json | 3210 +++++++++++++++++ research/rational-unitary-fft/VERIFY.json | 10 + research/rational-unitary-fft/exact.py | 13 + research/rational-unitary-fft/triangular.py | 199 + .../rational-unitary-fft/verify_triangle.py | 296 ++ 8 files changed, 4044 insertions(+), 11 deletions(-) create mode 100644 research/rational-unitary-fft/TRIANGLE-PROOF.md create mode 100644 research/rational-unitary-fft/TRIANGLE-RESULT.json create mode 100644 research/rational-unitary-fft/triangular.py create mode 100644 research/rational-unitary-fft/verify_triangle.py diff --git a/research/rational-unitary-fft/README.md b/research/rational-unitary-fft/README.md index d6ef1882dd..c44ab08cf5 100644 --- a/research/rational-unitary-fft/README.md +++ b/research/rational-unitary-fft/README.md @@ -1,9 +1,12 @@ -# Certified rational unitary FFT and rounded Toeplitz shear +# Certified rational FFT and rounded parity-triangular programs I connect the retained rational phase construction to the actual FFT and Toeplitz-shear gate schedule, and prove a complete coefficient/data error bound for both dirty banks. I execute the adapted one- and two-wire gates. This advances the child-execution obligation left by PR4198's macro replay. +I also execute the retained near-identity parity-triangular recursion and +its inverse under one coefficient and data-grid allocation. I preserve +the source's parallel even/odd branches, including reversed caller wires. For dyadic \(n=2^d\), the target is \((x,y)\mapsto(x,y+Tx)\), with rational Toeplitz \(T\). I use rational butterflies @@ -35,6 +38,22 @@ widths. Every exact contiguous circuit segment and inverse segment has norm at most \(H\); I do not multiply amplification once per layer. [PROOF.md](PROOF.md) gives the domains, oracle bounds and full derivation. +For the triangular program I admit a rational symbol \(f\) of dyadic +length \(N\) with component-l1 distance from the identity at most \(1/2\). +I round the scalar leaves and use exact reciprocals of those stored values +for the inverse. I prove forward and inverse endpoint error +\[ +eR+32DBN\,2^{-F}\le2^{-t},\qquad +e=2qB^3\,2^{-s}, +\] +where \(B=100N^3\), \(q=2\log_2N+1\) and +\(D=4(\log_2N)^2+12\log_2N+1\). I give explicit \(s,P,S,F\), preserve +the incoming grid, and bound intermediate states using arbitrary circuit +segments. The online records and gate temporaries need +\(O(F_0+t+\log N+\log R)\) bits. +[TRIANGLE-PROOF.md](TRIANGLE-PROOF.md) separates that claim from exact +offline preparation and physical coefficient delivery. + I start from canonical main `12fa8c5c844be21bfefb38a19503fbd867a11244`, priority 7, and pin the retained PR4100 source `337d04939571a637aeab54a0f58f54fde811663a`. @@ -51,7 +70,7 @@ python3.12 -B scripts/research_package.py \ --package research/rational-unitary-fft ``` -I declare two normal/optimized commands, with 240-second limits each. +I declare four normal/optimized commands, with 240-second limits each. I authenticate the source before parsing its schedule or executing its interval oracle. The adapter preserves bit reversal, gate ownership, disjoint layers, shear orientation and chronological calls. It replaces @@ -79,6 +98,19 @@ I compare the complete typed [RESULT.json](RESULT.json). Observed temporary heights describe completed complex components, not an exhaustive trace of internal `Fraction` temporaries. +The recursive verifier adds ten cases through 16 wires, 1,806 gates and +62 rounded states in forward and inverse directions. I check the exact +macro factorization against a direct Toeplitz matrix and the inverse +against separate Gauss-Jordan elimination. I include complex non-dyadic +leaves, the half-identity admission boundary, a zero odd polynomial, +reversed wires, a 100-bit incoming grid and small-grid data. +Controls reject missing Schur correction, swapped parity banks, serialized +siblings, wrong inverse order and leaf adjoints used as reciprocals. +I check rounded round trips under their separate error and second-pass +width allowances. [TRIANGLE-RESULT.json](TRIANGLE-RESULT.json) records +the complete recursive receipt. Exact dense endpoint arithmetic remains +a checker oracle, outside the online representation claim. + ## Overlap and attribution I inspected 100 recent PR records and 35 hub comments, the retained @@ -92,6 +124,12 @@ older comments, arbitrary diffs, unpublished work and literature priority remain outside the search. Before publication I rechecked unchanged main, the source head, 60 PR records and 12/708 hub comments. +Before the recursive extension I refreshed unchanged main and inspected +40 PR records and 12/713 hub comments, following a separate 100-PR/35-comment +orientation. I found no competing rounded parity-child claim in that +coverage. James Reed's claimed finite PR4198 check addresses the generator +identities, a separate question. I extend this PR rather than duplicate +the child package. I credit Chafik Boukhalfa/PR628 for exact rational unit phases and the certified interval machinery; Vlado Vojdanovski/PR1673 for the FFT/shear @@ -100,6 +138,17 @@ caller. PR3709 already supplies certified inward twiddles and rounded FFT execution under a different normalization. I claim the connection to this unitary child and its complete approximation/rounding contract, not the invention of certified phases or rounded FFTs. +Vlado also supplied the parity identity, recursion and qualitative +near-identity stability argument. I add the explicit global coefficient +perturbation, inverse, rounding and intermediate-width contract, with +source-bound recursive execution. + +I preserve the first completed revision `f02a4ab5ab70a8af80415ac35c5b48cd5a93a475` +and its FFT receipt +`2e203e42d3d617e828e3341ffeb44254c931db2aebb8a040d04d67338b5b4016`. +Its [hosted replay](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39442494617) +and 18-file export remain evidence for that revision. The recursive +extension has a separate receipt and verification commands. Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance, including source-analysis agents. I count those agents within the author @@ -110,7 +159,7 @@ Apache-2.0. ## Open next questions In [TASK.json](TASK.json) I specify the next steps and blockers: compose -this child inside the actual parity-triangular and bounded-generator -programs, then bind its coefficients and data widths to the physical +these children inside the actual bounded-generator program, then bind +their coefficients and data widths to the physical Gaussian caller. This package supplies neither complete fixed-tape lowering nor complete multiplication-theorem acceptance. diff --git a/research/rational-unitary-fft/TASK.json b/research/rational-unitary-fft/TASK.json index 7f161209ab..f5ec11e093 100644 --- a/research/rational-unitary-fft/TASK.json +++ b/research/rational-unitary-fft/TASK.json @@ -38,21 +38,21 @@ ] }, "scope_status": "refined", - "actual_scope": "Certify rational-unitary FFT and Toeplitz-shear gates in the retained schedule, including multiplier preparation, fixed-grid rounding in both dirty banks, incoming-grid preservation and explicit error/width budgets.", - "scope_change_reason": "PR4198 leaves numerical child execution open. PR628 already supplies rational unit phases, and PR3709 already supplies inward-twiddle rounded FFTs; this result connects the retained unitary child to a complete coefficient/data precision contract.", + "actual_scope": "Certify rational-unitary FFT/Toeplitz-shear gates and the retained near-identity parity-triangular recursion and stored inverse, including multiplier and leaf preparation, global fixed-grid rounding, incoming precision and explicit error/width budgets.", + "scope_change_reason": "PR4198 leaves numerical child execution open. PR628 already supplies rational phases and Vlado supplies the parity recursion; this result connects those source programs to one complete coefficient/data precision contract. The extension preserves the first FFT revision and its receipt.", "bottleneck_link": "The bounded-generator compiler needs an actual FFT/shear child with charged approximation, source-restoration error and data width before its full numerical composition can be admitted.", "budget_minutes": 120, "stop_rule": "Stop at an invalid phase enclosure, nonunitary adapted gate, missing multiplier/twist error, understated word width or source-schedule mismatch. Do not equate coefficient-level unitarity with exact rounded restoration or physical execution.", "outcome": "completed", - "evidence": "PROOF.md gives the quantified oracle, coefficient, prefix, rounding and width argument. RESULT.json records 12 cases, 1428 actual gates and 61 rounded states with exact Fourier/Toeplitz references and schedule/precision controls. Scientific acceptance remains separate.", + "evidence": "PROOF.md and TRIANGLE-PROOF.md give the quantified phase, recursion, coefficient, prefix/suffix, rounding and width arguments. RESULT.json preserves 12 FFT/shear cases, 1428 gates and 61 rounded states. TRIANGLE-RESULT.json adds 10 recursive cases, 1806 gates and 62 forward/inverse states with exact macro and separate inverse references. Scientific acceptance remains separate.", "author": "Jose Manuel (jmbarrancoml)", "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, source analysis, implementation and writing, including source-analysis agents. No independent human or formal-kernel acceptance.", "open_questions": [ { - "question": "Can the actual parity-triangular and bounded-generator programs consume this rounded child under one precision allocation?", - "why_it_matters": "The source calls this child inside a recursive triangular transform and a compensated generator schedule; a child bound alone does not finish that composition.", - "first_step": "Bind the literal parity recursion and its coefficient preparation, then propagate this child's defects through the previously bounded macro prefixes and inverse prefixes.", - "blocked_by": "A separate source-bound recursive prefix/error proof and complete emitted-gate replay." + "question": "Can the actual bounded-generator program consume these rounded triangular and shear children under one precision allocation?", + "why_it_matters": "The source composes a triangular transform, scaled shear, stored inverse and compensation; the individual child contracts do not finish that complete generator schedule.", + "first_step": "Combine the actual rational generator producer with the literal four-stage schedule and propagate child defects through its macro prefixes, including rounded restoration and compensation.", + "blocked_by": "A source-bound full-generator error/width proof and complete emitted-gate replay." }, { "question": "Can the coefficient preparation and canonical data widths be joined to the physical Gaussian caller?", diff --git a/research/rational-unitary-fft/TRIANGLE-PROOF.md b/research/rational-unitary-fft/TRIANGLE-PROOF.md new file mode 100644 index 0000000000..f923f4ff2d --- /dev/null +++ b/research/rational-unitary-fft/TRIANGLE-PROOF.md @@ -0,0 +1,256 @@ +# Rounded parity-triangular execution + +Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance. +Apache-2.0. + +## Domain and result + +I extend the FFT/shear argument in [PROOF.md](PROOF.md) to Vlado/PR1673's +retained `triangle_plan` and its exact stored-coefficient inverse. +For \(N=2^d\), let \(T=L(f)\) be lower triangular Toeplitz over +\(\mathbb Q(i)\). I admit +\[ +\sum_{j=0}^{N-1}(|\Re(f_j-\delta_{j0})| + +|\Im(f_j-\delta_{j0})|)\le\tfrac12. +\tag{1} +\] +This executable condition implies both induced \(1\)- and infinity-norm +bounds \(\|T-I\|\le1/2\). I do not extend the executable admission to +complex symbols that satisfy only the weaker modulus-sum condition. +The caller may supply a permutation of the \(N\) wires, including reversal. + +For target \(2^{-t}\), input norm at most \(R\ge1\), and an incoming dyadic +grid with \(F_0\) fractional bits, I construct one rational gate program +\(\widehat T\) and its inverse using the same stored coefficients. +After nearest component rounding on one common \(F\)-grid at each gate, +I prove error at most \(2^{-t}\) for either \(Tx\) or \(T^{-1}x\). +The input to each direction must satisfy the stated radius and grid. +I give a separate allowance for a rounded forward/inverse round trip. + +I use the following conservative bounds: +\[ +B=100N^3,\quad q=2d+1,\quad +D=4d^2+12d+1,\quad G=N(2d^2+8d+1). +\tag{2} +\] +Here \(q\) counts macro layers, \(D\) bounds physical gate layers and \(G\) +bounds logical one/two-wire gates. I preserve empty source layers and +zero shears. I claim logical execution and precision, with physical tape +delivery and full bounded-generator composition left for later work. + +## Descendants and ideal prefixes + +I use the predecessor's parity identity. The source executes the upper +shear, then the block diagonal, then the lower shear. After parity +permutation, their ordinary matrix product is +\[ +T\sim\begin{pmatrix}I&0\\K&I\end{pmatrix} +\begin{pmatrix}E&0\\0&S\end{pmatrix} +\begin{pmatrix}I&ZK\\0&I\end{pmatrix}, +\quad +K=OE^{-1},\quad S=E-ZOK. +\tag{3} +\] +All \(E,O,K\) are commuting lower +triangular Toeplitz matrices. The source computes their coefficients +with exact truncated series and convolution. + +Every recursive descendant is a principal Schur complement of a +principal restriction of the original \(T\), after a permutation. +For either induced norm, a principal restriction \(I+H\) still has +\(\|H\|\le1/2\). Its inverse has norm at most two. For a Schur complement +\(M\), the inverse is a principal block of that inverse, so +\(\|M^{-1}\|\le2\). The Schur formula also gives +\[ +\|M-I\|\le\tfrac12+ + \frac{(1/2)^2}{1-1/2}=1,\qquad \|M\|\le2. +\tag{4} +\] +Nested principal restrictions of Schur complements preserve this +description by retaining the previously eliminated set. I need the +original near-identity hypothesis on arbitrary principal restrictions; +conditioning only leading sections would not suffice. + +Thus \(\|O\|_{1,\infty},\|E^{-1}\|_{1,\infty}\le2\), and +\(\|K\|_{1,\infty},\|ZK\|_{1,\infty}\le4\). For a lower triangular +Toeplitz matrix, the full coefficient modulus sum equals its induced +1-norm. The component sum is therefore at most \(4\sqrt2<6\). +An ideal completed shear and its inverse have norm at most five. +An ideal middle block and its inverse have norm at most two. + +I apply the FFT/shear segment lemma to each child, whose cyclic and +negacyclic kernel component envelopes sum to less than six. +Both ideal and compiled child internal prefixes, suffixes and their +inverses have norm at most seven; for compiled children the additional +multiplier allowance \(2v\) still fits because \(v\le1/32\) and +\(4\sqrt2+2v<6\). + +For ideal physical prefixes and their inverses, I obtain +\[ +B(N)\le\max(7,\ 5B(N/2),\ 7\cdot2\cdot5),\qquad B(1)\le2. +\tag{5} +\] +The three cases stop in the first shear, recursive middle or last shear. +For inverse prefixes I reverse the factors in the inverse of that +prefix; the same norm products apply. Equation (2) dominates (5). +Disjoint siblings combine by the maximum of their operator norms. + +## Stored coefficients and aligned macro errors + +At each recursion depth, equal-size siblings emit identical schedules +regardless of their coefficients. I group their disjoint shear children +into one macro layer, recurse on the middle, and group their scalar leaves. +This gives \(q=2d+1\). The source keeps the two siblings in parallel. +The retained shear count gives +\[ +D(d)\le D(d-1)+2(4(d-1)+8),\quad D(0)=1, +\] +and summing the gates at each level gives the bounds in (2). + +Set +\[ +s=t+\left\lceil\log_2(4qB^3(R+1))\right\rceil,\quad +\varepsilon=2^{-s},\quad +P=s+\left\lceil\log_2(48(d+1))\right\rceil,\quad S=s+2. +\tag{6} +\] +I use \(P\) for the existing certified unit phases and \(S\) for stored +spectral multipliers. A child of size \(m\) has unrounded endpoint defect +\[ +4(\log_2m+1)M\,2^{-P}+2^{1-S}\le\varepsilon, +\quad M<6. +\tag{7} +\] +For the scalar leaves I store nearest \(s\)-grid values. Their complex +error is at most \(\varepsilon\). The ideal modulus lies between \(1/2\) +and two; the stored modulus is at least \(1/4\). In the reverse program +I use the exact reciprocal of that same stored scalar. I retain the +unrounded rational coefficients throughout recursive series preparation. +I do not apply leaf rounding to the unitary twist gates. + +For each ideal macro \(A_j\), its compiled counterpart has +\(\|\widehat A_j-A_j\|\le\varepsilon\), because its pieces act on disjoint +supports. Write \(Q_j=A_j\cdots A_1\), \(Q_0=I\), and +\[ +G_j=Q_j^{-1}(\widehat A_j-A_j)Q_{j-1},\qquad +Z_j=Q_j^{-1}\widehat A_j\cdots\widehat A_1. +\] +Then \(Z_j=(I+G_j)Z_{j-1}\) and +\[ +\|G_j\|\le B^2\varepsilon,\qquad +\theta=qB^2\varepsilon\le\tfrac14. +\tag{8} +\] +For a consecutive product \(W\) of these factors, the sum of the bounds +on its perturbations is at most \(\theta\). Expanding the product and +using \(\prod(1+a_j)\le1/(1-\sum a_j)\) gives +\[ +\|W\|,\|W^{-1}\|\le(1-\theta)^{-1}<2,\quad +\|W-I\|,\|W^{-1}-I\|\le\theta/(1-\theta)\le2\theta. +\tag{9} +\] +For the inverse inequality I use +\(\prod(1-a_j)\ge1-\sum a_j\), and the expansion of the inverse factors. +Since \(\widehat T=T Z_q\), both directions satisfy the loose common bound +\[ +\|\widehat T-T\|,\ \|\widehat T^{-1}-T^{-1}\| +\le 2qB^3\varepsilon=:e. +\tag{10} +\] +I use the exact inverse of the stored approximate circuit here. +The proof does not replace a nonunitary scalar's inverse with its adjoint. + +## Suffixes, accumulated rounding and word widths + +Actual macro prefixes and their inverses have norm at most \(2B\). +For a suffix after macro \(j\), I write it as +\[ +Q_q W Q_j^{-1}. +\] +Its inverse has the reversed inverse factors. Since +\(\|Q_q\|,\|Q_q^{-1}\|\le2\), (9) bounds both by \(4B\). +An internal child fragment adds a factor of at most seven. +Thus \(K=32B\) bounds physical prefixes, suffixes and their inverses +in either execution direction. + +An arbitrary physical segment may need the larger bound \(K^2\): +I express it as a prefix times an inverse prefix. I use this bound +for intermediate states after several rounding errors. +One physical layer contributes Euclidean rounding error at most +\(\sqrt{N/2}\,u\le Nu\), where \(u=2^{-F}\). +Therefore I obtain +\[ +\|\widehat z-T^{\pm1}x\|\le eR+DKNu, +\qquad +\max_{\text{intermediate states}}\|z\|\le +A:=KR+DK^2Nu. +\tag{11} +\] +Choose +\[ +F=\max\left(F_0,\ t+\left\lceil\log_2(64DBN)\right\rceil\right). +\tag{12} +\] +Equations (6), (10) and (12) assign at most \(2^{-t}/2\) to each +of coefficient and data error. Also \(A\le K(R+2^{-t}/2)\). +I retain finer incoming grids. + +If \(\widehat a=(a+ib)/2^s\) is a stored leaf, its exact reciprocal is +\(2^s(a-ib)/(a^2+b^2)\). With \(|\widehat a|<3\), its component height +is at most \(2s+6\). I use the still looser common bound +\[ +C_{\rm bits}=\max(4P+24,\ S+5,\ 2s+16),\quad +I=\max(1,\lceil\log_2\max(1,A)\rceil)+2. +\tag{13} +\] +The previous constant-size rational gate evaluation argument then gives +signed canonical grid width \(F+I\) and temporary allowance +\[ +2F+8C_{\rm bits}+2I+32. +\tag{14} +\] +All these online widths are \(O(F_0+t+\log N+\log R)\). +Exact offline series, Schur coefficients and dense checker matrices can +have longer denominators. I do not include them in the online word claim. +The original preprocessing proof supplies the algebraic preparation +context; complete setup and physical coefficient-delivery costs remain open. + +For a rounded forward/inverse round trip, let +\(E=eR+DKNu\). The forward output has norm at most \(2R+E\). +Applying the same inverse program gives the separate allowance +\[ +\|\widehat{\operatorname{inverse}}(\widehat{\operatorname{forward}}(x))-x\| +\le 2E+e(2R+E)+DKNu. +\tag{15} +\] +I do not promise exact restoration after data rounding. +For this extra second pass I use radius \(R'=2R+E\) in the amplitude +and integer-width formulas, keeping the same stored gates and \(F\). +The verifier records that enlarged second-pass width separately. + +## Executable evidence and attribution + +I extract the authenticated `toep`, `lower`, `invseries`, `conv` and +`triangle_plan` functions. I change only the floating diagnostic and the +scalar leaf return. I supply exact array/convolution operations and +certified child callbacks, keeping the source's `parallel` and `inverse`. +I retain even/odd wire order, reversed caller indices, empty layers and +zero shears. The verifier checks exact macro identities against a direct +Toeplitz matrix and the inverse against separate Gauss-Jordan elimination. + +The exact endpoint Frobenius checks corroborate consequences of the +operator proof; they do not independently establish its sharper norm +bound. I compare rounded vectors with exact endpoint images in both +directions and include a nonzero rounded round-trip error witness. +I test mutations of the Schur correction, parity banks, parallel middle +and inverse order, plus the nonunitary-leaf adjoint error. +I label observed temporary heights as completed rational components; +I do not claim exhaustive instrumentation of `Fraction` internals. + +I credit Vlado Vojdanovski/PR1673 for the parity factorization, recursion, +Schur descendant argument and circuit; Chafik Boukhalfa/PR628 for phases; +and Cinna Davis for the later caller. I add the explicit common +coefficient/grid allocation, inverse and intermediate-width accounting, +and source-bound rational execution. Internal source-analysis agents +belong to the author process. Independent mathematical review, +full generator/caller composition and physical lowering remain open. diff --git a/research/rational-unitary-fft/TRIANGLE-RESULT.json b/research/rational-unitary-fft/TRIANGLE-RESULT.json new file mode 100644 index 0000000000..4caa4aad37 --- /dev/null +++ b/research/rational-unitary-fft/TRIANGLE-RESULT.json @@ -0,0 +1,3210 @@ +{ + "schema_version": 1, + "scope": "retained parity recursion and inverse on one data grid", + "sources": [ + { + "path": "research/unconditional-batched-crt/GAUSSIAN.md", + "sha256": "2f63c91c58f2d1cf31294812c75970ea390c61866a8da7ed700d4316a033a242" + }, + { + "path": "research/unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md", + "sha256": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310" + }, + { + "path": "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_circuits.py", + "sha256": "06d1861bfe8af543908e26557ffda7e451ff176bceb9bacfaf0f04f234b336d5" + }, + { + "path": 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"17164491757516668297231984729208410815973446912584488421710554107679559507005038063978130063748639823536103/13624712545101046084835764100024738903333766019300779455441460946734401504491832023464759521781381116408513548183579860510128977561518080", + "parameter": "262539035482463537951359408951/633825300114114700748351602688" + }, + { + "denominator": 8, + "numerator": 3, + "unit": [ + "-332807765912685320299847905304416025354008184072102149904943/470661256216809817471133140866165275907093312819294267745745", + "332807765912685320299847905303823083578086411153578525720576/470661256216809817471133140866165275907093312819294267745745" + ], + "error_bound": "2145561469689583537153998091151051351996680867897183759475399030412901707913222624742399424625345903376191/1703089068137630760604470512503092362916720752412597431930182618341800188061479002933094940222672639551064193522947482563766122195189760", + "parameter": "-262539035482463537951359408951/633825300114114700748351602688" + } + ] + } + } + ], + "controls": [ + "near-identity-admission", + "nondyadic-size", + "floating-coefficients", + "zero-leaf-margin", + "leaf-adjoint-instead-of-reciprocal", + "inverse-layer-order", + "omitted-schur-correction", + "serialized-parallel-middle", + "swapped-parity-bank-routing", + "exact-vector-copy-zero-semantics", + "discarded-incoming-grid", + "underfunded-global-grid" + ], + "physical_tape_execution": false, + "full_generator_execution": false +} diff --git a/research/rational-unitary-fft/VERIFY.json b/research/rational-unitary-fft/VERIFY.json index 63b60051d6..3bbebe4796 100644 --- a/research/rational-unitary-fft/VERIFY.json +++ b/research/rational-unitary-fft/VERIFY.json @@ -23,6 +23,16 @@ "argv": ["{python}", "-I", "-O", "-B", "{package}/verify.py"], "timeout_seconds": 240, "checks": "Repeat the complete gate/rounding certificate under optimized Python, keeping authenticated source assertions and exact typed receipt comparison." + }, + { + "argv": ["{python}", "-I", "-B", "{package}/verify_triangle.py"], + "timeout_seconds": 240, + "checks": "Execute the retained parity recursion and exact stored inverse on one grid; check macro identities, independent exact endpoint inverses, global error/width budgets, reversed indices, nonunitary leaves and adversarial controls." + }, + { + "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_triangle.py"], + "timeout_seconds": 240, + "checks": "Repeat the complete recursive forward/inverse certificate under optimized Python with authenticated source assertions and complete typed receipt comparison." } ] } diff --git a/research/rational-unitary-fft/exact.py b/research/rational-unitary-fft/exact.py index 0475e7d803..9456b0bc39 100644 --- a/research/rational-unitary-fft/exact.py +++ b/research/rational-unitary-fft/exact.py @@ -85,9 +85,20 @@ def __init__(self, values): super().__init__(C.wrap(x) for x in values) def __mul__(self, other): + if isinstance(other, C) or type(other) in (int, Q): + return Vec(x * other for x in self) need(len(self) == len(other), "elementwise vector dimensions") return Vec(x * y for x, y in zip(self, other)) + __rmul__ = __mul__ + + def __sub__(self, other): + need(len(self) == len(other), "vector subtraction dimensions") + return Vec(x - y for x, y in zip(self, other)) + + def copy(self): + return Vec(self) + class Mat(list): def __init__(self, rows): @@ -129,6 +140,8 @@ def unitary(matrix): def inverse_gate(matrix): if unitary(matrix): return adjoint(matrix) + if matrix.shape == (1, 1): + return Mat([[1 / matrix[0][0]]]) if matrix.shape == (2, 2) and matrix[0][0] == matrix[1][1] == 1 and matrix[0][1] == 0: return Mat([[1, 0], [-matrix[1][0], 1]]) raise ValueError("unsupported nonunitary inverse gate") diff --git a/research/rational-unitary-fft/triangular.py b/research/rational-unitary-fft/triangular.py new file mode 100644 index 0000000000..0d045d04a9 --- /dev/null +++ b/research/rational-unitary-fft/triangular.py @@ -0,0 +1,199 @@ +"""Source-bound rational parity recursion with a common precision allocation. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. Parity recursion and its stability premise: Vlado/PR1673. +""" +import ast +import copy +from fractions import Fraction as Q +from hashlib import sha256 +from types import SimpleNamespace + +from exact import (C, Mat, Vec, ceil_log2, component_l1, inverse_gate, need, + rounded) +from engine import Builder, Phases, split_kernel + + +def allocation(n, radius, target, incoming=0): + need(type(n) is int and n > 0 and n & (n - 1) == 0, "dyadic triangle size") + need(type(radius) in (int, Q) and radius >= 1, "exact triangle radius") + need(type(target) is int and target >= 1, "positive triangle target") + need(type(incoming) is int and incoming >= 0, "incoming triangle grid") + d = n.bit_length() - 1 + q, depth, bound = 2 * d + 1, 4 * d * d + 12 * d + 1, 100 * n ** 3 + s = target + ceil_log2(4 * q * bound ** 3 * (radius + 1)) + p = s + ceil_log2(48 * (d + 1)) + multiplier = s + 2 + data = max(incoming, target + ceil_log2(64 * depth * bound * n)) + epsilon, u = Q(1, 1 << s), Q(1, 1 << data) + theta = q * bound ** 2 * epsilon + coefficient = 2 * q * bound ** 3 * epsilon + rounding = 32 * depth * bound * n * u + need(theta <= Q(1, 4) and coefficient * radius + rounding <= Q(1, 1 << target), + "complete triangular error allocation") + amplitude = 32 * bound * radius + 1024 * depth * bound ** 2 * n * u + integer = max(1, ceil_log2(max(Q(1), amplitude))) + 2 + height = max(4 * p + 24, multiplier + 5, 2 * s + 16) + return dict(n=n, radius=radius, target_bits=target, incoming_bits=incoming, + ideal_bound=bound, macro_layers=q, layer_bound=depth, + gate_bound=n * (2 * d * d + 8 * d + 1), + leaf_bits=s, phase_bits=p, multiplier_bits=multiplier, + data_bits=data, epsilon=epsilon, theta=theta, + coefficient_error=coefficient, rounding_error=rounding, + total_error=coefficient * radius + rounding, + physical_prefix_suffix_bound=32 * bound, + segment_bound=1024 * bound ** 2, amplitude_bound=amplitude, + integer_bits=integer, coefficient_height_bound=height, + temporary_bits=2 * data + 8 * height + 2 * integer + 32) + + +class Plan(list): + """Actual layers plus consecutive, disjoint-support macro partitions.""" + def __init__(self, layers, blocks): + super().__init__(layers) + self.blocks = blocks + + def __add__(self, other): + need(isinstance(other, Plan), "typed triangular composition") + offset = len(self) + blocks = [dict(block) for block in self.blocks] + blocks += [dict(start=b["start"] + offset, stop=b["stop"] + offset, + pieces=list(b["pieces"])) for b in other.blocks] + return Plan(list(self) + list(other), blocks) + + +class Concatenate: + def __getitem__(self, parts): + values = [] + for part in parts: + values.extend(part if isinstance(part, (list, tuple)) else [part]) + return Vec(values) + + +def convolution(a, b): + return Vec(sum((a[j] * b[k - j] for j in range(len(a)) + if 0 <= k - j < len(b)), C()) + for k in range(len(a) + len(b) - 1)) + + +class AdaptTriangle(ast.NodeTransformer): + def __init__(self): + self.changes = [] + + def visit_Assign(self, node): + if (len(node.targets) == 1 and isinstance(node.targets[0], ast.Subscript) + and isinstance(node.targets[0].slice, ast.Constant) + and node.targets[0].slice.value == "max_triangle_coefficient"): + self.changes.append("exact-component-diagnostic") + node.value = ast.parse( + "max(STATS['max_triangle_coefficient'], component_l1(f))", + mode="eval").body + return node + return self.generic_visit(node) + + def visit_Return(self, node): + if (isinstance(node.value, ast.Call) and isinstance(node.value.func, ast.Name) + and node.value.func.id == "diag"): + self.changes.append("stored-rounded-leaf") + node.value.func.id = "leaf_diag" + return self.generic_visit(node) + + +class Triangle: + def __init__(self, bound, circuit_source, oracle_source): + self.bound, self.circuit_source = bound, circuit_source + self.phases = Phases(bound["phase_bits"], oracle_source) + base = Builder(self.phases, bound["multiplier_bits"], circuit_source) + self.source_parallel = base.namespace["parallel"] + self.source_inverse = base.namespace["inverse"] + self.children, self.leaves = [], [] + selected = {"toep", "lower", "invseries", "conv", "triangle_plan"} + tree = ast.parse(circuit_source) + tree.body = [copy.deepcopy(node) for node in tree.body + if isinstance(node, ast.FunctionDef) and node.name in selected] + need({node.name for node in tree.body} == selected, "complete parity source") + self.original_ast = sha256(ast.dump(tree).encode()).hexdigest() + adapter = AdaptTriangle() + tree = ast.fix_missing_locations(adapter.visit(tree)) + need(adapter.changes == ["exact-component-diagnostic", "stored-rounded-leaf"], + "exact declared parity adaptations") + self.adapted_ast = sha256(ast.dump(tree).encode()).hexdigest() + facade = SimpleNamespace(asarray=lambda a, dtype=None: Vec(a), + array=lambda a, dtype=None: Mat(a), + zeros=lambda n, dtype=None: Vec([0] * n), + convolve=convolution, r_=Concatenate()) + self.stats = {"triangle_nodes": 0, "max_triangle_coefficient": Q(0)} + self.namespace = dict(np=facade, STATS=self.stats, component_l1=component_l1, + leaf_diag=self.leaf, parallel=self.parallel, + toeplitz_shear=self.shear) + exec(compile(tree, "", "exec", optimize=0), + self.namespace) + + def leaf(self, indices, values): + need(len(indices) == len(values) == 1, "scalar triangular leaf") + wanted = values[0] + stored = rounded(wanted, self.bound["leaf_bits"]) + epsilon = self.bound["epsilon"] + need(Q(1, 4) <= wanted.norm2() <= 4, "ideal scalar leaf margin") + need((stored - wanted).norm2() <= epsilon ** 2 + and stored.norm2() >= Q(1, 16), "stored scalar leaf margin") + gate = Mat([[stored]]) + inverse = inverse_gate(gate) + need(stored * inverse[0][0] == 1, "same stored scalar reciprocal") + piece = dict(kind="leaf", ids=list(indices), ideal=Mat([[wanted]]), + error=epsilon, stored=stored, reciprocal=inverse[0][0]) + self.leaves.append(piece) + return Plan([[(tuple(indices), gate)]], + [dict(start=0, stop=1, pieces=[piece])]) + + def shear(self, target, source, matrix): + n = len(source) + need(len(set(source + target)) == 2 * n, "owned disjoint parity banks") + child = Builder(self.phases, self.bound["multiplier_bits"], self.circuit_source) + local = child.plan(matrix) + g, h = split_kernel(matrix) + envelope = component_l1(g) + component_l1(h) + v = Q(1, 1 << self.bound["multiplier_bits"]) + defect = 4 * (n.bit_length()) * envelope * self.phases.nu + 2 * v + need(envelope + 2 * v <= 6 and defect <= self.bound["epsilon"], + "bounded parity child and allocated operator defect") + mapping = source + target + layers = [[(tuple(mapping[i] for i in ids), gate) for ids, gate in layer] + for layer in local] + piece = dict(kind="shear", ids=list(mapping), ideal=matrix, error=defect, + envelope=envelope, source_ast=child.original_ast, + adapted_ast=child.adapted_ast, local_layers=len(local)) + self.children.append(piece) + return Plan(layers, [dict(start=0, stop=len(layers), pieces=[piece])]) + + def parallel(self, left, right): + need([(b["start"], b["stop"]) for b in left.blocks] + == [(b["start"], b["stop"]) for b in right.blocks], + "equal sibling macro boundary partitions") + blocks = [] + for a, b in zip(left.blocks, right.blocks): + pieces = a["pieces"] + b["pieces"] + ids = [i for piece in pieces for i in piece["ids"]] + need(len(ids) == len(set(ids)), "disjoint sibling macro supports") + blocks.append(dict(start=a["start"], stop=a["stop"], pieces=pieces)) + return Plan(self.source_parallel(left, right), blocks) + + def plan(self, symbol, indices=None): + symbol = Vec(symbol) + n = len(symbol) + need(n == self.bound["n"], "allocated triangle dimension") + need(component_l1(symbol - Vec([1] + [0] * (n - 1))) <= Q(1, 2), + "component-l1 near-identity admission") + indices = list(range(n)) if indices is None else list(indices) + need(len(indices) == n and sorted(indices) == list(range(n)), "triangle wire permutation") + result = self.namespace["triangle_plan"](indices, symbol) + need(self.stats["triangle_nodes"] == 2 * n - 1, "complete parity recursion tree") + need(len(result.blocks) == self.bound["macro_layers"] + and len(result) <= self.bound["layer_bound"] + and sum(map(len, result)) <= self.bound["gate_bound"], "complete schedule bounds") + return result + + def inverse(self, plan): + # Forward pieces certify forward defects. Inverse endpoint bounds use + # the inverse-product proof, not relabelled forward macro certificates. + return self.source_inverse(plan) diff --git a/research/rational-unitary-fft/verify_triangle.py b/research/rational-unitary-fft/verify_triangle.py new file mode 100644 index 0000000000..3b2797942f --- /dev/null +++ b/research/rational-unitary-fft/verify_triangle.py @@ -0,0 +1,296 @@ +"""Check parity gates, macro identities, both directions and one data grid. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +from exact import (C, Mat, Vec, adjoint, apply, ceil_log2, component_l1, eye, inverse_gate, + multiply, need, norm2, qheight, upper, vector_hash) +from triangular import Plan, Triangle, allocation +from verify import (CIRCUIT, ORACLE, authenticate, canonical, mv, operator, + plan_hash, rejected) + + +def reference(symbol, indices): + n = len(symbol) + result = eye(n) + for i in range(n): + for j in range(n): + result[indices[i]][indices[j]] = symbol[i - j] if i >= j else C() + return result + + +def eliminate(matrix): + """Separate exact Gauss-Jordan inverse; no series routine from the compiler.""" + n = len(matrix) + rows = [list(row) + list(unit) for row, unit in zip(matrix, eye(n))] + for j in range(n): + pivot = next(i for i in range(j, n) if rows[i][j] != 0) + rows[j], rows[pivot] = rows[pivot], rows[j] + scale = rows[j][j] + rows[j] = [x / scale for x in rows[j]] + for i in range(n): + if i != j: + scale = rows[i][j] + rows[i] = [a - scale * b for a, b in zip(rows[i], rows[j])] + need(Mat([row[:n] for row in rows]) == eye(n), "Gauss-Jordan inverse reference") + return Mat([row[n:] for row in rows]) + + +def frobenius_error(left, right): + return sum(((a - b).norm2() for x, y in zip(left, right) + for a, b in zip(x, y)), Q(0)) + + +def macro_reference(block, n): + result, used = eye(n), set() + for piece in block["pieces"]: + ids, wanted = piece["ids"], piece["ideal"] + need(not used.intersection(ids), "disjoint macro reference supports") + used.update(ids) + if piece["kind"] == "leaf": + result[ids[0]][ids[0]] = wanted[0][0] + else: + m = len(ids) // 2 + for i in range(m): + for j in range(m): + result[ids[m + i]][ids[j]] = wanted[i][j] + need(len(used) == n, "complete macro wire coverage") + return result + + +def macro_check(plan, wanted, bound): + n, cursor, product = len(wanted), 0, eye(len(wanted)) + partitions = [] + for block in plan.blocks: + need(block["start"] == cursor and block["stop"] > cursor, + "consecutive nonempty macro partitions") + cursor = block["stop"] + need(max(piece["error"] for piece in block["pieces"]) <= bound["epsilon"], + "direct-sum macro defect") + product = multiply(macro_reference(block, n), product) + partitions.append([block["start"], block["stop"], + [piece["ids"] for piece in block["pieces"]]]) + need(cursor == len(plan) and product == wanted, "exact retained parity factorization") + need(len(partitions) == bound["macro_layers"], "macro depth certificate") + return partitions + + +def structure(plan, bound): + height = 0 + for layer in plan: + used = set() + for ids, gate in layer: + need(len(ids) == len(gate) in (1, 2), "one/two-wire parity gate") + need(len(set(ids)) == len(ids) and not used.intersection(ids), + "disjoint physical parity layer") + need(all(0 <= i < bound["n"] for i in ids), "owned parity wires") + used.update(ids) + need(multiply(inverse_gate(gate), gate) == eye(len(gate)), "stored gate inverse") + for row in gate: + for entry in row: + height = max(height, qheight(entry.re), qheight(entry.im)) + need(height <= bound["coefficient_height_bound"], "forward/inverse coefficient width") + return height + + +def one_case(symbol, raw, reverse=False, fine=False, label=""): + n = len(symbol) + indices = list(reversed(range(n))) if reverse else list(range(n)) + vectors = [ + Vec(C(Q((3 * i + 1) % 7 - 3, 8), Q((5 * i + 2) % 9 - 4, 8)) for i in range(n)), + Vec(C((-1) ** i * (1 << 12), (1 << 10) * (i % 2)) for i in range(n)), + ] + if fine: + vectors.append(Vec([C(Q(1, 1 << 100))] + [C()] * (n - 1))) + radius = max(Q(1), *(component_l1(v) for v in vectors)) + incoming = max(x.denominator.bit_length() - 1 for v in vectors for z in v + for x in (z.re, z.im)) + bound = allocation(n, radius, 20, incoming) + builder = Triangle(bound, raw[CIRCUIT], raw[ORACLE]) + plan = builder.plan(symbol, indices) + phase_count = len(builder.phases.certificates) + inverse = builder.inverse(plan) + need(len(builder.phases.certificates) == phase_count, "inverse reuses phase coefficients") + wanted = reference(symbol, indices) + intended_inverse = eliminate(wanted) + partitions = macro_check(plan, wanted, bound) + height = max(structure(plan, bound), structure(inverse, bound)) + actual = operator(plan, n) + actual_inverse = operator(inverse, n) + need(multiply(actual_inverse, actual) == eye(n), "exact stored circuit inverse") + forward_error = frobenius_error(actual, wanted) + inverse_error = frobenius_error(actual_inverse, intended_inverse) + need(max(forward_error, inverse_error) <= n * bound["coefficient_error"] ** 2, + "two-sided independent endpoint references") + vectors.append(Vec([C(Q(2, 1 << bound["data_bits"]))] + [C()] * (n - 1))) + final_bound = allocation(n, radius, 20, bound["data_bits"]) + need(all(final_bound[key] == value for key, value in bound.items() + if key != "incoming_bits"), "micro fixture preserves triangular allocation") + bound = final_bound + states = [] + for direction, program, actual_matrix, ideal_matrix in ( + ("forward", plan, actual, wanted), + ("inverse", inverse, actual_inverse, intended_inverse)): + for vector in vectors: + need(norm2(vector) <= radius ** 2, "triangle input radius") + output, stats = apply(program, vector, bound["data_bits"]) + rational = mv(actual_matrix, vector) + ideal = mv(ideal_matrix, vector) + coeff = norm2(rational - ideal) + rounding = norm2(output - rational) + total = norm2(output - ideal) + need(coeff <= (bound["coefficient_error"] * radius) ** 2, + "triangle coefficient error") + need(rounding <= bound["rounding_error"] ** 2, "triangle global data error") + need(total <= bound["total_error"] ** 2, "triangle complete target error") + need(stats["max_observed_height"] <= bound["temporary_bits"], + "triangle observed temporary width") + need(stats["max_grid_numerator_bits"] + 1 <= bound["data_bits"] + bound["integer_bits"] + and stats["max_reduced_fractional_bits"] <= bound["data_bits"], + "triangle signed canonical grid width") + states.append(dict(direction=direction, input_sha256=vector_hash(vector), + output_sha256=vector_hash(output), + coefficient_squared_upper=str(upper(coeff)), + rounding_squared_upper=str(upper(rounding)), + total_squared_upper=str(upper(total)), statistics=stats)) + rounded_forward, _ = apply(plan, vectors[0], bound["data_bits"]) + restored, roundtrip_stats = apply(inverse, rounded_forward, bound["data_bits"]) + roundtrip = norm2(restored - vectors[0]) + allowance = (2 * bound["total_error"] + bound["coefficient_error"] + * (2 * radius + bound["total_error"]) + bound["rounding_error"]) + need(roundtrip <= allowance ** 2, "charged rounded round trip") + roundtrip_radius = 2 * radius + bound["total_error"] + roundtrip_amplitude = (bound["physical_prefix_suffix_bound"] * roundtrip_radius + + bound["layer_bound"] * bound["segment_bound"] * n + * Q(1, 1 << bound["data_bits"])) + roundtrip_integer = max(1, ceil_log2(max(Q(1), roundtrip_amplitude))) + 2 + roundtrip_temporary = (2 * bound["data_bits"] + 8 * bound["coefficient_height_bound"] + + 2 * roundtrip_integer + 32) + need(roundtrip_stats["max_grid_numerator_bits"] + 1 + <= bound["data_bits"] + roundtrip_integer + and roundtrip_stats["max_observed_height"] <= roundtrip_temporary, + "enlarged second-pass round-trip width") + record_bound = {k: str(v) if isinstance(v, Q) else v for k, v in bound.items()} + return dict(label=label, n=n, indices=indices, symbol=[x.record() for x in symbol], + budget=record_bound, partitions=partitions, layers=len(plan), + gates=sum(map(len, plan)), children=len(builder.children), + leaves=len(builder.leaves), triangle_nodes=builder.stats["triangle_nodes"], + source_ast_sha256=builder.original_ast, adapted_ast_sha256=builder.adapted_ast, + plan_sha256=plan_hash(plan), inverse_sha256=plan_hash(inverse), + max_coefficient_height=height, + forward_frobenius_squared_upper=str(upper(forward_error)), + inverse_frobenius_squared_upper=str(upper(inverse_error)), + rounded_roundtrip_squared_upper=str(upper(roundtrip)), + rounded_roundtrip_bound=str(allowance), + roundtrip_width=dict(radius=str(roundtrip_radius), + integer_bits=roundtrip_integer, + temporary_bits=roundtrip_temporary, + statistics=roundtrip_stats), states=states, + phases=builder.phases.record()) + + +def controls(raw): + symbol = Vec([C(Q(3, 4), Q(1, 16)), C(Q(1, 16)), C(Q(-1, 32)), C(Q(1, 32))]) + bound = allocation(4, Q(4), 20) + def new(): + return Triangle(bound, raw[CIRCUIT], raw[ORACLE]) + out = [] + out.append(rejected(lambda: new().plan(Vec([Q(1, 4), 0, 0, 0])), "near-identity-admission")) + out.append(rejected(lambda: allocation(3, 1, 20), "nondyadic-size")) + out.append(rejected(lambda: new().plan([1.0, 0, 0, 0]), "floating-coefficients")) + out.append(rejected(lambda: new().leaf([0], Vec([0])), "zero-leaf-margin")) + builder = new() + plan = builder.plan(symbol) + wanted = reference(symbol, list(range(4))) + wrong_inverse = [[(ids, adjoint(gate) if len(ids) == 1 else inverse_gate(gate)) + for ids, gate in layer] for layer in reversed(plan)] + need(frobenius_error(operator(wrong_inverse, 4), eliminate(wanted)) + > 4 * bound["coefficient_error"] ** 2, "adjoint leaf must fail inverse reference") + out.append("leaf-adjoint-instead-of-reciprocal") + wrong_order = [[(ids, inverse_gate(gate)) for ids, gate in layer] for layer in plan] + need(frobenius_error(operator(wrong_order, 4), eliminate(wanted)) + > 4 * bound["coefficient_error"] ** 2, "wrong inverse order must fail") + out.append("inverse-layer-order") + builder = new() + original_conv = builder.namespace["conv"] + calls = [0] + def missing_shift(a, b, n): + result = original_conv(a, b, n) + calls[0] += 1 + # The second convolution feeds [0, conv(...)[:-1]] in s. + # Changing it changes the exact Schur complement while retaining topology. + return Vec(result) * 0 if calls[0] == 2 else result + builder.namespace["conv"] = missing_shift + mutant = builder.plan(symbol) + out.append(rejected(lambda: macro_check(mutant, wanted, bound), "omitted-schur-correction")) + builder = new() + original_parallel = builder.parallel + def serial(a, b): + merged = original_parallel(a, b) + return Plan(list(a) + list(b), merged.blocks) + builder.namespace["parallel"] = serial + out.append(rejected(lambda: macro_check(builder.plan(symbol), wanted, bound), + "serialized-parallel-middle")) + builder = new() + original_shear = builder.shear + builder.namespace["toeplitz_shear"] = lambda target, source, matrix: original_shear(source, target, matrix) + out.append(rejected(lambda: macro_check(builder.plan(symbol), wanted, bound), + "swapped-parity-bank-routing")) + need(Vec([1, 2]).copy() * 0 == Vec([0, 0]), "NumPy copy/scalar semantics") + out.append("exact-vector-copy-zero-semantics") + out.append(rejected(lambda: apply(plan, Vec([Q(1, 1 << 90), 0, 0, 0]), 20), + "discarded-incoming-grid")) + out.append(rejected(lambda: need( + 32 * bound["layer_bound"] * bound["ideal_bound"] * 4 * Q(1, 1 << 4) + <= Q(1, 1 << 20), "rounding budget"), "underfunded-global-grid")) + return out + + +def run(): + raw, sources = authenticate() + cases = [] + for n in (1, 2, 4, 8): + symbol = Vec([C(Q(4, 5), Q(1, 35))] + + [C(Q((-1) ** j, 32 * n), Q(1, 64 * n)) for j in range(1, n)]) + cases.append(one_case(symbol, raw, reverse=n >= 4, fine=n == 8, label="complex-rational")) + boundary = Vec([Q(1, 2)] + [0] * (n - 1)) + cases.append(one_case(boundary, raw, label="half-identity-boundary")) + n = 16 + symbol = Vec([1] + [C(Q((-1) ** j, 128), Q(1, 256)) for j in range(1, n)]) + cases.append(one_case(symbol, raw, reverse=True, label="deep-parity")) + symbol = Vec([1, 0, Q(1, 8), 0, Q(-1, 16), 0, Q(1, 32), 0]) + cases.append(one_case(symbol, raw, label="zero-odd-polynomial")) + need(any(Q(row["rounded_roundtrip_squared_upper"]) > 0 for row in cases), + "nonzero rounded inverse-restoration witness") + return dict(schema_version=1, scope="retained parity recursion and inverse on one data grid", + sources=sources, cases=cases, controls=controls(raw), + physical_tape_execution=False, full_generator_execution=False) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-result", action="store_true") + args = parser.parse_args() + result = run() + path = HERE / "TRIANGLE-RESULT.json" + if args.write_result: + path.write_text(json.dumps(result, indent=2) + "\n") + else: + need(canonical(result) == canonical(json.loads(path.read_text())), "complete typed triangular receipt") + print(json.dumps(dict(status="PASS_RATIONAL_PARITY_TRIANGLE", cases=len(result["cases"]), + gates=sum(row["gates"] for row in result["cases"]), + rounded_states=sum(len(row["states"]) for row in result["cases"]), + result_sha256=sha256(canonical(result)).hexdigest()), sort_keys=True)) + + +if __name__ == "__main__": + main() From 0e181260108dd940a594fad6b912588bdee62522 Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 17:43:34 +0200 Subject: [PATCH 3/9] Retain exact generator source for circuit integration --- .../vendor/STEIN-PROVENANCE.json | 14 ++ .../vendor/stein_rational.py | 196 ++++++++++++++++++ 2 files changed, 210 insertions(+) create mode 100644 research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json create mode 100644 research/rational-unitary-fft/vendor/stein_rational.py diff --git a/research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json b/research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json new file mode 100644 index 0000000000..c602fb5bb3 --- /dev/null +++ b/research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json @@ -0,0 +1,14 @@ +{ + "schema_version": 1, + "source_pr": "https://github.com/CrocSwap/integer-mult-bounds/pull/4198", + "source_commit": "e457d5e912a27307d725827aeca7198169320b84", + "source_path": "research/rational-stein-generators/rational.py", + "source_git_blob": "71b240a460a185ee2fe926cd189b56aeac6f997f", + "sha256": "3065fc5d6d22ec44de533b08fb0c639603449456383668e561dc46629e641502", + "retained_path": "research/rational-unitary-fft/vendor/stein_rational.py", + "byte_comparison": "I compared the retained bytes with git show at the original source commit before the local archive commit.", + "usage": "Execute the exact bounded rational generator producer as a coefficient-preparation dependency of the complete gate program.", + "proof": "https://github.com/CrocSwap/integer-mult-bounds/blob/e457d5e912a27307d725827aeca7198169320b84/research/rational-stein-generators/PROOF.md", + "credit": "Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. The producer refines Vlado Vojdanovski/PR1673's rational Gram-Schmidt route; Cinna Davis supplied the later caller. Apache-2.0.", + "history_boundary": "The original sibling PR commit is not an ancestor of PR4205. SOURCES.json will pin these local retained bytes at their archive commit, separately from this original-source identity. The later integration does not claim original sibling-branch ancestry." +} diff --git a/research/rational-unitary-fft/vendor/stein_rational.py b/research/rational-unitary-fft/vendor/stein_rational.py new file mode 100644 index 0000000000..71b240a460 --- /dev/null +++ b/research/rational-unitary-fft/vendor/stein_rational.py @@ -0,0 +1,196 @@ +"""Exact dyadically scaled Stein generators over Q(i). + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. The construction refines the retained rational Gram-Schmidt route. +""" +from fractions import Fraction as Q + +MAX_HEIGHT = 0 + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def observe(x): + global MAX_HEIGHT + x = Q(x) + MAX_HEIGHT = max(MAX_HEIGHT, abs(x.numerator).bit_length(), x.denominator.bit_length()) + return x + + +class C: + __slots__ = ("re", "im") + + def __init__(self, re=0, im=0): + need(type(re) in (int, Q) and type(im) in (int, Q), "exact complex rational components") + self.re, self.im = observe(re), observe(im) + + @staticmethod + def wrap(x): + return x if isinstance(x, C) else C(x) + + def __add__(self, other): + other = C.wrap(other) + return C(self.re + other.re, self.im + other.im) + + __radd__ = __add__ + + def __neg__(self): + return C(-self.re, -self.im) + + def __sub__(self, other): + return self + -C.wrap(other) + + def __rsub__(self, other): + return C.wrap(other) + -self + + def __mul__(self, other): + other = C.wrap(other) + return C(self.re * other.re - self.im * other.im, + self.re * other.im + self.im * other.re) + + __rmul__ = __mul__ + + def conjugate(self): + return C(self.re, -self.im) + + def norm2(self): + return observe(self.re ** 2 + self.im ** 2) + + def __truediv__(self, other): + other = C.wrap(other) + denominator = other.norm2() + need(denominator != 0, "nonzero complex divisor") + product = self * other.conjugate() + return C(product.re / denominator, product.im / denominator) + + def __rtruediv__(self, other): + return C.wrap(other) / self + + def __bool__(self): + return bool(self.re or self.im) + + def __eq__(self, other): + if not isinstance(other, C) and type(other) not in (int, Q): + return False + other = C.wrap(other) + return self.re == other.re and self.im == other.im + + def record(self): + return [str(self.re), str(self.im)] + + +class Vector(list): + def __truediv__(self, scalar): + return Vector(C.wrap(x) / scalar for x in self) + + +class Matrix(list): + def __init__(self, rows): + super().__init__([[C.wrap(x) for x in row] for row in rows]) + + def __rmul__(self, scalar): + return Matrix([[C.wrap(x) * scalar for x in row] for row in self]) + + +def inner(left, right): + return sum((C.wrap(x).conjugate() * C.wrap(y) for x, y in zip(left, right)), C()) + + +def norm2(vector): + return observe(sum((C.wrap(x).norm2() for x in vector), Q(0))) + + +def component_l1(vector): + return observe(sum((abs(C.wrap(x).re) + abs(C.wrap(x).im) for x in vector), Q(0))) + + +def pow2(exponent): + return Q(1 << exponent) if exponent >= 0 else Q(1, 1 << -exponent) + + +def norm_scale(squared): + need(squared > 0, "positive residual squared norm") + exponent = (squared.numerator.bit_length() - squared.denominator.bit_length()) // 2 + scale = pow2(exponent) + while scale * scale < squared: + exponent += 1 + scale *= 2 + while scale * scale / 4 >= squared: + exponent -= 1 + scale /= 2 + need(squared <= scale * scale < 4 * squared, "dyadic norm bracket") + observe(scale) + return scale, exponent + + +def upper_dyadic(value): + need(value > 0, "positive dyadic upper target") + exponent = value.numerator.bit_length() - value.denominator.bit_length() + result = pow2(exponent) + while result < value: + result *= 2 + while result / 2 >= value: + result /= 2 + return observe(result) + + +def height(matrix): + return max([1] + [max(abs(x.numerator).bit_length(), x.denominator.bit_length()) + for row in matrix for entry in row + for x in (C.wrap(entry).re, C.wrap(entry).im)]) + + +def generators(displacement): + """Return column g, explicit coefficient ROW r, and a dyadic admission scale.""" + global MAX_HEIGHT + displacement = Matrix(displacement) + size = len(displacement) + need(size > 0 and all(len(row) == size for row in displacement), "nonempty square displacement") + input_height = height(displacement) + MAX_HEIGHT = input_height + columns = [[displacement[i][j] for i in range(size)] for j in range(size)] + residuals, result = [], [] + for column_index, column in enumerate(columns): + residual = list(column) + for old, squared in residuals: + factor = inner(old, residual) / squared + residual = [x - factor * y for x, y in zip(residual, old)] + squared = norm2(residual) + if squared == 0: + continue + scale, exponent = norm_scale(squared) + g = Vector(x / scale for x in residual) + row = Vector(inner(residual, c) * (scale / squared) for c in columns) + gamma = upper_dyadic(max(Q(1), 2 * component_l1(row))) + need(norm2(g) <= 1 and 2 * component_l1(row) <= gamma, "bounded generator admission") + residuals.append((residual, squared)) + result.append({"column": column_index, "residual": residual, "squared_norm": squared, + "scale": scale, "scale_exponent": exponent, "g": g, "row": row, + "gamma": gamma}) + return result, {"input_height": input_height, "max_observed_height": MAX_HEIGHT} + + +def lower(vector): + return Matrix([[vector[i - j] if i >= j else C() for j in range(len(vector))] + for i in range(len(vector))]) + + +def transpose(matrix): + return Matrix(zip(*matrix)) + + +def frobenius2(matrix): + return observe(sum((C.wrap(x).norm2() for row in matrix for x in row), Q(0))) + + +def norm_bound(matrix): + """Rational upper bound for the induced 2-norm.""" + return max(max(component_l1(row) for row in matrix), + max(component_l1(column) for column in zip(*matrix))) + + +def exact_record(matrix): + return [[C.wrap(x).record() for x in row] for row in matrix] From fdcdc2690910392f7b42dbae4155b7d7b5790abd Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 18:03:26 +0200 Subject: [PATCH 4/9] Execute certified compensated generator circuits --- .../rational-unitary-fft/GENERATOR-PROOF.md | 277 + .../GENERATOR-RESULT.json | 12181 ++++++++++++++++ research/rational-unitary-fft/README.md | 70 +- research/rational-unitary-fft/SOURCES.json | 5 +- research/rational-unitary-fft/TASK.json | 16 +- research/rational-unitary-fft/VERIFY.json | 23 +- research/rational-unitary-fft/exact.py | 6 + research/rational-unitary-fft/generator.py | 232 + research/rational-unitary-fft/triangular.py | 5 +- research/rational-unitary-fft/verify.py | 12 +- .../rational-unitary-fft/verify_generator.py | 316 + 11 files changed, 13112 insertions(+), 31 deletions(-) create mode 100644 research/rational-unitary-fft/GENERATOR-PROOF.md create mode 100644 research/rational-unitary-fft/GENERATOR-RESULT.json create mode 100644 research/rational-unitary-fft/generator.py create mode 100644 research/rational-unitary-fft/verify_generator.py diff --git a/research/rational-unitary-fft/GENERATOR-PROOF.md b/research/rational-unitary-fft/GENERATOR-PROOF.md new file mode 100644 index 0000000000..8bef383eec --- /dev/null +++ b/research/rational-unitary-fft/GENERATOR-PROOF.md @@ -0,0 +1,277 @@ +# Complete rounded bounded-generator circuits + +Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance. +Apache-2.0. + +## 1. Input, output and inherited generator identity + +I consider a rational complex matrix \(T\) of dyadic size \(m=2^d\). +Let \(Z\) be the nilpotent lower shift and \(E=T-ZTZ^T\), with rank \(r\). +The target is the dirty shear +\[ +S(T)(x,y)=(x,y+Tx) +\] +on exactly \(2m\) existing complex wires. I require no zero bank. +The input lies on an \(F_0\)-bit dyadic grid and has norm at most \(R\ge1\); +the requested output error is \(2^{-t}\), with integer \(t\ge1\). +I use a rational bound \(B_E\ge\|E\|_2\). The executable takes the maximum +of the row and column component sums, which bounds both induced norms. + +I execute the frozen PR4198 rational producer. For each nonzero orthogonal +residual \(v_j\) of \(E\), it chooses a dyadic \(c_j\) with +\(\|v_j\|\le c_j<2\|v_j\|\), then emits +\[ +g_j=v_j/c_j,\qquad +a_j=\frac{c_j}{v_j^*v_j}v_j^*E. +\] +The orthogonal projectors sum to the identity on the column space of \(E\). +Thus +\[ +E=\sum_{j=1}^r g_ja_j,\quad +\|g_j\|\le1,\quad \|a_j\|<2B_E,\quad +T=\sum_{j=1}^r L(g_j)U(a_j). +\tag{1} +\] +The last identity follows by telescoping +\(T=\sum_{k=0}^{m-1}Z^kE(Z^T)^k\). The \(a_j\) are coefficient rows; +I apply no further conjugation. + +I retain the producer's scale +\[ +\Gamma_j=\min\{2^e:2^e\ge\max(1,2\ell(a_j))\},\quad +\ell(a)=\sum_k(|\Re a_k|+|\Im a_k|), +\] +and put \(P_j=I+U(a_j)/\Gamma_j\), \(G_j=\Gamma_jL(g_j)\). +Then \(P_j\) satisfies the component-l1 admission used in +[TRIANGLE-PROOF.md](TRIANGLE-PROOF.md), and +\(\|P_j\|\le3/2,\ \|P_j^{-1}\|\le2\). +The source realizes the upper triangular map by reversing its source wires. + +I use the rational bounds +\[ +\alpha=2rmB_E,\qquad K_G=2m+16mB_E. +\tag{2} +\] +Indeed \(\|L(g_j)U(a_j)\|\le2mB_E\), and +\[ +\Gamma_j\sqrt m + <2\sqrt m+8\sqrt2\,mB_E\le K_G. +\] +The scaled shear's coefficient component sum is at most +\(\sqrt2K_G<2K_G\). I budget the approximation of \(G_j\) itself; +approximating \(L(g_j)\) first and multiplying by an uncharged +\(\Gamma_j\) would not meet this contract. + +## 2. Paired compensation and additive defects + +I retain the source's four calls: +\[ +P_j;\quad S(G_j);\quad P_j^{-1};\quad S(-G_j). +\tag{3} +\] +I compile \(\widehat P_j\) and reuse its exact stored inverse. +For the positive shear I use the certified FFT construction. +For its compensation I reuse the same unitary tables and negate the +stored spectral multipliers, preserving the source's cyclic-then-negacyclic +order. The two unrounded shear endpoints are exactly +\(S(\widehat G_j)\) and \(S(-\widehat G_j)\). + +Negating a nearest-grid value gives a nearest-grid value of the negative +exact quantity, so this paired rule retains the same multiplier-error +bound. At half-grid ties it need not agree with independent rounding +toward positive infinity. The verifier includes that discrepancy as a +control. I do not replace the negative child with a reversed whole +positive-shear program. + +For \(\|\widehat P_j-P_j\|,\|\widehat G_j-G_j\|\le\eta\le1/4\), +the exact stored-coefficient program has endpoint +\[ +(x,y)\longmapsto(x,y+\widehat G_j(\widehat P_j-I)x). +\] +Write \(T_j=G_j(P_j-I)\) and +\(\widehat T_j=\widehat G_j(\widehat P_j-I)\). Then +\[ +\|\widehat T_j-T_j\| +\le K_G\eta+(1/2+\eta)\eta\le(K_G+1)\eta. +\] +The completed terms restore the same source bank, so their endpoints +compose to \(S(\sum_j\widehat T_j)\). Consequently +\[ +\boxed{\|\sum_j\widehat T_j-T\|\le +\Delta:=r(K_G+1)\eta.} +\tag{4} +\] +I need no generic product-of-perturbed-matrices bound across completed +generator terms. This additive estimate uses exact stored-coefficient +restoration; data-rounding errors receive a separate bound below. + +## 3. Checkpoint prefixes and suffixes + +Assume \(\Delta\le1\). Any preceding or remaining sum of approximate terms +has norm at most \(a_*=\alpha+1\). +For the current term write \(p=\widehat P_j\), \(h=\widehat G_j\), +\(A=\sum_{i", "exec", optimize=0), namespace) + return namespace + + +def lower(vector): + return Mat([[vector[i - j] if i >= j else C() for j in range(len(vector))] + for i in range(len(vector))]) + + +def prepare(matrix, producer): + matrix = Mat(matrix) + m = len(matrix) + need(m > 0 and m & (m - 1) == 0 and matrix.shape == (m, m), "dyadic generator matrix") + displacement = Mat([[matrix[i][j] - (matrix[i - 1][j - 1] if i and j else C()) + for j in range(m)] for i in range(m)]) + old_c = producer["C"] + old_e = producer["Matrix"]([[old_c(x.re, x.im) for x in row] for row in displacement]) + old_items, heights = producer["generators"](old_e) + items = [dict(column=item["column"], gamma=item["gamma"], + scale=item["scale"], scale_exponent=item["scale_exponent"], + g=Vec(C(x.re, x.im) for x in item["g"]), + row=Vec(C(x.re, x.im) for x in item["row"])) for item in old_items] + bound_e = max(max(component_l1(row) for row in displacement), + max(component_l1(column) for column in zip(*displacement))) + need(all(norm2(item["g"]) <= 1 and norm2(item["row"]) < 4 * bound_e ** 2 + and 2 * component_l1(item["row"]) <= item["gamma"] for item in items), + "actual bounded rational generator output") + return items, displacement, bound_e, heights + + +def allocation(m, rank, bound_e, radius, target, incoming): + need(type(m) is int and m > 0 and m & (m - 1) == 0, "dyadic macro size") + need(type(rank) is int and 0 <= rank <= m and type(bound_e) in (int, Q) + and bound_e >= 0, "generator count and exact displacement bound") + need(type(radius) in (int, Q) and radius >= 1 and type(target) is int + and target >= 1 and type(incoming) is int and incoming >= 0, "global generator input") + d = m.bit_length() - 1 + kg, alpha = 2 * m + 16 * m * bound_e, 2 * rank * m * bound_e + checkpoint = 10 + 4 * alpha + 2 * kg + child = max(Q(3200 * m ** 3), 2 + 2 * kg) + execution = child * checkpoint + depth = rank * (8 * d * d + 32 * d + 18) + gates = rank * m * (4 * d * d + 24 * d + 22) + s = target + ceil_log2(4 * (rank + 1) * (kg + 1) * (radius + 1)) + eta = Q(1, 1 << s) + triangle = triangle_allocation(m, 1, s) + envelope = max(Q(6), 2 * kg) + p = max(triangle["phase_bits"], s + ceil_log2(32 * (d + 1) * (envelope + 1))) + mult = max(triangle["multiplier_bits"], s + 4) + triangle = dict(triangle, phase_bits=p, multiplier_bits=mult, + coefficient_height_bound=max(4 * p + 24, mult + 5, + 2 * triangle["leaf_bits"] + 16)) + data = max(incoming, target + ceil_log2(max(Q(1), 2 * depth * execution * m))) + u = Q(1, 1 << data) + defect = rank * (kg + 1) * eta + rounding = depth * execution * m * u + need(defect <= 1 and defect * radius + rounding <= Q(1, 1 << target), + "complete compensated-generator target budget") + amplitude = execution * radius + depth * execution ** 2 * m * u + integer = max(1, ceil_log2(max(Q(1), amplitude))) + 2 + height = max(4 * p + 24, mult + ceil_log2(envelope + 2) + 2, + 2 * triangle["leaf_bits"] + 16) + return dict(m=m, rank=rank, displacement_bound=bound_e, radius=radius, + target_bits=target, incoming_bits=incoming, kg=kg, alpha=alpha, + checkpoint_bound=checkpoint, child_bound=child, execution_bound=execution, + segment_bound=execution ** 2, layer_bound=depth, gate_bound=gates, + child_target_bits=s, eta=eta, phase_bits=p, multiplier_bits=mult, + leaf_bits=triangle["leaf_bits"], data_bits=data, + multiplier_envelope=envelope, coefficient_error=defect, + rounding_error=rounding, total_error=defect * radius + rounding, + amplitude_bound=amplitude, integer_bits=integer, + coefficient_height_bound=height, + temporary_bits=2 * data + 8 * height + 2 * integer + 32), triangle + + +def macro_template(source): + tree = ast.parse(source) + function = next(n for n in tree.body if isinstance(n, ast.FunctionDef) + and n.name == "lowdisp_shear") + loop = next(n for n in function.body if isinstance(n, ast.For)) + start = next(i for i, n in enumerate(loop.body) if isinstance(n, ast.Assign) + and isinstance(n.targets[0], ast.Name) and n.targets[0].id == "p") + tail = copy.deepcopy(loop.body[start:]) + need([type(n).__name__ for n in tail] == + ["Assign", "AugAssign", "Assign", "AugAssign", "AugAssign", "AugAssign", "AugAssign"], + "retained seven-statement macro") + identity = sha256(ast.dump(ast.Module(body=tail, type_ignores=[])).encode()).hexdigest() + need(identity == "2a2e0f16a779597fa555857307451831bace11e66861fdc3caf26d2373a21210", + "unchanged source macro identity") + function = ast.FunctionDef( + name="emit", args=ast.arguments(posonlyargs=[], args=[ast.arg(arg=n) for n in + ("target", "source", "g", "h", "gamma")], vararg=None, kwonlyargs=[], + kw_defaults=[], kwarg=None, defaults=[]), + body=[ast.Assign(targets=[ast.Name(id="result", ctx=ast.Store())], + value=ast.List(elts=[], ctx=ast.Load()))] + tail + + [ast.Return(value=ast.Name(id="result", ctx=ast.Load()))], + decorator_list=[], type_params=[]) + module = ast.fix_missing_locations(ast.Module(body=[function], type_ignores=[])) + return compile(module, "", "exec", optimize=0), identity + + +class PairedNegative(Builder): + def __init__(self, positive, source): + super().__init__(positive.phases, positive.multiplier_bits, source) + self.positive = positive + + def spectrum(self, kernel): + row = self.positive.spectra[len(self.spectra)] + need(Vec(kernel) == Vec(-x for x in row["kernel"]), "paired negative spectral kernel") + result = Vec(-x for x in row["multipliers"]) + bound = self.bounds[len(self.spectra)] + need(bound == row["bound"], "paired spectral envelope") + v = Q(1, 1 << self.multiplier_bits) + need(all(x.norm2() <= (bound + v) ** 2 for x in result), "negative stored multiplier bound") + self.spectra.append(dict(kernel=Vec(kernel), multipliers=result, bound=bound)) + return result + + +class Program: + def __init__(self, bound, triangle_bound, raw): + self.bound, self.triangle_bound, self.raw = bound, triangle_bound, raw + self.phases = Phases(bound["phase_bits"], raw[ORACLE]) + self.source = list(range(bound["m"])) + self.target = list(range(bound["m"], 2 * bound["m"])) + self.stages, self.position, self.calls = [], 0, [] + self.pending = None + self.last_triangle = None + self.triangles, self.shears = [], [] + code, self.macro_ast = macro_template(raw[SOURCE + "check_circuits.py"]) + namespace = dict(triangle_plan=self.triangle, toeplitz_shear=self.shear, + inverse=self.inverse, lower=lower) + exec(code, namespace) + self.emit = namespace["emit"] + + def record(self, kind, plan, **details): + self.calls.append(kind) + self.stages.append(dict(kind=kind, start=self.position, stop=self.position + len(plan), + **details)) + self.position += len(plan) + return plan + + def triangle(self, indices, coefficients): + builder = Triangle(self.triangle_bound, self.raw[SOURCE + "check_circuits.py"], + self.raw[ORACLE], self.phases) + local_indices = [self.source.index(i) for i in indices] + local = builder.plan(coefficients, local_indices) + plan = [[(tuple(self.source[i] for i in ids), gate) for ids, gate in layer] for layer in local] + need(builder.bound["coefficient_error"] <= self.bound["eta"], "triangle operator allowance") + self.last_triangle = plan + self.triangles.append(builder) + return self.record("triangle", plan, symbol=Vec(coefficients), indices=list(indices), + operator_error=builder.bound["coefficient_error"]) + + def inverse(self, plan): + need(plan is self.last_triangle, "inverse of the same stored triangle") + result = [[(ids, inverse_gate(gate)) for ids, gate in layer] for layer in reversed(plan)] + return self.record("inverse", result) + + def shear(self, target, source, matrix): + need(target == self.target and source == self.source, "literal complete dirty banks") + if self.pending is None: + builder = Builder(self.phases, self.bound["multiplier_bits"], + self.raw[SOURCE + "check_circuits.py"]) + local = builder.plan(matrix) + self.pending = (builder, matrix, local) + kind = "positive" + else: + positive, original, positive_plan = self.pending + need(matrix == -1 * original, "literal negative compensation matrix") + builder = PairedNegative(positive, self.raw[SOURCE + "check_circuits.py"]) + local = builder.plan(matrix) + m = len(source) + need(len(local) == len(positive_plan), "same compensation layer schedule") + for a, b in zip(positive_plan, local): + need(len(a) == len(b), "same compensation gate schedule") + for (ids, g), (other_ids, h) in zip(a, b): + crossing = len(ids) == 2 and ids[0] < m <= ids[1] + expected = Mat([[1, 0], [-g[1][0], 1]]) if crossing else g + need(ids == other_ids and h == expected, "paired coefficients preserve source order") + self.pending = None + kind = "compensation" + g, h = split_kernel(matrix) + envelope = component_l1(g) + component_l1(h) + defect = (4 * len(source).bit_length() * envelope * self.phases.nu + + Q(2, 1 << self.bound["multiplier_bits"])) + need(envelope <= self.bound["multiplier_envelope"] and defect <= self.bound["eta"], + "scaled shear absolute operator allowance") + self.shears.append(builder) + mapping = source + target + plan = [[(tuple(mapping[i] for i in ids), gate) for ids, gate in layer] for layer in local] + return self.record(kind, plan, matrix=matrix, operator_error=defect, envelope=envelope) + + def plan(self, items): + result = [] + for item in items: + before = Vec(item["row"]) + result += self.emit(self.target, self.source, item["g"], item["row"], item["gamma"]) + need(before == item["row"], "fresh division preserves generator row") + need(self.pending is None and self.calls == + ["triangle", "positive", "inverse", "compensation"] * len(items), + "complete retained compensated call sequence") + need(self.position == len(result) <= self.bound["layer_bound"] + and sum(map(len, result)) <= self.bound["gate_bound"], "full program gate/layer budget") + return result diff --git a/research/rational-unitary-fft/triangular.py b/research/rational-unitary-fft/triangular.py index 0d045d04a9..2bd8de01a7 100644 --- a/research/rational-unitary-fft/triangular.py +++ b/research/rational-unitary-fft/triangular.py @@ -100,9 +100,10 @@ def visit_Return(self, node): class Triangle: - def __init__(self, bound, circuit_source, oracle_source): + def __init__(self, bound, circuit_source, oracle_source, phase_oracle=None): self.bound, self.circuit_source = bound, circuit_source - self.phases = Phases(bound["phase_bits"], oracle_source) + self.phases = phase_oracle if phase_oracle is not None else Phases(bound["phase_bits"], oracle_source) + need(self.phases.bits == bound["phase_bits"], "common triangular phase precision") base = Builder(self.phases, bound["multiplier_bits"], circuit_source) self.source_parallel = base.namespace["parallel"] self.source_inverse = base.namespace["inverse"] diff --git a/research/rational-unitary-fft/verify.py b/research/rational-unitary-fft/verify.py index 60d02a6f0d..e4de89f896 100644 --- a/research/rational-unitary-fft/verify.py +++ b/research/rational-unitary-fft/verify.py @@ -33,7 +33,7 @@ def rejected(operation, label): raise ValueError("control unexpectedly passed: " + label) -def authenticate(): +def authenticate(include_generator=False): rows = [r for r in json.loads((HERE / "SOURCES.json").read_text())["files"] if r["usage"] == "input"] raw = {} for row in rows: @@ -42,7 +42,15 @@ def authenticate(): rejected(lambda: need(sha256(content + b"\0").hexdigest() == row["sha256"], "changed source"), "source corruption") raw[row["path"]] = content - return raw, [{"path": r["path"], "sha256": r["sha256"]} for r in rows] + # Preserve the completed child receipts' mathematical input inventory. + # All declared inputs above still receive the same byte-integrity check. + added = { + "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py", + "research/rational-unitary-fft/vendor/stein_rational.py", + "research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json", + } + return raw, [{"path": r["path"], "sha256": r["sha256"]} for r in rows + if include_generator or r["path"] not in added] def operator(plan, n): diff --git a/research/rational-unitary-fft/verify_generator.py b/research/rational-unitary-fft/verify_generator.py new file mode 100644 index 0000000000..266ad80e44 --- /dev/null +++ b/research/rational-unitary-fft/verify_generator.py @@ -0,0 +1,316 @@ +"""Source-bound complete compensated-generator gate replay. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +import argparse +import ast +import copy +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +from exact import (C, Mat, Vec, apply, component_l1, eye, inverse_gate, multiply, need, + norm2, upper, vector_hash) +from engine import Builder, Phases +from generator import (ORACLE, SOURCE, PairedNegative, Program, allocation, + load_producer, lower, prepare) +from verify import authenticate, canonical, mv, operator, plan_hash, rejected +from verify_triangle import frobenius_error, structure +from triangular import allocation as triangle_allocation + + +def collect(raw, producer): + source = raw[SOURCE + "check_exact_algebra.py"] + namespace = {"__name__": "_retained_exact_subject", "__file__": ""} + exec(compile(source, "", "exec", optimize=0), namespace) + tree = ast.parse(source) + main = next(n for n in tree.body if isinstance(n, ast.FunctionDef) and n.name == "main") + walk = copy.deepcopy(next(n for n in main.body if isinstance(n, ast.FunctionDef) and n.name == "walk")) + index = next(i for i, node in enumerate(walk.body) if isinstance(node, ast.Assign) + and isinstance(node.targets[0], ast.Name) and node.targets[0].id == "S") + walk.body.insert(index + 1, ast.Expr(value=ast.Call(func=ast.Name(id="capture", ctx=ast.Load()), + args=[ast.Name(id=x, ctx=ast.Load()) for x in ("K", "L", "n", "depth")], keywords=[]))) + collected = [] + def capture(k, l, n, depth): + for label, matrix in (("K", k), ("L", l)): + collected.append((f"source-{label}-n{n}-d{depth}", + Mat([[C(x.re, x.im) if isinstance(x, producer["C"]) else C(x) + for x in row] for row in matrix]))) + namespace.update(trace=[], capture=capture) + exec(compile(ast.fix_missing_locations(ast.Module(body=[walk], type_ignores=[])), + "", "exec", optimize=0), namespace) + old_c, old_matrix = producer["C"], producer["Matrix"] + for size in (2, 4, 8): + for complex_case in (False, True): + column = [old_c(1)] + [old_c(Q((-1) ** i, 64 * size), + Q(i, 128 * size) if complex_case else 0) + for i in range(1, size)] + row = [old_c(1)] + [old_c(Q(1, 64 * size), + Q((-1) ** j, 128 * size) if complex_case else 0) + for j in range(1, size)] + matrix = old_matrix([[column[i - j] if i >= j else row[j - i] + for j in range(size)] for i in range(size)]) + namespace["walk"](matrix) + matrix = old_matrix([[int(i == j) for j in range(4)] for i in range(4)]) + matrix[0][2] = matrix[1][3] = old_c(Q(1, 16)) + matrix[0][3] = old_c(0, Q(1, 16)) + namespace["walk"](matrix) + return collected, namespace["trace"], sha256(ast.dump(walk).encode()).hexdigest() + +def exact_rank(matrix): + rows, pivot_row = [list(row) for row in matrix], 0 + for column in range(len(matrix)): + pivot = next((i for i in range(pivot_row, len(rows)) if rows[i][column] != 0), None) + if pivot is None: + continue + rows[pivot_row], rows[pivot] = rows[pivot], rows[pivot_row] + divisor = rows[pivot_row][column] + rows[pivot_row] = [x / divisor for x in rows[pivot_row]] + for i in range(pivot_row + 1, len(rows)): + multiplier = rows[i][column] + rows[i] = [x - multiplier * y for x, y in zip(rows[i], rows[pivot_row])] + pivot_row += 1 + return pivot_row + + +def shear_matrix(matrix): + m = len(matrix) + return Mat([[int(i == j) if i < m or j >= m else matrix[i - m][j] + for j in range(2 * m)] for i in range(2 * m)]) + + +def reconstruct(items, displacement, matrix): + m = len(matrix) + e = Mat([[0] * m for _ in range(m)]) + t = Mat([[0] * m for _ in range(m)]) + for item in items: + g, row = item["g"], item["row"] + term = multiply(lower(g), Mat(zip(*lower(row)))) + for i in range(m): + for j in range(m): + e[i][j] = e[i][j] + g[i] * row[j] + t[i][j] = t[i][j] + term[i][j] + need(e == displacement and t == matrix, "exact producer and Stein reconstruction") + +def checkpoint_check(program, plan, actual, bound): + n = 2 * bound["m"] + prefix, inverse_prefix = eye(n), eye(n) + final_inverse = Mat([[int(i == j) if i < n // 2 or j >= n // 2 else -actual[i][j] + for j in range(n)] for i in range(n)]) + peak = Q(0) + for stage in program.stages: + fragment = plan[stage["start"]:stage["stop"]] + reverse = [[(ids, inverse_gate(gate)) for ids, gate in layer] for layer in reversed(fragment)] + prefix = multiply(operator(fragment, n), prefix) + inverse_prefix = multiply(inverse_prefix, operator(reverse, n)) + need(multiply(prefix, inverse_prefix) == eye(n), "exact checkpoint inverse") + suffix = multiply(actual, inverse_prefix) + inverse_suffix = multiply(prefix, final_inverse) + for matrix in (prefix, inverse_prefix, suffix, inverse_suffix): + squared = sum((x.norm2() for row in matrix for x in row), Q(0)) + peak = max(peak, squared) + need(squared <= n * bound["checkpoint_bound"] ** 2, + "checkpoint Frobenius consequence") + need(prefix == actual, "checkpoint sequence reaches actual endpoint") + return dict(checkpoints=len(program.stages), max_frobenius_squared_upper=str(upper(peak))) + + +def one_case(label, matrix, raw, producer, fine=False): + m = len(matrix) + items, displacement, bound_e, heights = prepare(matrix, producer) + need(len(items) == exact_rank(displacement), "separate exact rank reference") + reconstruct(items, displacement, matrix) + vectors = [ + Vec(C(Q((5 * i + 2) % 11 - 5, 8), Q((3 * i + 1) % 7 - 3, 8)) + for i in range(2 * m)), + Vec(C((-1) ** i * (1 << 12), (i % 2) * (1 << 10)) for i in range(2 * m)), + ] + if fine: + vectors.append(Vec([C(Q(1, 1 << 100))] + [C()] * (2 * m - 1))) + radius = max(Q(1), *(component_l1(v) for v in vectors)) + incoming = max(part.denominator.bit_length() - 1 for v in vectors + for entry in v for part in (entry.re, entry.im)) + bound, triangular = allocation(m, len(items), bound_e, radius, 20, incoming) + program = Program(bound, triangular, raw) + plan = program.plan(items) + actual = operator(plan, 2 * m) + expected = shear_matrix(matrix) + for i in range(2 * m): + for j in range(2 * m): + if i < m or j >= m: + need(actual[i][j] == int(i == j), "exact unrounded source restoration") + coefficient_squared = frobenius_error(actual, expected) + need(coefficient_squared <= m * bound["coefficient_error"] ** 2, + "independent full-matrix endpoint reference") + height = structure(plan, dict(bound, n=2 * m)) + checkpoints = (checkpoint_check(program, plan, actual, bound) + if label in ("dense-complex", "large-scale", "source-K-n8-d0") + else None) + need(len(program.stages) == 4 * len(items), "complete actual macro checkpoints") + cursor = 0 + stage_records = [] + for stage in program.stages: + need(stage["start"] == cursor and stage["stop"] > cursor, "consecutive nonempty macro stages") + cursor = stage["stop"] + entry = {k: str(v) if isinstance(v, Q) else v for k, v in stage.items() + if k not in ("matrix", "symbol")} + if "matrix" in stage: + entry["matrix_sha256"] = vector_hash([x for row in stage["matrix"] for x in row]) + if "symbol" in stage: + entry["symbol"] = [x.record() for x in stage["symbol"]] + stage_records.append(entry) + need(cursor == len(plan), "complete actual stage coverage") + vectors.append(Vec([C(Q(2, 1 << bound["data_bits"]))] + [C()] * (2 * m - 1))) + final, _ = allocation(m, len(items), bound_e, radius, 20, bound["data_bits"]) + need(all(final[k] == v for k, v in bound.items() if k != "incoming_bits"), + "micro fixture preserves global allocation") + bound = final + states = [] + for vector in vectors: + need(norm2(vector) <= radius ** 2, "dirty input radius") + output, stats = apply(plan, vector, bound["data_bits"]) + unrounded, intended = mv(actual, vector), mv(expected, vector) + coefficient = norm2(unrounded - intended) + rounding = norm2(output - unrounded) + total = norm2(output - intended) + source_error = norm2(Vec(output[:m]) - Vec(vector[:m])) + need(coefficient <= (bound["coefficient_error"] * radius) ** 2, "additive coefficient defect") + need(rounding <= bound["rounding_error"] ** 2, "global rounded gate error") + need(total <= bound["total_error"] ** 2 and source_error <= bound["rounding_error"] ** 2, + "complete compensated error and rounded restoration") + need(stats["max_observed_height"] <= bound["temporary_bits"] + and stats["max_grid_numerator_bits"] + 1 <= bound["data_bits"] + bound["integer_bits"] + and stats["max_reduced_fractional_bits"] <= bound["data_bits"], + "global signed grid and temporary widths") + states.append(dict(input_sha256=vector_hash(vector), output_sha256=vector_hash(output), + coefficient_squared_upper=str(upper(coefficient)), + rounding_squared_upper=str(upper(rounding)), + total_squared_upper=str(upper(total)), + source_squared_upper=str(upper(source_error)), + source_restoration_exact=source_error == 0, statistics=stats)) + return dict(label=label, m=m, rank=len(items), + input_sha256=vector_hash([x for row in matrix for x in row]), + stein_sha256=vector_hash([x for row in displacement for x in row]), + generators=[dict(column=x["column"], gamma=str(x["gamma"]), scale=str(x["scale"]), + g=[z.record() for z in x["g"]], row=[z.record() for z in x["row"]]) + for x in items], + producer_heights=heights, macro_ast_sha256=program.macro_ast, + budget={k: str(v) if isinstance(v, Q) else v for k, v in bound.items()}, + layers=len(plan), gates=sum(map(len, plan)), stages=stage_records, + plan_sha256=plan_hash(plan), max_coefficient_height=height, + endpoint_frobenius_squared_upper=str(upper(coefficient_squared)), + checkpoints=checkpoints, phases=program.phases.record(), states=states), program, plan, expected + + +def controls(raw, producer): + matrix = Mat([[C(Q(1, 8)), C(0, Q(1, 16))], [C(), C(Q(1, 8))]]) + items, e, be, _ = prepare(matrix, producer) + bound, triangular = allocation(2, len(items), be, 4, 20, 0) + program = Program(bound, triangular, raw) + plan = program.plan(items) + wanted = shear_matrix(matrix) + result = [] + for kind in ("inverse", "compensation"): + stage = next(s for s in program.stages if s["kind"] == kind) + mutant = plan[:stage["start"]] + plan[stage["stop"]:] + need(frobenius_error(operator(mutant, 4), wanted) > 2 * bound["coefficient_error"] ** 2, + "omitted generator stage changes endpoint") + result.append("omitted-" + kind) + wrong = [dict(item, row=Vec(x.conjugate() for x in item["row"])) for item in items] + result.append(rejected(lambda: reconstruct(wrong, e, matrix), "extra-row-conjugation")) + tiny = Mat([[Q(1, 1 << 200), 0], [0, Q(1, 1 << 200)]]) + tiny_items, tiny_e, _, _ = prepare(tiny, producer) + need(len(tiny_items) == 1, "retain tiny exact rank before approximation") + result.append(rejected(lambda: reconstruct([], tiny_e, tiny), "omitted-tiny-exact-generator")) + result.append(rejected(lambda: prepare([[1.0]], producer), "floating-generator-input")) + result.append(rejected(lambda: prepare([[1, 0, 0], [0, 1, 0], [0, 0, 1]], producer), + "nondyadic-generator-size")) + result.append(rejected(lambda: apply(plan, Vec([Q(1, 1 << 100), 0, 0, 0]), 20), + "discarded-generator-input-grid")) + bad = dict(bound, phase_bits=4) + bad_triangle = dict(triangular, phase_bits=4) + result.append(rejected(lambda: Program(bad, bad_triangle, raw).plan(items), + "underfunded-common-phases")) + large = Mat([[1 << 80, 1 << 80], [0, 1 << 80]]) + large_items, _, large_e, _ = prepare(large, producer) + large_bound, large_triangle = allocation(2, len(large_items), large_e, 4, 20, 0) + triangle_only = triangle_allocation(2, 1, large_bound["child_target_bits"])["phase_bits"] + need(triangle_only < large_bound["phase_bits"], "large shear needs additional phase precision") + try: + Program(dict(large_bound, phase_bits=triangle_only), + dict(large_triangle, phase_bits=triangle_only), raw).plan(large_items) + except ValueError as error: + need(str(error) == "scaled shear absolute operator allowance", + "scale control reaches and fails the scaled shear contract") + else: + raise ValueError("uncharged scale control unexpectedly passed") + result.append("uncharged-scaled-shear-phase-precision") + # At a half tie, independent nearest-up rounding of +/- is not odd. + bits = 8 + phases = Phases(12, raw[ORACLE]) + source = raw[SOURCE + "check_circuits.py"] + kernel = Mat([[Q(1, 1 << bits)]]) + positive = Builder(phases, bits, source) + plus = positive.plan(kernel) + negative = PairedNegative(positive, source) + paired = negative.plan(-1 * kernel) + independent = Builder(phases, bits, source).plan(-1 * kernel) + need(operator(plus + paired, 2) == eye(2), "paired half-tie compensation") + need(operator(plus + independent, 2) != eye(2), "independent half ties lose exact compensation") + result.append("independent-half-tie-compensation") + need(Vec([1, 2]) / 2 == Vec([Q(1, 2), 1]), "fresh exact macro vector division") + result.append("exact-fresh-vector-division") + need(canonical({"rank": 1}) != canonical({"rank": True}), "typed full receipt") + return result + ["typed-generator-receipt"] + + +def run(): + raw, sources = authenticate(include_generator=True) + producer = load_producer(raw) + fixtures, trace, collector_ast = collect(raw, producer) + tiny = Q(1, 1 << 200) + fixtures += [ + ("zero", Mat([[0, 0], [0, 0]])), + ("scalar", Mat([[Q(1, 32)]])), + ("tiny-diagonal", Mat([[tiny, 0], [0, tiny]])), + ("near-dependent", Mat([[1, 1], [0, 1 + tiny]])), + ("complex-row", Mat([[C(Q(1, 16)), C(0, Q(1, 16))], [0, Q(1, 16)]])), + ("large-scale", Mat([[1 << 80, 1 << 80], [0, 1 << 80]])), + ("dense-complex", Mat([[C(Q(1 + (3 * i + j) % 7, 32), + Q((i + 2 * j) % 5 - 2, 64)) + for j in range(4)] for i in range(4)])), + ] + records = [one_case(label, matrix, raw, producer, fine=label == "zero")[0] + for label, matrix in fixtures] + need(any(not state["source_restoration_exact"] for row in records for state in row["states"]), + "nonzero full-program rounded restoration witness") + return dict(schema_version=1, scope="actual compensated generator gates on one data grid", + sources=sources, collector_ast_sha256=collector_ast, source_schur_trace=trace, + cases=records, controls=controls(raw, producer), + physical_tape_execution=False, complete_gaussian_execution=False) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-result", action="store_true") + args = parser.parse_args() + result = run() + path = HERE / "GENERATOR-RESULT.json" + if args.write_result: + path.write_text(json.dumps(result, indent=2) + "\n") + else: + need(canonical(result) == canonical(json.loads(path.read_text())), "complete typed generator receipt") + print(json.dumps(dict(status="PASS_RATIONAL_GENERATOR_GATES", cases=len(result["cases"]), + generators=sum(x["rank"] for x in result["cases"]), + gates=sum(x["gates"] for x in result["cases"]), + rounded_states=sum(len(x["states"]) for x in result["cases"]), + result_sha256=sha256(canonical(result)).hexdigest()), sort_keys=True)) + + +if __name__ == "__main__": + main() From 7ab1bc8973cc9e606053c65f094ca3abc4a42fee Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 19:01:42 +0200 Subject: [PATCH 5/9] Certify full Toeplitz recursion with polynomial prefix bounds --- research/rational-unitary-fft/FULL-PROOF.md | 374 + .../rational-unitary-fft/FULL-RESULT.json | 11717 ++++++++++++++++ research/rational-unitary-fft/README.md | 70 +- research/rational-unitary-fft/SOURCES.json | 3 +- research/rational-unitary-fft/TASK.json | 16 +- research/rational-unitary-fft/VERIFY.json | 25 +- research/rational-unitary-fft/frontier.py | 207 + research/rational-unitary-fft/full.py | 243 + research/rational-unitary-fft/generator.py | 5 +- research/rational-unitary-fft/verify.py | 6 +- research/rational-unitary-fft/verify_full.py | 274 + 11 files changed, 12909 insertions(+), 31 deletions(-) create mode 100644 research/rational-unitary-fft/FULL-PROOF.md create mode 100644 research/rational-unitary-fft/FULL-RESULT.json create mode 100644 research/rational-unitary-fft/frontier.py create mode 100644 research/rational-unitary-fft/full.py create mode 100644 research/rational-unitary-fft/verify_full.py diff --git a/research/rational-unitary-fft/FULL-PROOF.md b/research/rational-unitary-fft/FULL-PROOF.md new file mode 100644 index 0000000000..e650f46863 --- /dev/null +++ b/research/rational-unitary-fft/FULL-PROOF.md @@ -0,0 +1,374 @@ +# Full Toeplitz recursion with a polynomial conditioning envelope + +Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance. +Apache-2.0. + +## 1. Domain, result and predecessor scope + +Let \(N=2^d\), and let \(T\in\mathbb Q(i)^{N\times N}\) be Toeplitz. +Write \(T_b=T[0:b,0:b]\). I assume +\[ +\Lambda,Q\ge1,\qquad \|T\|_2\le\Lambda,\qquad +\|T_b^{-1}\|_2\le Q\quad(1\le b\le N). +\tag{1} +\] +The full inverse bound alone would not suffice: the Toeplitz matrix +\(\left[\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right]\) is invertible +but has a singular first section. I include it as an admission control. +I allow indefinite and non-Hermitian matrices satisfying (1). + +I execute the retained `full_plan` on exactly \(N\) logical complex wires, +using the completed generator circuit and the same stored gates in reverse +for inversion. For input norm at most \(R\ge1\), incoming grid +\(F_0\in\mathbb Z_{\ge0}\) and target \(2^{-t}\), \(t\in\mathbb Z_{\ge1}\), +I prove forward and inverse error at most \(2^{-t}\) +with online widths +\[ +O(F_0+t+\log N+\log\Lambda+\log Q+\log R). +\tag{2} +\] +The circuit has \(O(\log^4N)\) layers and \(O(N\log^4N)\) gates. +I give explicit constants below. + +Vlado/PR1673 supplied the recursive compiler and displacement identities. +The retained `GAUGEFREE-INTERIORS.md`, Theorem 4.1, supplies a +conditioning-dependent guard of +\(O(\log N\log(2+\Lambda/\mu))\) for its accretive domain. +I use the scalar unit-LDU factors of legal completed-call sets to obtain +a joint polynomial envelope in dimension and conditioning. This +sharpens the sufficient guard's conditioning dependence under (1). +I leave the selected multiplication exponent unchanged. + +ToikY's [PR4209](https://github.com/CrocSwap/integer-mult-bounds/pull/4209) +at `ccc33bfde8baadef7979fac81e28c0a6f6f278c4` supplies sharper local +Schur closure under a row/column near-identity majorant. I retain that +distinct credit. I do not infer its hypothesis from (1) or use its +sharper constants in the present allocation. + +### Executable admission + +`admission()` computes exact inverses of the same root's leading sections. +It returns \(\lambda_c,Q_c\), each at least one, using the maximum row +and column sums of \(|\Re z|+|\Im z|\). These bound the corresponding +operator norms and also support the executable component-envelope guards. +The verifier binds them to the unchanged root passed to `Full.plan()`. +That lower-level factory does not repeat the complete admission. + +If (1) is supplied as operator-norm information, then +\[ +\lambda_c\le\sqrt{2N}\Lambda,\qquad +Q_c\le\sqrt{2N}Q. +\tag{3} +\] +I use the executable envelopes in its explicit constants. The inflation +in (3) preserves (2). The written operator lemma below uses any valid +\(\Lambda,Q\); passing uninflated operator bounds directly to the +component-guarded factory need not satisfy its stronger guards. + +Exact admission and Schur preparation are offline rational computations. +I do not equate their denominators with online data words or claim a +complete physical preparation/delivery tariff. + +## 2. Contiguous descendants, shears and rank + +The node on logical interval \([a,b)\) has reference matrix +\[ +M_{a,b}=S(a,b)=T_b/T_a, +\] +with the empty prefix convention at \(a=0\). Its children are \(S(a,c)\) +and \(S(c,b)\), where \(c=(a+b)/2\), by the Schur quotient identity. +Consequently +\[ +\|M_{a,b}\|\le U:=\Lambda+\Lambda^2Q,\qquad +\|M_{a,b}^{-1}\|\le Q. +\tag{4} +\] +The inverse is a principal block of \(T_b^{-1}\); the direct bound follows +from the Schur formula and restrictions of the original \(T\). + +At a split, write \(M=\left[\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right]\), +\(K=A^{-1}B\), \(L=CA^{-1}\). Let \(R_a\) select the coordinates +\([a,c)\) from \([0,c)\). The block inverse identities give +\[ +K=R_aT_c^{-1}T[0:c,c:b],\qquad +L=T[c:b,0:c]T_c^{-1}R_a^T. +\] +Thus, with \(\chi=\Lambda Q\), +\[ +\|K\|,\|L\|\le\chi. +\tag{5} +\] +The component-envelope versions follow under the executable admission. +The source computes \(L\) as `solve(A.T, C.T).T`. I preserve ordinary +transpose in this complex identity. + +The retained `SHALLOW-TOEPLITZ.md`, Sections 4–6, gives commutator rank +at most \(2+\text{depth}\) for the Schur nodes. Its rectangular identities +bound the shear commutator ranks by \(2\,\text{depth}+5\) and +\(2\,\text{depth}+7\). Conversion to +\(E_X=X-ZXZ^T\) adds at most one rank. I use the uniform allowance +\[ +r_*=2d+8. +\tag{6} +\] +The exact rational producer retains that rank; I do not use the numerical +source's floating SVD threshold as a universal rank argument. +The executable also checks exact generator rank against separate +elimination in its finite fixtures. + +## 3. Uniform scalar-LDU bounds + +Write \(M_J=\ell_J\delta_Ju_J\) for the scalar unit-LDU factors of a +contiguous descendant. All its leading sections are further \(S(a,c)\), +so (4) also applies during scalar elimination. +For \(A_k=M_J[0:k,0:k]\), the last row of +\(A_k^{-1}M_J[0:k,k:]\) is the corresponding unit-upper row tail. +The last column of \(M_J[k:,0:k]A_k^{-1}\) is the unit-lower column tail. +Each therefore has norm at most +\[ +UQ=\chi+\chi^2\le2\chi^2. +\] +The Frobenius estimate therefore permits +\[ +\|\ell_J\|,\|u_J\|\le W:=3N\chi^2. +\] +The pivot diagonal has norm at most \(U\) and inverse norm at most \(Q\). +From +\[ +u_J^{-1}=M_J^{-1}\ell_J\delta_J,\qquad +\ell_J^{-1}=\delta_Ju_JM_J^{-1} +\] +I obtain +\[ +\|\ell_J^{-1}\|,\|u_J^{-1}\|\le V:=6N\chi^4, +\tag{7} +\] +using \(QU\le2\chi^2\). A diagonal containing some completed pivots +and otherwise ones has direct and inverse norms at most \(2\chi^2\). + +## 4. Exact downsets and the upper frontier + +At an internal node \(v\), denote the upper and lower atomic calls by +\[ +R_v=\begin{pmatrix}I&K_v\\0&I\end{pmatrix},\qquad +B_v=\begin{pmatrix}I&0\\L_v&I\end{pmatrix}. +\] +The source executes \(R_v\), its two child programs in parallel, then +\(B_v\). A leaf executes its scalar pivot. +The atomic partial order places \(R_v\) before both children and both +children before \(B_v\). Incomparable calls act on disjoint intervals +and commute. I count identity shears as calls even when they emit zero layers. + +For a downset \(I\), let \(Q(I)\) be its product in any legal linearization. +At a node, the restricted downset is empty; or it contains \(R_v\) and +child downsets \(I_0,I_1\) but not \(B_v\); or it contains the whole node. +For an empty node set all three factors to identity. At a completed leaf +set them to \(I,d_i,I\). Otherwise define +\[ +\begin{aligned} +\mathcal L(I)&=B_v^\epsilon + \operatorname{diag}(\mathcal L(I_0),\mathcal L(I_1)),\\ +\mathcal D(I)&=\operatorname{diag}(\mathcal D(I_0),\mathcal D(I_1)),\\ +\mathcal U(I)&=\operatorname{diag}(\mathcal U(I_0),\mathcal U(I_1))R_v, +\end{aligned} +\tag{8} +\] +where \(\epsilon=1\) exactly when the lower call has completed. +Then both children are complete. Substituting the child identities gives +\[ +\boxed{Q(I)=\mathcal L(I)\mathcal D(I)\mathcal U(I).} +\tag{9} +\] +This induction never commutes overlapping factors. + +Uniqueness of scalar unit-LDU gives +\[ +\ell_v=B_v\operatorname{diag}(\ell_{v0},\ell_{v1}),\quad +\delta_v=\operatorname{diag}(\delta_{v0},\delta_{v1}),\quad +u_v=\operatorname{diag}(u_{v0},u_{v1})R_v. +\] +Hence \(\mathcal L(I)\) is the direct sum of full lower factors on the +maximal completed subtrees, with identity elsewhere. The partial diagonal +contains completed pivots and otherwise ones. + +For the upper frontier, stop at a node whose upper call has not completed; +otherwise descend to its two children. Retain singleton leaves. +The resulting intervals partition the coordinates. With +\(U_{\mathcal F}=\bigoplus_{v\in\mathcal F}u_v\), induction gives +\[ +u_{\rm root}=U_{\mathcal F}\mathcal U(I),\qquad +\mathcal U(I)=U_{\mathcal F}^{-1}u_{\rm root}. +\tag{10} +\] +At an unvisited node the identity is \(u_v=u_vI\); at a visited node, +substitute the two child identities and multiply by \(R_v\). + +Equations (7)–(10) bound the lower factor and its inverse by \(V\), +the partial diagonal and its inverse by \(2\chi^2\), and the upper +factor and its inverse by \(VW\). Thus +\[ +\boxed{\|Q(I)\|,\|Q(I)^{-1}\| +\le216N^3\chi^{12}<\mathcal B:=256N^3\chi^{12}.} +\tag{11} +\] +I use a fixed number of polynomial factors, rather than multiplying +conditioning factors through the recursion depth. + +## 5. Approximate atomic calls and physical cuts + +There are \(J=3N-2\) atomic calls: two shears per internal node and one +scalar per leaf. Approximate each by operator error at most \(\eta\). +For any legal linearization, set +\[ +G_j=Q_j^{-1}(\widehat A_j-A_j)Q_{j-1},\qquad +\theta=J\mathcal B^2\eta. +\] +The downset bound (11) applies to its prefixes. +If \(\theta\le1/4\), the product of \(I+G_j\), or its inverse, has norm +below two and distance from identity at most \(2\theta\). +Using \(\Lambda,Q\le\mathcal B\), I obtain +\[ +\|\widehat T-T\|,\|\widehat T^{-1}-T^{-1}\| +\le e:=2J\mathcal B^3\eta. +\tag{12} +\] +Completed approximate prefixes and inverse prefixes have norm at most +\(2\mathcal B\). Remaining suffixes and inverse suffixes have norm at most +\(2L_0\mathcal B\), where \(L_0=\max(1,\Lambda,Q)\). +I apply the exact LDU identities to the ideal calls, not to the +approximate endpoints. + +At a physical gate cut, completed calls form a downset \(I\). +Started but unfinished calls form a pairwise incomparable set \(A\): +the source finishes an upper call before starting its descendants and +finishes those descendants before its lower call. Active supports are +therefore disjoint, even when sibling ranks and lengths differ. + +Write each active stored endpoint as \(\widehat A_a=T_aF_a\), with executed +prefix \(F_a\) and remaining suffix \(T_a\). A completed call executed +after an active call began must act on a disjoint subtree, so I may commute +only that disjoint work. I obtain the exact identities +\[ +C_{\rm prefix}=(\bigoplus_{a\in A}F_a\oplus I)\widehat Q(I),\qquad +C_{\rm suffix}=\widehat Q_{\rm remaining} + (\bigoplus_{a\in A}T_a\oplus I). +\tag{13} +\] +The set \(I\cup A\) is also a downset, so I may order \(I\), then \(A\), +then the remaining calls in the suffix argument. +A direct sum contributes the largest fragment norm. I do not require +aligned child macro boundaries. A zero-layer identity completes at its +position in the source schedule and contributes no active fragment. + +## 6. Child, scalar and data bounds + +The component envelope used by a shear's exact producer satisfies +\(B_E\le4N\chi\): its displacement norm is at most \(2\chi\), and a +component row/column sum is at most \(\sqrt{2m}\) times that norm. +Substituting \(m\le N\), \(r\le r_*\), \(B_E\le4N\chi\) in +[GENERATOR-PROOF.md](GENERATOR-PROOF.md), put +\[ +\alpha_*=8r_*N^2\chi,\quad K_{G,*}=2N+64N^2\chi, +\] +\[ +H_*=\max(3200N^3,2+2K_{G,*})(10+4\alpha_*+2K_{G,*}). +\] +The generator's radius-one allocation supplies an unrounded operator +defect. I discard its local data grid and execute the flattened full +program on one global grid. + +A scalar pivot has \(1/Q\le|a|\le U\). I round it at \(s\) bits, +with \(\eta=2^{-s}\le1/(2Q)\), and invert the same stored value. +Then \(|\widehat a|\ge1/(2Q)\), its direct norm is at most \(U+1\), +and its inverse norm is at most \(2Q\). +Consequently +\[ +\mathcal H=\max(H_*,U+1,2Q),\qquad +K_{\rm phys}=2L_0\mathcal B\mathcal H +\tag{14} +\] +bounds physical prefixes, suffixes and their inverses. +An arbitrary segment needs the conservative bound \(K_{\rm phys}^2\). + +The unchanged source schedule permits +\[ +D_*=1+2dr_*(8d^2+32d+18),\qquad +G_*=N[1+dr_*(4d^2+24d+22)]. +\tag{15} +\] +I sum two generator calls per level and the \(N\) scalar leaves; parallel +siblings contribute their maximum depth. +Choose +\[ +s=t+\lceil\log_2(4J\mathcal B^3(R+1))\rceil,\quad +F=\max(F_0,t+\lceil\log_2(2D_*K_{\rm phys}N)\rceil). +\tag{16} +\] +These choices ensure the perturbation and scalar margins above. +One layer's rounding error is at most \(N2^{-F}\). Thus either direction +has error at most +\[ +eR+D_*K_{\rm phys}N2^{-F}\le2^{-t}, +\tag{17} +\] +and intermediate amplitude at most +\[ +A=K_{\rm phys}R+D_*K_{\rm phys}^2N2^{-F}. +\tag{18} +\] +The inverse direction uses the exact stored inverse program on its own +input satisfying the same \(R,F_0\) contract. I claim no exact rounded +forward/inverse round trip. + +If \(\widehat a=(a+ib)/2^s\) is a scalar leaf, its reciprocal +\(2^s(a-ib)/(a^2+b^2)\) has component height at most +\(2s+2\lceil\log_2(U+2)\rceil+8\). +I take the maximum of this bound and the generator children's declared +coefficient bounds, all fixed before online execution. +With \(I=\max(1,\lceil\log_2\max(1,A)\rceil)+2\), signed grid words fit +\(F+I\) bits and constant-size gate temporaries fit +\[ +2F+8C_{\rm bits}+2I+32. +\] +The constants in (6), (11), (14) and (15) are polynomial in \(N,\Lambda,Q\). +The child precision formulas and (16) therefore give (2), including the +executable envelope inflation (3). + +## 7. Source binding, finite evidence and limits + +I authenticate and execute the unmodified `full_plan`, `parallel` and +`inverse` ASTs. I supply exact rational matrix slicing, ordinary transpose, +solve, subtraction and multiplication; wrap recursive calls only to +record their exact reference matrices; and bind `lowdisp_shear` to the +completed generator program. I preserve source assertions in its +dependencies under optimized Python. + +The verifier checks direct full matrices and separate inverse references, +exact stored inversion, leading-section bounds, scalar-LDU identities, +legal downsets and upper-frontier identities. It enumerates all downsets +through size four and checks source-timed downsets on larger fixtures. +Selected physical cuts compare both identities in (13), including +simultaneous nonempty active fragments. Timing checks explicitly cover +unequal sibling lengths and zero-layer calls. + +I test indefinite and half-shift Gaussian matrices, complex inputs, +small and large scalar pivots, reversed wire labels, a nonzero 512-bit +incoming fixture and a size-16 identity with zero-layer shear calls. +The finite Frobenius inequalities include the dimension factor \(N\); +they corroborate consequences of the written operator proof rather than +independently establish its sharper bounds. +Exact dense matrices remain checker oracles outside the online +representation claim. Observed heights describe completed rational +components rather than exhaustive `Fraction` internals. + +I preserve the earlier FFT, triangular and generator receipts and the +fixed-head scope of James Reed's finite budget/source-seal audit at +`fdcdc269`. That audit excludes gate replay and universal-proof acceptance. +I credit Vlado Vojdanovski/PR1673 and the retained gauge-free source lineage +for the original recursive, displacement and conditioning arguments; +Chafik Boukhalfa/PR628 for certified phases; Cinna Davis for the later caller; +ToikY for the distinct same-rho Schur theorem; and classical LU/Schur methods. +Internal agents belong to the author process. + +Complete Gaussian assembly, physical coefficient setup/delivery, fixed-tape +costs and independent mathematical acceptance remain open. diff --git a/research/rational-unitary-fft/FULL-RESULT.json b/research/rational-unitary-fft/FULL-RESULT.json new file mode 100644 index 0000000000..bb362b60bd --- /dev/null +++ b/research/rational-unitary-fft/FULL-RESULT.json @@ -0,0 +1,11717 @@ +{ + "schema_version": 1, + "scope": "actual full Toeplitz recursion and stored inverse", + "sources": [ + { + "path": "research/unconditional-batched-crt/GAUSSIAN.md", + "sha256": 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"1/5070602400912917605986812821504", + "theta": "23/2305843009213693952", + "data_bits": 103, + "coefficient_error": "23/1099511627776", + "rounding_error": "71861946525/288230376151711744", + "total_error": "182994225309/288230376151711744", + "amplitude_bound": "10885246589300138535689822025/131072", + "integer_bits": 79, + "scalar_height_bound": 216, + "coefficient_height_bound": 708, + "temporary_bits": 6060 + }, + "max_coefficient_height": 1, + "forward_frobenius_squared_upper": "0", + "inverse_frobenius_squared_upper": "0", + "phase_oracles": [ + { + "phase_bits": 136, + "oracle_bits": 336, + "certificates": [] + }, + { + "phase_bits": 149, + "oracle_bits": 362, + "certificates": [] + }, + { + "phase_bits": 161, + "oracle_bits": 386, + "certificates": [] + }, + { + "phase_bits": 171, + "oracle_bits": 406, + "certificates": [] + } + ], + "states": [ + { + "direction": "forward", + "input_sha256": "4a809fba078380280ec6ecb27b5270f59f70de6e9fad2616dd7b6b2da66b1785", + "output_sha256": "4a809fba078380280ec6ecb27b5270f59f70de6e9fad2616dd7b6b2da66b1785", + "coefficient_squared_upper": "0", + "rounding_squared_upper": "0", + "total_squared_upper": "0", + "statistics": { + "max_observed_height": 4, + "max_grid_numerator_bits": 103, + "max_reduced_fractional_bits": 3, + "rounded_outputs": 16 + } + }, + { + "direction": "forward", + "input_sha256": "a16fd8393b3ad05eec45929d2c2139d94269bbc5540f17c7403af38af161db86", + "output_sha256": "a16fd8393b3ad05eec45929d2c2139d94269bbc5540f17c7403af38af161db86", + "coefficient_squared_upper": "0", + "rounding_squared_upper": "0", + "total_squared_upper": "0", + "statistics": { + "max_observed_height": 11, + "max_grid_numerator_bits": 114, + "max_reduced_fractional_bits": 0, + "rounded_outputs": 16 + } + }, + { + "direction": "forward", + "input_sha256": "8a960c740c5be521ec0308172fa83f0e8a74303df69902cb310385d42f78c0f0", + "output_sha256": "8a960c740c5be521ec0308172fa83f0e8a74303df69902cb310385d42f78c0f0", + "coefficient_squared_upper": "0", + "rounding_squared_upper": "0", + "total_squared_upper": "0", + "statistics": { + "max_observed_height": 103, + "max_grid_numerator_bits": 2, + "max_reduced_fractional_bits": 102, + "rounded_outputs": 16 + } + }, + { + "direction": "inverse", + "input_sha256": "4a809fba078380280ec6ecb27b5270f59f70de6e9fad2616dd7b6b2da66b1785", + "output_sha256": "4a809fba078380280ec6ecb27b5270f59f70de6e9fad2616dd7b6b2da66b1785", + "coefficient_squared_upper": "0", + "rounding_squared_upper": "0", + "total_squared_upper": "0", + "statistics": { + "max_observed_height": 4, + "max_grid_numerator_bits": 103, + "max_reduced_fractional_bits": 3, + "rounded_outputs": 16 + } + }, + { + "direction": "inverse", + "input_sha256": "a16fd8393b3ad05eec45929d2c2139d94269bbc5540f17c7403af38af161db86", + "output_sha256": "a16fd8393b3ad05eec45929d2c2139d94269bbc5540f17c7403af38af161db86", + "coefficient_squared_upper": "0", + "rounding_squared_upper": "0", + "total_squared_upper": "0", + "statistics": { + "max_observed_height": 11, + "max_grid_numerator_bits": 114, + "max_reduced_fractional_bits": 0, + "rounded_outputs": 16 + } + }, + { + "direction": "inverse", + "input_sha256": "8a960c740c5be521ec0308172fa83f0e8a74303df69902cb310385d42f78c0f0", + "output_sha256": "8a960c740c5be521ec0308172fa83f0e8a74303df69902cb310385d42f78c0f0", + "coefficient_squared_upper": "0", + "rounding_squared_upper": "0", + "total_squared_upper": "0", + "statistics": { + "max_observed_height": 103, + "max_grid_numerator_bits": 2, + "max_reduced_fractional_bits": 102, + "rounded_outputs": 16 + } + } + ] + } + ], + "controls": [ + "invertible-full-matrix-with-singular-leading-section", + "non-Toeplitz-root", + "nondyadic-full-size", + "floating-full-input", + "omitted-root-upper-call", + "full-inverse-layer-order", + "one-sided-conjugation-in-lower-solve", + "serialized-full-siblings", + "delayed-zero-layer-identity-call", + "full-scalar-adjoint-instead-of-reciprocal", + "underfunded-full-data-grid", + "rounded-zero-full-pivot", + "discarded-full-incoming-grid", + "missing-upper-frontier-inverse", + "typed-full-receipt" + ], + "full_toeplitz_gate_execution": true, + "physical_tape_execution": false, + "complete_gaussian_execution": false, + "universal_proof_kernel": false +} diff --git a/research/rational-unitary-fft/README.md b/research/rational-unitary-fft/README.md index 5684ea59f3..d3f09fc837 100644 --- a/research/rational-unitary-fft/README.md +++ b/research/rational-unitary-fft/README.md @@ -1,4 +1,20 @@ -# Certified rational bounded-generator circuits +# Certified full Toeplitz circuits with polynomial conditioning bounds + +I execute the retained full Toeplitz recursion and its stored inverse, +using the rational generator, triangular and FFT circuits below. +For dyadic size \(N\), I assume \(\|T\|\le\Lambda\) and +\(\|T_b^{-1}\|\le Q\) for every leading section. +I prove a completed-call prefix/inverse bound +\(256N^3(\Lambda Q)^{12}\) through scalar unit-LDU factors and legal +completed-call sets. I handle unequal parallel branches with disjoint +active fragments and obtain sufficient online widths +\[ +O(F_0+t+\log N+\log\Lambda+\log Q+\log R). +\] +I compute stronger rational component envelopes for executable admission; +their inflation preserves this width bound. +[FULL-PROOF.md](FULL-PROOF.md) gives the domain, frontier induction, +explicit allocation and comparison with the earlier guard. I connect PR4198's exact rational generators to the retained compensated gate program, replacing its exact matrix callbacks with certified FFT, @@ -7,8 +23,8 @@ one- and two-wire schedule on both dirty banks and prove a common coefficient/data error and storage budget. I preserve parallel even/odd branches, reversed caller wires and the source's compensation order. -For dyadic \(n=2^d\), the target is \((x,y)\mapsto(x,y+Tx)\), with rational -Toeplitz \(T\). I use rational butterflies +For the Toeplitz-shear child of dyadic size \(n=2^d\), the target is +\((x,y)\mapsto(x,y+Tx)\), with rational Toeplitz \(T\). I use rational butterflies \[ \begin{pmatrix}a&bw\\b&-aw\end{pmatrix}, \qquad a^2+b^2=|w|^2=1, @@ -17,7 +33,7 @@ and adjoints of the same stored gates. I certify their approximation to the ideal FFT, prepare the **unnormalized** DFT multipliers, round those coefficients before online use, and round data after each gate. -The exact rational circuit restores the source. Fixed-grid execution +The exact rational shear circuit restores its source bank. Fixed-grid execution generally has nonzero source-restoration error, which I include in the budget rather than treating exact coefficient unitarity as exact data recovery. @@ -86,8 +102,8 @@ python3.12 -B scripts/research_package.py \ --package research/rational-unitary-fft ``` -I declare six normal/optimized commands, with limits of 120, 120, 180, -180, 300 and 300 seconds, totaling 1,200 seconds. +I declare eight normal/optimized commands, with limits of 60, 60, 90, +90, 150, 150, 300 and 300 seconds, totaling 1,200 seconds. I authenticate the source before parsing its schedule or executing its interval oracle. The adapter preserves bit reversal, gate ownership, disjoint layers, shear orientation and chronological calls. It replaces @@ -145,6 +161,25 @@ I freeze the executed producer with its original [provenance](vendor/STEIN-PROVENANCE.json) and pin its local archive separately from the original sibling PR commit. +The full-recursion replay adds 11 cases, 3,659 gates and 68 rounded +forward/inverse states. Nontrivial cases reach size eight; a size-16 +identity exercises zero-layer shear calls. I include indefinite and +half-shift Gaussian matrices, complex scalar pivots, a small pivot, +reversed wires, a nonzero 512-bit incoming fixture and an identity left +subtree paired with a longer right subtree. + +I check 151 ideal completed-call sets and 12 selected physical cuts. +I enumerate all legal sets through size four and use actual atom-boundary +sets for larger cases. Ownership/precedence checks cover every layer cut; +selected operator checks compare both active-fragment factorizations, +including two simultaneously unfinished, nonempty fragments. +I check complete matrix and separate inverse references, stored-circuit +inversion, global rounding errors and signed word widths. +Fifteen controls cover leading-section admission, source order, ordinary +transpose, parallel timing, zero-layer calls, scalar reciprocals, incoming +precision and the upper-frontier inverse. +[FULL-RESULT.json](FULL-RESULT.json) records the complete receipt. + ## Overlap and attribution I inspected 100 recent PR records and 35 hub comments, the retained @@ -169,6 +204,12 @@ Before the generator extension I refreshed unchanged main, inspected searches. The card-wide helper searched 5,236 local records but had no usable remote cache. I found no competing complete generator-gate claim in that coverage. PR4208 addresses CRT stencil production. +Before the full extension I refreshed unchanged main and inspected +25 PR records and 8/733 hub comments. ToikY's PR4209 supplies a distinct +same-rho Schur-closure theorem under row/column majorant admission, +excluding full gate execution and global allocation. I credit that +sharper local result without transferring its hypothesis to the present +leading-section domain. I credit Chafik Boukhalfa/PR628 for exact rational unit phases and the certified interval machinery; Vlado Vojdanovski/PR1673 for the FFT/shear @@ -193,6 +234,13 @@ its [hosted replay](https://github.com/CrocSwap/integer-mult-bounds/actions/runs and receipt `d430fced712daa2e620b53a49dd09f0ba10ddddb4a4ff6b79d2abbbfd3afc1e6`. The generator extension retains both earlier scientific receipts and adds a third record and two commands. +I preserve the completed generator head +`fdcdc2690910392f7b42dbae4155b7d7b5790abd`, its +[hosted replay](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39454327335) +and receipt `f74e0efc459e9c3d44d8ca8b3a33dc0fccc31731d0829419c3d515fe7f0f1e02`. +James Reed's [finite audit](https://github.com/CrocSwap/integer-mult-bounds/pull/4205#issuecomment-6295524563) +at that fixed head checks budgets and source seals; he excludes circuit +re-execution and universal-proof acceptance. I retain that scoped credit. Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance, including source-analysis agents. I count those agents within the author @@ -202,8 +250,8 @@ Apache-2.0. ## Open next questions -In [TASK.json](TASK.json) I specify the next steps and blockers: compose -the completed generator program inside the actual full Toeplitz recursion, -then bind its coefficients and data widths to the physical Gaussian caller. -Full setup/delivery, fixed-tape lowering and independent acceptance of -the complete multiplication theorem remain open. +In [TASK.json](TASK.json) I specify the next steps and blockers: bind the +full circuit's conditioning and precision to the actual Gaussian block +producer, then connect its coefficients and data widths to physical +delivery. Complete setup, fixed-tape lowering and independent acceptance +of the complete multiplication theorem remain open. diff --git a/research/rational-unitary-fft/SOURCES.json b/research/rational-unitary-fft/SOURCES.json index 3ffbb6b5ce..23ccb215f5 100644 --- a/research/rational-unitary-fft/SOURCES.json +++ b/research/rational-unitary-fft/SOURCES.json @@ -14,6 +14,7 @@ {"path":"research/physical-ring-product/README.md","usage":"input","git_blob":"823f72e1b26aa8c71c5bd7141c5f93b2d59847fe","sha256":"dc4769ad9d7f3179659e28dc02bacc2822931a9bb8daaf9ad67cd935c1bb7094"}, {"path":"research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py","usage":"input","git_blob":"4f0180c88f6ea25b85814883ef7cf2ad41e67110","sha256":"5ff7f25c55b370a1d0cda6c84ab9cdf2c7b0de3266d3f2e7a43280a6491074ab"}, {"path":"research/rational-unitary-fft/vendor/stein_rational.py","usage":"input","pin":"head:0e181260108dd940a594fad6b912588bdee62522:research/rational-unitary-fft/vendor/stein_rational.py","git_blob":"71b240a460a185ee2fe926cd189b56aeac6f997f","sha256":"3065fc5d6d22ec44de533b08fb0c639603449456383668e561dc46629e641502"}, - {"path":"research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json","usage":"input","pin":"head:0e181260108dd940a594fad6b912588bdee62522:research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json","git_blob":"c602fb5bb332dbe098950b14d55da05ce2cf4312","sha256":"e7f0e8dc517387e302b1e8bf5dff7e12b29da8aa91abc6ea597f33c1ac08ba62"} + {"path":"research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json","usage":"input","pin":"head:0e181260108dd940a594fad6b912588bdee62522:research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json","git_blob":"c602fb5bb332dbe098950b14d55da05ce2cf4312","sha256":"e7f0e8dc517387e302b1e8bf5dff7e12b29da8aa91abc6ea597f33c1ac08ba62"}, + {"path":"research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md","usage":"input","git_blob":"20585e44fa4e49b13b5d8495bbf6ed254459a362","sha256":"d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116"} ] } diff --git a/research/rational-unitary-fft/TASK.json b/research/rational-unitary-fft/TASK.json index 107d8aa2e4..e3600a8e5e 100644 --- a/research/rational-unitary-fft/TASK.json +++ b/research/rational-unitary-fft/TASK.json @@ -38,21 +38,21 @@ ] }, "scope_status": "refined", - "actual_scope": "Execute the retained compensated bounded-generator program with exact rational generators and certified FFT/shear/parity children, paired compensation multipliers, additive endpoint defects, one global data grid and explicit intermediate-width bounds.", - "scope_change_reason": "PR4198 used exact matrix callbacks for its children. The completed child contracts now support the literal compensated generator schedule. I preserve both earlier revisions and receipts, pin the exact producer bytes, and add one complete generator-level error/width proof and gate replay.", - "bottleneck_link": "The full Toeplitz compiler needs the actual compensated generator program, including cleanup and rounded dirty-source restoration, before its full numerical composition can be admitted.", + "actual_scope": "Execute the retained full Toeplitz recursion and stored inverse with exact rational reference coefficients, certified generator/FFT/triangle gates, a polynomial completed-downset/frontier bound, one global grid and explicit online widths.", + "scope_change_reason": "The completed generator contract supports the full recursive caller. I preserve the three earlier receipts and add a global LDU/downset proof that avoids multiplying conditioning losses through recursion depth, including unequal sibling schedules and active fragments.", + "bottleneck_link": "The Gaussian caller needs full circuit execution and a quantified conditioning/rounding contract before physical delivery and the complete numerical composition can be admitted.", "budget_minutes": 120, "stop_rule": "Stop at an invalid phase enclosure, nonunitary adapted gate, missing multiplier/twist error, understated word width or source-schedule mismatch. Do not equate coefficient-level unitarity with exact rounded restoration or physical execution.", "outcome": "completed", - "evidence": "PROOF.md, TRIANGLE-PROOF.md and GENERATOR-PROOF.md give the phase, recursive, compensated coefficient/error and width arguments. I preserve the 12-case FFT and 10-case triangular receipts. GENERATOR-RESULT.json adds 57 matrices, 71 generators, 5614 gates and 172 rounded states, with all 50 predecessor source Schur matrices, separate exact rank and full endpoint references, paired half-tie compensation and selected checkpoint checks. Scientific acceptance remains separate.", + "evidence": "I preserve the completed FFT, triangle and generator proofs/receipts. FULL-PROOF.md gives the leading-section, polynomial downset/frontier, perturbation, scalar and width arguments. FULL-RESULT.json records 11 cases, 3659 gates, 68 rounded states, 151 completed-call sets and 12 physical cuts. Schedule checks cover every cut; operator factorizations cover the selected cuts. Nontrivial fixtures reach size8; size16 is an identity/zero-call case. Scientific acceptance remains separate.", "author": "Jose Manuel (jmbarrancoml)", "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, source analysis, implementation and writing, including source-analysis agents. No independent human or formal-kernel acceptance.", "open_questions": [ { - "question": "Can the actual full Toeplitz recursion consume this rounded compensated-generator program under one precision allocation?", - "why_it_matters": "The source composes Schur factorization and recursive solves beyond each individual dirty shear. The generator contract does not finish that full matrix program.", - "first_step": "Bind the literal full_plan source, bound its leading-section/Schur coefficients and global prefixes, then execute the emitted generator and inverse plans on one grid.", - "blocked_by": "A source-bound full-recursion error/width proof, its conditioning hypotheses and complete emitted-gate replay." + "question": "Can the actual Gaussian block producer supply the full circuit's leading-section conditioning and reference-error contract?", + "why_it_matters": "The full circuit theorem assumes a rational Toeplitz root and bounded leading inverses. The physical Gaussian caller must supply the same rounded/truncated root and pay its approximation error.", + "first_step": "Connect PR4193's conditioning estimates and the retained Gaussian coefficient producer to exact rational roots, then allocate truncation/coefficient/circuit errors in the actual block call.", + "blocked_by": "A source-bound Gaussian-to-full-circuit interface and complete preparation/error ledger." }, { "question": "Can the coefficient preparation and canonical data widths be joined to the physical Gaussian caller?", diff --git a/research/rational-unitary-fft/VERIFY.json b/research/rational-unitary-fft/VERIFY.json index 2b598479d5..bb1d4242be 100644 --- a/research/rational-unitary-fft/VERIFY.json +++ b/research/rational-unitary-fft/VERIFY.json @@ -14,38 +14,49 @@ "research/physical-ring-product/README.md", "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py", "research/rational-unitary-fft/vendor/stein_rational.py", - "research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json" + "research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json", + "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md" ], "commands": [ { "argv": ["{python}", "-I", "-B", "{package}/verify.py"], - "timeout_seconds": 120, + "timeout_seconds": 60, "checks": "Authenticate and adapt the actual FFT/shear schedule, certify rational phases and unitary gates, prepare charged spectral coefficients, execute fixed-grid dirty data, compare independent Fourier/Toeplitz references and reject schedule/precision controls." }, { "argv": ["{python}", "-I", "-O", "-B", "{package}/verify.py"], - "timeout_seconds": 120, + "timeout_seconds": 60, "checks": "Repeat the complete gate/rounding certificate under optimized Python, keeping authenticated source assertions and exact typed receipt comparison." }, { "argv": ["{python}", "-I", "-B", "{package}/verify_triangle.py"], - "timeout_seconds": 180, + "timeout_seconds": 90, "checks": "Execute the retained parity recursion and exact stored inverse on one grid; check macro identities, independent exact endpoint inverses, global error/width budgets, reversed indices, nonunitary leaves and adversarial controls." }, { "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_triangle.py"], - "timeout_seconds": 180, + "timeout_seconds": 90, "checks": "Repeat the complete recursive forward/inverse certificate under optimized Python with authenticated source assertions and complete typed receipt comparison." }, { "argv": ["{python}", "-I", "-B", "{package}/verify_generator.py"], - "timeout_seconds": 300, + "timeout_seconds": 150, "checks": "Execute the exact rational producer and retained compensated generator gate program on actual Schur fixtures; check paired spectral coefficients, additive endpoint defects, dirty-source rounding, intermediate widths and cancellation controls." }, { "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_generator.py"], - "timeout_seconds": 300, + "timeout_seconds": 150, "checks": "Repeat the complete generator program with source assertions enabled under optimized Python and compare its full typed receipt." + }, + { + "argv": ["{python}", "-I", "-B", "{package}/verify_full.py"], + "timeout_seconds": 300, + "checks": "Execute the retained full Toeplitz recursion and stored inverse, check leading-section admission, legal downsets/frontiers and active fragments, and compare exact endpoints plus rounded global error/width budgets." + }, + { + "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_full.py"], + "timeout_seconds": 300, + "checks": "Repeat full recursive execution and discriminating conditioning/schedule controls under optimized Python with authenticated source assertions and a complete typed receipt." } ] } diff --git a/research/rational-unitary-fft/frontier.py b/research/rational-unitary-fft/frontier.py new file mode 100644 index 0000000000..d37cae2aaf --- /dev/null +++ b/research/rational-unitary-fft/frontier.py @@ -0,0 +1,207 @@ +"""Exact completed-downset and active-fragment checks for the full schedule. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +from fractions import Fraction as Q + +from exact import C, Mat, eye, multiply, need, upper, vector_hash +from verify import operator +from verify_triangle import eliminate + + +def blockdiag(left, right): + a, b = len(left), len(right) + return Mat([[left[i][j] if i < a and j < a + else right[i - a][j - a] if i >= a and j >= a else C() + for j in range(a + b)] for i in range(a + b)]) + + +def shear(matrix, upper_bank=False): + m = len(matrix) + result = eye(2 * m) + for i in range(m): + for j in range(m): + result[i if upper_bank else m + i][m + j if upper_bank else j] = matrix[i][j] + return result + + +def embed(matrix, indices, n): + result = eye(n) + for i, row in enumerate(indices): + for j, column in enumerate(indices): + result[row][column] = matrix[i][j] + return result + + +def frobenius2(matrix): + return sum((x.norm2() for row in matrix for x in row), Q(0)) + + +class Frontier: + def __init__(self, builder, plan): + self.builder, self.plan, self.n = builder, plan, builder.bound["n"] + self.atoms = {a["id"]: a for a in plan.atoms} + self.subsets, self.factors, self.actual_atoms = {}, {}, {} + self._factor(0, self.n) + + def _factor(self, a, b): + k = b - a + ids = {key for key, atom in self.atoms.items() + if a <= atom["interval"][0] and atom["interval"][1] <= b} + self.subsets[a, b] = ids + if k == 1: + need(ids == {f"D:{a}"}, "one scalar leaf call") + factors = (eye(1), self.atoms[f"D:{a}"]["ideal"], eye(1)) + else: + c = (a + b) // 2 + left, right = self._factor(a, c), self._factor(c, b) + need({f"R:{a}:{b}", f"L:{a}:{b}"}.issubset(ids), "two full node shears") + lower = shear(self.atoms[f"L:{a}:{b}"]["ideal"]) + upper_bank = shear(self.atoms[f"R:{a}:{b}"]["ideal"], True) + factors = (multiply(lower, blockdiag(left[0], right[0])), + blockdiag(left[1], right[1]), + multiply(blockdiag(left[2], right[2]), upper_bank)) + need(multiply(multiply(factors[0], factors[1]), factors[2]) + == self.builder.nodes[a, b]["matrix"], "exact scalar LDU factorization") + chi = self.builder.bound["conditioning"] + w, v = 3 * self.n * chi ** 2, 6 * self.n * chi ** 4 + for factor in (factors[0], factors[2]): + need(frobenius2(factor) <= k * w ** 2 + and frobenius2(eliminate(factor)) <= k * v ** 2, + "scalar LDU norm consequences") + self.factors[a, b] = factors + return factors + + def endpoints(self, a, b): + if b - a == 1: + atom = self.atoms[f"D:{a}"] + return atom["start"], atom["stop"] + return self.atoms[f"R:{a}:{b}"]["start"], self.atoms[f"L:{a}:{b}"]["stop"] + + def schedule(self): + n = self.n + need(len(self.plan.atoms) == 3 * n - 2 and len(self.atoms) == 3 * n - 2, + "full atomic call inventory") + for (a, b), node in self.builder.nodes.items(): + if b - a == 1: + start, stop = self.endpoints(a, b) + need(stop == start + 1, "one full scalar layer") + continue + c = (a + b) // 2 + right, left = self.atoms[f"R:{a}:{b}"], self.atoms[f"L:{a}:{b}"] + lstart, lend = self.endpoints(a, c) + rstart, rend = self.endpoints(c, b) + need(lstart == rstart == right["stop"] and left["start"] == max(lend, rend), + "literal full parallel timing") + need(left["stop"] - right["start"] == node["layers"], "source node layer count") + max_active, multi_nonempty = 0, False + for cut in range(len(self.plan) + 1): + completed = {a["id"] for a in self.plan.atoms if a["stop"] <= cut} + active = [a for a in self.plan.atoms if a["start"] <= cut < a["stop"]] + used = set() + for atom in active: + need(not used.intersection(atom["indices"]), "disjoint active full calls") + used.update(atom["indices"]) + max_active = max(max_active, len(active)) + multi_nonempty |= sum(a["start"] < cut for a in active) >= 2 + # Parent upper calls precede a completed or active descendant. + for atom in active + [self.atoms[x] for x in completed]: + a, b = atom["interval"] + for x, y in self.builder.nodes: + if x <= a and b <= y and (x, y) != (a, b): + need(f"R:{x}:{y}" in completed, "completed ancestor upper call") + if atom["kind"] == "lower": + need(self.subsets[a, b] - {atom["id"]} <= completed, + "lower call waits for its subtree") + unequal = sum(self.endpoints(a, (a + b) // 2)[1] - self.endpoints(a, (a + b) // 2)[0] + != self.endpoints((a + b) // 2, b)[1] - self.endpoints((a + b) // 2, b)[0] + for a, b in self.builder.nodes if b - a > 1) + return dict(max_active_calls=max_active, simultaneous_nonempty_fragments=multi_nonempty, + unequal_sibling_nodes=unequal, + zero_layer_calls=sum(a["start"] == a["stop"] for a in self.plan.atoms)) + + def downsets(self, a=0, b=None): + b = self.n if b is None else b + if b - a == 1: + return [set(), {f"D:{a}"}] + c = (a + b) // 2 + return [set()] + [ + left | right | {f"R:{a}:{b}"} + for left in self.downsets(a, c) for right in self.downsets(c, b) + ] + [self.subsets[a, b].copy()] + + def _partial(self, completed, a, b): + ids = self.subsets[a, b] & completed + if not ids: + return eye(b - a), eye(b - a), eye(b - a), [(a, b)] + if b - a == 1: + need(ids == {f"D:{a}"}, "legal leaf downset") + return eye(1), self.atoms[f"D:{a}"]["ideal"], eye(1), [(a, b)] + need(f"R:{a}:{b}" in ids, "downset includes upper predecessor") + c = (a + b) // 2 + l = self._partial(completed, a, c) + r = self._partial(completed, c, b) + low = blockdiag(l[0], r[0]) + if f"L:{a}:{b}" in ids: + need(ids == self.subsets[a, b], "completed lower closes its subtree") + low = multiply(shear(self.atoms[f"L:{a}:{b}"]["ideal"]), low) + return (low, blockdiag(l[1], r[1]), + multiply(blockdiag(l[2], r[2]), shear(self.atoms[f"R:{a}:{b}"]["ideal"], True)), + l[3] + r[3]) + + def check_downset(self, completed): + need(completed <= self.atoms.keys(), "known downset calls") + low, diagonal, up, frontier = self._partial(completed, 0, self.n) + product = eye(self.n) + for atom in self.plan.atoms: + if atom["id"] in completed: + local = atom["ideal"] if atom["kind"] == "leaf" else shear(atom["ideal"]) + ids = [self.builder.positions[i] for i in atom["indices"]] + product = multiply(embed(local, ids, self.n), product) + need(multiply(multiply(low, diagonal), up) == product, "exact downset LDU identity") + front = eye(self.n) + for a, b in frontier: + part = self.factors[a, b][2] + for i in range(a, b): + for j in range(a, b): + front[i][j] = part[i - a][j - a] + need(multiply(front, up) == self.factors[0, self.n][2], "exact upper frontier identity") + bound = self.builder.bound["completed_prefix_bound"] + need(max(frobenius2(product), frobenius2(eliminate(product))) <= self.n * bound ** 2, + "polynomial completed-prefix consequence") + return dict(completed=sorted(completed), frontier=[list(x) for x in frontier], + prefix_sha256=vector_hash([x for row in product for x in row])) + + def actual_endpoint(self, atom): + if atom["id"] not in self.actual_atoms: + local = operator(atom["local_plan"], len(atom["indices"])) + self.actual_atoms[atom["id"]] = embed(local, atom["indices"], self.n) + return self.actual_atoms[atom["id"]] + + def physical_cut(self, cut): + completed = {a["id"] for a in self.plan.atoms if a["stop"] <= cut} + active = [a for a in self.plan.atoms if a["start"] <= cut < a["stop"]] + active_ids = {a["id"] for a in active} + done, remaining = eye(self.n), eye(self.n) + for atom in self.plan.atoms: + if atom["id"] in completed: + done = multiply(self.actual_endpoint(atom), done) + elif atom["id"] not in active_ids: + remaining = multiply(self.actual_endpoint(atom), remaining) + fragments, tails = eye(self.n), eye(self.n) + for atom in active: + offset, k = cut - atom["start"], len(atom["indices"]) + front = operator(atom["local_plan"][:offset], k) + tail = operator(atom["local_plan"][offset:], k) + fragments = multiply(embed(front, atom["indices"], self.n), fragments) + tails = multiply(embed(tail, atom["indices"], self.n), tails) + prefix, suffix = operator(self.plan[:cut], self.n), operator(self.plan[cut:], self.n) + need(prefix == multiply(fragments, done), "exact active-prefix factorization") + need(suffix == multiply(remaining, tails), "exact active-suffix factorization") + bound = self.builder.bound["execution_bound"] + peak = max(frobenius2(x) for x in (prefix, suffix, eliminate(prefix), eliminate(suffix))) + need(peak <= self.n * bound ** 2, "physical prefix/suffix norm consequences") + return dict(cut=cut, completed=len(completed), active=sorted(active_ids), + nonempty_active_prefixes=sum(a["start"] < cut for a in active), + max_frobenius_squared_upper=str(upper(peak))) diff --git a/research/rational-unitary-fft/full.py b/research/rational-unitary-fft/full.py new file mode 100644 index 0000000000..7f85ad1a89 --- /dev/null +++ b/research/rational-unitary-fft/full.py @@ -0,0 +1,243 @@ +"""Retained full Toeplitz recursion under leading-section conditioning. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. Recursive source and displacement identities: Vlado/PR1673. +""" +import ast +import copy +from fractions import Fraction as Q +from hashlib import sha256 +from types import SimpleNamespace + +from exact import (C, Mat, Vec, ceil_log2, component_l1, eye, inverse_gate, + multiply, need, rounded) +from engine import Phases +from generator import (ORACLE, SOURCE, Program, allocation as generator_allocation, + load_producer, prepare) + + +class Matrix(Mat): + @property + def T(self): + return Matrix(zip(*self)) + + def __getitem__(self, key): + value = super().__getitem__(key) + return Matrix(value) if isinstance(value, Mat) else value + + def __sub__(self, other): + need(self.shape == other.shape, "full matrix subtraction shape") + return Matrix([[x - y for x, y in zip(a, b)] for a, b in zip(self, other)]) + + def __matmul__(self, other): + return Matrix(multiply(self, other)) + + +def solve(left, right): + n = len(left) + need(left.shape == (n, n) and len(right) == n, "exact solve shapes") + width = len(right[0]) + rows = [list(a) + list(b) for a, b in zip(left, right)] + for j in range(n): + pivot = next((i for i in range(j, n) if rows[i][j] != 0), None) + need(pivot is not None, "singular leading/Schur section") + rows[j], rows[pivot] = rows[pivot], rows[j] + divisor = rows[j][j] + rows[j] = [x / divisor for x in rows[j]] + for i in range(n): + if i != j: + factor = rows[i][j] + rows[i] = [x - factor * y for x, y in zip(rows[i], rows[j])] + need(all(rows[i][j] == int(i == j) for i in range(n) for j in range(n)), + "exact solve elimination") + return Matrix([row[n:n + width] for row in rows]) + + +def norm_upper(matrix): + return max(max(component_l1(row) for row in matrix), + max(component_l1(column) for column in zip(*matrix))) + + +def admission(matrix): + matrix = Matrix(matrix) + n = len(matrix) + need(n > 0 and n & (n - 1) == 0 and matrix.shape == (n, n), "dyadic full Toeplitz shape") + need(all(matrix[i][j] == (matrix[i - j][0] if i >= j else matrix[0][j - i]) + for i in range(n) for j in range(n)), "exact full Toeplitz admission") + leading = [] + for b in range(1, n + 1): + inverse = solve(matrix[:b, :b], Matrix(eye(b))) + leading.append((b, norm_upper(inverse))) + return max(Q(1), norm_upper(matrix)), max(Q(1), *(v for _, v in leading)), leading + + +def allocation(n, lam, inverse_bound, radius, target, incoming): + need(type(n) is int and n > 0 and n & (n - 1) == 0, "full dyadic dimension") + need(all(type(x) in (int, Q) and x >= 1 for x in (lam, inverse_bound, radius)), + "exact full conditioning and radius bounds") + need(type(target) is int and target >= 1 and type(incoming) is int and incoming >= 0, + "full target and incoming precision") + lam, inverse_bound, radius = (Q(x) for x in (lam, inverse_bound, radius)) + d, chi = n.bit_length() - 1, lam * inverse_bound + descendant = lam + lam ** 2 * inverse_bound + b = 256 * n ** 3 * chi ** 12 + calls, rank = 3 * n - 2, 2 * d + 8 + alpha, kg = 8 * rank * n ** 2 * chi, 2 * n + 64 * n ** 2 * chi + generator = max(Q(3200 * n ** 3), 2 + 2 * kg) * (10 + 4 * alpha + 2 * kg) + fragment = max(generator, descendant + 1, 2 * inverse_bound) + execution = 2 * max(Q(1), lam, inverse_bound) * b * fragment + layers = 1 + 2 * d * rank * (8 * d * d + 32 * d + 18) + gates = n * (1 + d * rank * (4 * d * d + 24 * d + 22)) + s = target + ceil_log2(4 * calls * b ** 3 * (radius + 1)) + eta = Q(1, 1 << s) + theta = calls * b ** 2 * eta + data = max(incoming, target + ceil_log2(2 * layers * execution * n)) + coefficient = 2 * calls * b ** 3 * eta + rounding = layers * execution * n * Q(1, 1 << data) + need(theta <= Q(1, 4) and eta <= 1 / (2 * inverse_bound) + and coefficient * radius + rounding <= Q(1, 1 << target), "complete full precision budget") + amplitude = execution * radius + layers * execution ** 2 * n * Q(1, 1 << data) + integer = max(1, ceil_log2(max(Q(1), amplitude))) + 2 + scalar_height = 2 * s + 2 * ceil_log2(descendant + 2) + 8 + return dict(n=n, lam=lam, inverse_bound=inverse_bound, radius=radius, + target_bits=target, incoming_bits=incoming, conditioning=chi, + descendant_bound=descendant, completed_prefix_bound=b, + atomic_calls=calls, rank_bound=rank, displacement_bound=4 * n * chi, + generator_fragment_bound=generator, fragment_bound=fragment, + execution_bound=execution, segment_bound=execution ** 2, + layer_bound=layers, gate_bound=gates, child_target_bits=s, eta=eta, + theta=theta, data_bits=data, coefficient_error=coefficient, + rounding_error=rounding, total_error=coefficient * radius + rounding, + amplitude_bound=amplitude, integer_bits=integer, + scalar_height_bound=scalar_height, coefficient_height_bound=scalar_height) + + +class Plan(list): + def __init__(self, layers, atoms): + super().__init__(layers) + self.atoms = atoms + + def __add__(self, other): + need(isinstance(other, Plan), "typed full-plan composition") + return Plan(list(self) + list(other), [dict(a) for a in self.atoms] + + [dict(a, start=a["start"] + len(self), stop=a["stop"] + len(self)) + for a in other.atoms]) + + +class Full: + def __init__(self, bound, raw): + self.bound, self.raw = dict(bound), raw + self.producer = load_producer(raw) + self.phase_pool, self.nodes = {}, {} + selected = {"full_plan", "parallel", "inverse"} + tree = ast.parse(raw[SOURCE + "check_circuits.py"]) + tree.body = [copy.deepcopy(node) for node in tree.body + if isinstance(node, ast.FunctionDef) and node.name in selected] + need({node.name for node in tree.body} == selected, "complete full source function set") + self.source_ast = sha256(ast.dump(tree).encode()).hexdigest() + self.stats = {"full_nodes": 0} + self.namespace = dict(STATS=self.stats, + np=SimpleNamespace(linalg=SimpleNamespace(solve=solve, inv=inverse_gate)), + lowdisp_shear=self.shear, diag=self.leaf) + exec(compile(tree, "", "exec", optimize=0), + self.namespace) + self.source_full = self.namespace["full_plan"] + self.source_parallel = self.namespace["parallel"] + self.source_inverse = self.namespace["inverse"] + self.namespace["parallel"] = self.parallel + self.namespace["full_plan"] = self.node + + def interval(self, indices): + positions = [self.positions[i] for i in indices] + a, b = min(positions), max(positions) + 1 + need(positions == list(range(a, b)), "ordered contiguous logical interval") + return a, b + + def node(self, indices, matrix): + interval = self.interval(indices) + need(interval not in self.nodes, "one exact Schur reference per node") + self.nodes[interval] = dict(indices=list(indices), matrix=Matrix(matrix)) + need(norm_upper(matrix) <= self.bound["descendant_bound"], "component descendant bound") + result = self.source_full(indices, matrix) + self.nodes[interval]["layers"] = len(result) + return result + + def leaf(self, indices, values): + need(len(indices) == len(values) == 1, "scalar full leaf") + intended = values[0] + need(intended.norm2() >= 1 / self.bound["inverse_bound"] ** 2 + and intended.norm2() <= self.bound["descendant_bound"] ** 2, "full pivot margins") + stored = rounded(intended, self.bound["child_target_bits"]) + need((stored - intended).norm2() <= self.bound["eta"] ** 2 + and stored.norm2() >= 1 / (4 * self.bound["inverse_bound"] ** 2), "stored full pivot margin") + gate = Mat([[stored]]) + need(stored * inverse_gate(gate)[0][0] == 1, "same stored full scalar inverse") + a, b = self.interval(indices) + atom = dict(id=f"D:{a}", kind="leaf", interval=(a, b), indices=list(indices), + start=0, stop=1, ideal=Mat([[intended]]), + local_plan=[[((0,), gate)]], error=self.bound["eta"], + coefficient_height=self.bound["scalar_height_bound"]) + return Plan([[(tuple(indices), gate)]], [atom]) + + def shear(self, target, source, matrix): + m = len(source) + positions = sorted(self.positions[i] for i in source + target) + a, b = min(positions), max(positions) + 1 + need(positions == list(range(a, b)) and b - a == 2 * m, "owned full shear interval") + upper = self.positions[target[0]] < self.positions[source[0]] + need([self.positions[i] for i in target] == list(range(a if upper else a + m, + a + m if upper else b)) + and [self.positions[i] for i in source] == list(range(a + m if upper else a, + b if upper else a + m)), + "source full shear orientation") + need(norm_upper(matrix) <= self.bound["conditioning"], "full K/L conditioning") + items, displacement, be, heights = prepare(matrix, self.producer) + need(len(items) <= self.bound["rank_bound"] and be <= self.bound["displacement_bound"], + "retained displacement rank and envelope") + child, triangle = generator_allocation(m, len(items), be, 1, + self.bound["child_target_bits"], 0) + need(child["coefficient_error"] <= self.bound["eta"] + and child["execution_bound"] <= self.bound["generator_fragment_bound"], + "full atomic generator contract") + p = child["phase_bits"] + if p not in self.phase_pool: + self.phase_pool[p] = Phases(p, self.raw[ORACLE]) + program = Program(child, triangle, self.raw, self.phase_pool[p]) + local = program.plan(items) + mapping = source + target + plan = [[(tuple(mapping[i] for i in ids), gate) for ids, gate in layer] for layer in local] + kind = "upper" if upper else "lower" + atom = dict(id=f"{'R' if upper else 'L'}:{a}:{b}", kind=kind, + interval=(a, b), indices=list(mapping), start=0, stop=len(plan), + ideal=Mat(matrix), local_plan=local, error=child["coefficient_error"], + coefficient_height=child["coefficient_height_bound"], + program=program, generator_items=items, displacement=displacement, + producer_heights=heights) + return Plan(plan, [atom]) + + def parallel(self, left, right): + left_ids = {i for a in left.atoms for i in a["indices"]} + right_ids = {i for a in right.atoms for i in a["indices"]} + need(not left_ids.intersection(right_ids), "disjoint full sibling supports") + return Plan(self.source_parallel(left, right), + [dict(a) for a in left.atoms] + [dict(a) for a in right.atoms]) + + def plan(self, matrix, indices=None): + n = len(matrix) + need(n == self.bound["n"], "allocated full dimension") + indices = list(range(n)) if indices is None else list(indices) + need(all(type(i) is int for i in indices) and sorted(indices) == list(range(n)), + "full root wire permutation") + self.positions = {wire: pos for pos, wire in enumerate(indices)} + result = self.node(indices, Matrix(matrix)) + need(self.stats["full_nodes"] == 2 * n - 1 and len(result.atoms) == 3 * n - 2 + and len({a["id"] for a in result.atoms}) == len(result.atoms), "complete full recursion") + need(len(result) <= self.bound["layer_bound"] + and sum(map(len, result)) <= self.bound["gate_bound"], "full source schedule bounds") + self.bound["coefficient_height_bound"] = max(a["coefficient_height"] for a in result.atoms) + self.bound["temporary_bits"] = (2 * self.bound["data_bits"] + + 8 * self.bound["coefficient_height_bound"] + 2 * self.bound["integer_bits"] + 32) + return result + + def inverse(self, plan): + return self.source_inverse(plan) diff --git a/research/rational-unitary-fft/generator.py b/research/rational-unitary-fft/generator.py index a1217fac59..d761d3c88b 100644 --- a/research/rational-unitary-fft/generator.py +++ b/research/rational-unitary-fft/generator.py @@ -145,9 +145,10 @@ def spectrum(self, kernel): class Program: - def __init__(self, bound, triangle_bound, raw): + def __init__(self, bound, triangle_bound, raw, phase_oracle=None): self.bound, self.triangle_bound, self.raw = bound, triangle_bound, raw - self.phases = Phases(bound["phase_bits"], raw[ORACLE]) + self.phases = phase_oracle if phase_oracle is not None else Phases(bound["phase_bits"], raw[ORACLE]) + need(self.phases.bits == bound["phase_bits"], "common generator phase precision") self.source = list(range(bound["m"])) self.target = list(range(bound["m"], 2 * bound["m"])) self.stages, self.position, self.calls = [], 0, [] diff --git a/research/rational-unitary-fft/verify.py b/research/rational-unitary-fft/verify.py index e4de89f896..60ecb552d8 100644 --- a/research/rational-unitary-fft/verify.py +++ b/research/rational-unitary-fft/verify.py @@ -33,7 +33,7 @@ def rejected(operation, label): raise ValueError("control unexpectedly passed: " + label) -def authenticate(include_generator=False): +def authenticate(include_generator=False, include_full=False): rows = [r for r in json.loads((HERE / "SOURCES.json").read_text())["files"] if r["usage"] == "input"] raw = {} for row in rows: @@ -49,8 +49,10 @@ def authenticate(include_generator=False): "research/rational-unitary-fft/vendor/stein_rational.py", "research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json", } + full = {"research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md"} return raw, [{"path": r["path"], "sha256": r["sha256"]} for r in rows - if include_generator or r["path"] not in added] + if (include_generator or r["path"] not in added) + and (include_full or r["path"] not in full)] def operator(plan, n): diff --git a/research/rational-unitary-fft/verify_full.py b/research/rational-unitary-fft/verify_full.py new file mode 100644 index 0000000000..ecee33aa15 --- /dev/null +++ b/research/rational-unitary-fft/verify_full.py @@ -0,0 +1,274 @@ +"""Exact full-recursion frontiers and rounded forward/inverse execution. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +from exact import (C, Mat, Vec, adjoint, apply, component_l1, eye, inverse_gate, + multiply, need, norm2, upper, vector_hash) +from full import Full, Matrix, Plan, admission, allocation, solve +from frontier import Frontier, embed +from verify import authenticate, canonical, mv, operator, plan_hash, rejected +from verify_generator import exact_rank +from verify_triangle import eliminate, frobenius_error, structure + + +def one_case(label, matrix, raw, reverse=False, fine=False): + n = len(matrix) + lam, q, leading = admission(matrix) + vectors = [ + Vec(C(Q((3 * i + 1) % 7 - 3, 8), Q((5 * i + 2) % 11 - 5, 8)) for i in range(n)), + Vec(C((-1) ** i * (1 << 10), (i % 2) * (1 << 8)) for i in range(n)), + ] + if fine: + vectors.append(Vec([C(Q(1, 1 << 512))] + [C()] * (n - 1))) + radius = max(Q(1), *(component_l1(v) for v in vectors)) + incoming = max(part.denominator.bit_length() - 1 for v in vectors for z in v + for part in (z.re, z.im)) + bound = allocation(n, lam, q, radius, 20, incoming) + if fine: + coarse = allocation(n, lam, q, radius, 20, 0) + need(bound["data_bits"] > coarse["data_bits"], "nonzero fine-grid caller overrides required grid") + builder = Full(bound, raw) + indices = list(reversed(range(n))) if reverse else list(range(n)) + plan = builder.plan(matrix, indices) + inverse = builder.inverse(plan) + bound = builder.bound + need(all(len(a["generator_items"]) == exact_rank(a["displacement"]) + for a in plan.atoms if a["kind"] != "leaf"), "full-call exact generator ranks") + wanted = embed(matrix, indices, n) + wanted_inverse = eliminate(wanted) + actual, actual_inverse = operator(plan, n), operator(inverse, n) + need(multiply(actual, actual_inverse) == eye(n), "full exact stored-program inverse") + forward_squared = frobenius_error(actual, wanted) + inverse_squared = frobenius_error(actual_inverse, wanted_inverse) + need(max(forward_squared, inverse_squared) <= n * bound["coefficient_error"] ** 2, + "full matrix and separate inverse references") + height = max(structure(plan, bound), structure(inverse, bound)) + frontier = Frontier(builder, plan) + timing = frontier.schedule() + downsets = (frontier.downsets() if n <= 4 else + [{a["id"] for a in plan.atoms if a["stop"] <= cut} + for cut in sorted({0, len(plan)} | + {a["start"] for a in plan.atoms} | + {a["stop"] for a in plan.atoms})]) + unique = {tuple(sorted(x)): x for x in downsets} + downset_records = [frontier.check_downset(unique[key]) for key in sorted(unique)] + physical = [] + if n in (2, 4) or label == "unequal-subtrees": + candidates = list(range(len(plan) + 1)) + def score(cut): + active = [a for a in plan.atoms if a["start"] < cut < a["stop"]] + return len(active), sum(cut - a["start"] for a in active) + selected = max(candidates, key=score) + cuts = {len(plan) // 2, selected} + physical = [frontier.physical_cut(cut) for cut in sorted(cuts)] + vectors.append(Vec([C(Q(2, 1 << bound["data_bits"]))] + [C()] * (n - 1))) + final = allocation(n, lam, q, radius, 20, bound["data_bits"]) + need(all(final[k] == value for k, value in bound.items() + if k not in ("incoming_bits", "coefficient_height_bound", "temporary_bits")), + "micro data preserves full allocation") + bound["incoming_bits"] = bound["data_bits"] + states = [] + for direction, program, stored, intended in ( + ("forward", plan, actual, wanted), + ("inverse", inverse, actual_inverse, wanted_inverse)): + for vector in vectors: + need(norm2(vector) <= radius ** 2, "full input radius") + output, stats = apply(program, vector, bound["data_bits"]) + exact_stored, exact_target = mv(stored, vector), mv(intended, vector) + coefficient = norm2(exact_stored - exact_target) + rounding = norm2(output - exact_stored) + total = norm2(output - exact_target) + need(coefficient <= (bound["coefficient_error"] * radius) ** 2, + "full coefficient error") + need(rounding <= bound["rounding_error"] ** 2 and total <= bound["total_error"] ** 2, + "full rounded forward/inverse budget") + need(stats["max_observed_height"] <= bound["temporary_bits"] + and stats["max_grid_numerator_bits"] + 1 <= bound["data_bits"] + bound["integer_bits"] + and stats["max_reduced_fractional_bits"] <= bound["data_bits"], + "full signed grid and temporary widths") + states.append(dict(direction=direction, input_sha256=vector_hash(vector), + output_sha256=vector_hash(output), + coefficient_squared_upper=str(upper(coefficient)), + rounding_squared_upper=str(upper(rounding)), + total_squared_upper=str(upper(total)), statistics=stats)) + atoms = [] + for atom in plan.atoms: + row = {k: atom[k] for k in ("id", "kind", "indices", "start", "stop", "coefficient_height")} + row.update(interval=list(atom["interval"]), operator_error=str(atom["error"]), + ideal_sha256=vector_hash([x for r in atom["ideal"] for x in r])) + if atom["kind"] != "leaf": + row.update(rank=len(atom["generator_items"]), + child_budget={k: str(v) if isinstance(v, Q) else v + for k, v in atom["program"].bound.items()}) + atoms.append(row) + return dict(label=label, n=n, indices=indices, + input_sha256=vector_hash([x for row in matrix for x in row]), + leading_bounds=[dict(size=b, inverse_component_bound=str(v)) for b, v in leading], + source_ast_sha256=builder.source_ast, nodes=len(builder.nodes), + layers=len(plan), gates=sum(map(len, plan)), atoms=atoms, timing=timing, + downsets=downset_records, physical_cuts=physical, + plan_sha256=plan_hash(plan), inverse_sha256=plan_hash(inverse), + budget={k: str(v) if isinstance(v, Q) else v for k, v in bound.items()}, + max_coefficient_height=height, + forward_frobenius_squared_upper=str(upper(forward_squared)), + inverse_frobenius_squared_upper=str(upper(inverse_squared)), + phase_oracles=[builder.phase_pool[p].record() for p in sorted(builder.phase_pool)], + states=states) + + +def controls(raw): + results = [] + results.append(rejected(lambda: admission(Mat([[0, 1], [1, 0]])), + "invertible-full-matrix-with-singular-leading-section")) + results.append(rejected(lambda: admission(Mat([[1, 2], [3, 4]])), "non-Toeplitz-root")) + results.append(rejected(lambda: admission(Mat([[1, 0, 0], [0, 1, 0], [0, 0, 1]])), + "nondyadic-full-size")) + results.append(rejected(lambda: admission([[1.0]]), "floating-full-input")) + matrix = Mat([[C(1, Q(1, 4)), C(Q(1, 16), Q(1, 8))], + [C(Q(1, 8), Q(-1, 16)), C(1, Q(1, 4))]]) + lam, q, _ = admission(matrix) + bound = allocation(2, lam, q, 4, 20, 0) + builder = Full(bound, raw) + plan = builder.plan(matrix) + first = next(a for a in plan.atoms if a["id"] == "R:0:2") + mutant = plan[first["stop"]:] + need(frobenius_error(operator(mutant, 2), matrix) > 2 * bound["coefficient_error"] ** 2, + "missing root upper shear changes full operator") + results.append("omitted-root-upper-call") + wrong_order = [[(ids, inverse_gate(gate)) for ids, gate in layer] for layer in plan] + need(frobenius_error(operator(wrong_order, 2), eliminate(matrix)) + > 2 * bound["coefficient_error"] ** 2, "wrong full inverse order") + results.append("full-inverse-layer-order") + builder = Full(bound, raw) + calls = [0] + def wrong_solve(left, right): + calls[0] += 1 + if calls[0] == 2: + left = Matrix([[x.conjugate() for x in row] for row in left]) + return solve(left, right) + builder.namespace["np"].linalg.solve = wrong_solve + wrong = builder.plan(matrix) + need(frobenius_error(operator(wrong, 2), matrix) > 2 * bound["coefficient_error"] ** 2, + "one-sided conjugation in complex lower solve") + results.append("one-sided-conjugation-in-lower-solve") + builder = Full(bound, raw) + builder.namespace["parallel"] = lambda a, b: a + b + serial = builder.plan(matrix) + results.append(rejected(lambda: Frontier(builder, serial).schedule(), "serialized-full-siblings")) + identity = Mat(eye(4)) + identity_bound = allocation(4, 1, 1, 4, 20, 0) + builder = Full(identity_bound, raw) + identity_plan = builder.plan(identity) + changed = [dict(a, stop=1) if a["id"] == "R:0:4" else dict(a) for a in identity_plan.atoms] + results.append(rejected(lambda: Frontier(builder, Plan(identity_plan, changed)).schedule(), + "delayed-zero-layer-identity-call")) + scalar = Mat([[C(Q(1, 3), Q(1, 7))]]) + lam, q, _ = admission(scalar) + scalar_bound = allocation(1, lam, q, 1, 20, 0) + scalar_plan = Full(scalar_bound, raw).plan(scalar) + adjoints = [[(ids, adjoint(gate)) for ids, gate in layer] for layer in reversed(scalar_plan)] + need(frobenius_error(operator(adjoints, 1), eliminate(scalar)) + > scalar_bound["coefficient_error"] ** 2, "full scalar adjoint is not reciprocal") + results.append("full-scalar-adjoint-instead-of-reciprocal") + coarse, _ = apply(scalar_plan, Vec([1]), 4) + need(norm2(coarse - Vec([scalar[0][0]])) > Q(1, 1 << 40), "coarse full grid misses target") + results.append("underfunded-full-data-grid") + tiny = Mat([[Q(1, 1 << 40)]]) + lam, q, _ = admission(tiny) + tiny_bound = dict(allocation(1, lam, q, 1, 20, 0), child_target_bits=4, eta=Q(1, 16)) + results.append(rejected(lambda: Full(tiny_bound, raw).plan(tiny), "rounded-zero-full-pivot")) + results.append(rejected(lambda: apply(scalar_plan, Vec([Q(1, 1 << 512)]), 20), + "discarded-full-incoming-grid")) + matrix4 = Mat([[C(1, Q(1, 8)) if i == j + else C(Q(1, 32 * (1 + abs(i - j))), Q(i - j, 128)) + for j in range(4)] for i in range(4)]) + lam, q, _ = admission(matrix4) + builder = Full(allocation(4, lam, q, 4, 20, 0), raw) + p4 = builder.plan(matrix4) + f = Frontier(builder, p4) + _, _, up, intervals = f._partial({"R:0:4"}, 0, 4) + front = eye(4) + for a, b in intervals: + for i in range(a, b): + for j in range(a, b): + front[i][j] = f.factors[a, b][2][i - a][j - a] + need(multiply(front, f.factors[0, 4][2]) != up, "upper frontier requires its inverse") + results.append("missing-upper-frontier-inverse") + need(canonical({"nodes": 1}) != canonical({"nodes": True}), "typed full receipt") + return results + ["typed-full-receipt"] + + +def run(): + raw, sources = authenticate(include_generator=True, include_full=True) + fixtures = [ + ("complex-scalar", Mat([[C(Q(1, 3), Q(1, 7))]]), False, False), + ("small-pivot", Mat([[Q(1, 1 << 80)]]), False, False), + ("large-scalar", Mat([[C(1 << 30, -(1 << 28))]]), False, False), + ] + for n in (2, 4, 8): + matrix = Mat([[C(1, Q(1, 16)) if i == j + else C(Q(-1 if (i - j) % 2 else 1, 32 * n), Q(i - j, 64 * n)) + for j in range(n)] for i in range(n)]) + fixtures.append((f"complex-near-identity-{n}", matrix, n >= 4, n == 4)) + for n in (2, 4): + matrix = Mat([[1 if i == j else 2 if abs(i - j) == 1 else 0 + for j in range(n)] for i in range(n)]) + fixtures.append((f"indefinite-{n}", matrix, True, False)) + n = 4 + matrix = Mat([[Q(1, 2) ** ((i - j) ** 2 - (i - j)) for j in range(n)] for i in range(n)]) + fixtures.append(("half-shift-Gaussian", matrix, False, False)) + n = 8 + matrix = Mat([[1 if i == j else Q(1, 16) if abs(i - j) == 4 + else Q(1, 32) if abs(i - j) == 5 else 0 + for j in range(n)] for i in range(n)]) + fixtures.append(("unequal-subtrees", matrix, False, False)) + fixtures.append(("zero-call-identity-16", Mat(eye(16)), True, False)) + records = [] + for label, matrix, reverse, fine in fixtures: + records.append(one_case(label, matrix, raw, reverse, fine)) + print("CHECK_FULL_CASE", label, flush=True) + need(any(row["timing"]["unequal_sibling_nodes"] for row in records), "unequal sibling execution witness") + need(any(row["timing"]["simultaneous_nonempty_fragments"] for row in records), + "simultaneous unfinished child fragments") + need(any(cut["nonempty_active_prefixes"] >= 2 + for row in records for cut in row["physical_cuts"]), + "checked exact factorization with simultaneous nonempty fragments") + need(next(row for row in records if row["label"] == "unequal-subtrees") + ["timing"]["unequal_sibling_nodes"] > 0, "designed unequal-subtree fixture") + need(any(row["timing"]["zero_layer_calls"] for row in records), "zero-call schedule witness") + return dict(schema_version=1, scope="actual full Toeplitz recursion and stored inverse", + sources=sources, cases=records, controls=controls(raw), + full_toeplitz_gate_execution=True, physical_tape_execution=False, + complete_gaussian_execution=False, universal_proof_kernel=False) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-result", action="store_true") + args = parser.parse_args() + result = run() + path = HERE / "FULL-RESULT.json" + if args.write_result: + path.write_text(json.dumps(result, indent=2) + "\n") + else: + need(canonical(result) == canonical(json.loads(path.read_text())), "complete typed full receipt") + print(json.dumps(dict(status="PASS_RATIONAL_FULL_TOEPLITZ", cases=len(result["cases"]), + gates=sum(x["gates"] for x in result["cases"]), + rounded_states=sum(len(x["states"]) for x in result["cases"]), + downsets=sum(len(x["downsets"]) for x in result["cases"]), + physical_cuts=sum(len(x["physical_cuts"]) for x in result["cases"]), + result_sha256=sha256(canonical(result)).hexdigest()), sort_keys=True)) + + +if __name__ == "__main__": + main() From 66b95cc916552f487203be14fd99a7aea0bdd78e Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 19:52:22 +0200 Subject: [PATCH 6/9] Certify normalized Gaussian coefficients and full circuit error --- .../rational-unitary-fft/GAUSSIAN-PROOF.md | 288 +++ .../rational-unitary-fft/GAUSSIAN-RESULT.json | 1883 +++++++++++++++++ research/rational-unitary-fft/README.md | 55 +- research/rational-unitary-fft/SOURCES.json | 7 +- research/rational-unitary-fft/TASK.json | 14 +- research/rational-unitary-fft/VERIFY.json | 15 +- research/rational-unitary-fft/gaussian.py | 274 +++ research/rational-unitary-fft/verify.py | 12 +- research/rational-unitary-fft/verify_full.py | 4 + .../rational-unitary-fft/verify_gaussian.py | 209 ++ 10 files changed, 2740 insertions(+), 21 deletions(-) create mode 100644 research/rational-unitary-fft/GAUSSIAN-PROOF.md create mode 100644 research/rational-unitary-fft/GAUSSIAN-RESULT.json create mode 100644 research/rational-unitary-fft/gaussian.py create mode 100644 research/rational-unitary-fft/verify_gaussian.py diff --git a/research/rational-unitary-fft/GAUSSIAN-PROOF.md b/research/rational-unitary-fft/GAUSSIAN-PROOF.md new file mode 100644 index 0000000000..b7c9cad252 --- /dev/null +++ b/research/rational-unitary-fft/GAUSSIAN-PROOF.md @@ -0,0 +1,288 @@ +# Normalized Gaussian coefficients through the full logical circuit + +Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance. +Apache-2.0. + +## 1. Domain and source boundary + +For \(N=2^d\), rational \(c>0\), and rational \(|s|\le1/2\), write +\[ + k(u)=e^{-\pi c(u^2-2su)},\qquad T_{ij}=k(j-i),\quad0\le i,j0\) and one for \(m=0\). +Substitution into \(L^{-1}L\) proves (2). + +Let \(D_s[i,i]=q^{2si}\). Equation (1) gives \(T_n(s)=D_sGD_s^{-1}\). +At \(s=1/2\), each shifted lower-inverse column has absolute sum at most +\[ + \sum_{r=0}^{n-1}z^r/P_r=1/P_{n-1}, +\] +since \(z^r/P_r=P_r^{-1}-P_{r-1}^{-1}\) for \(r\ge1\). +Each shifted upper-inverse entry is at most \(P_{n-1}^{-1}\); +each inverse pivot has the same bound. I multiply the three factors +to obtain \(|T_n(1/2)^{-1}_{ij}|\le P_{n-1}^{-3}\). +Transpose gives the negative endpoint. Using the appropriate endpoint +on each side of the diagonal then gives +\[ + |G^{-1}_{ij}|\le q^{|i-j|}P_{n-1}^{-3},\qquad + |T_n(s)^{-1}_{ij}| + \le q^{|i-j|+2s(i-j)}P_{n-1}^{-3}\le P_{n-1}^{-3}. +\tag{3} +\] +I expand the logarithm of the finite product and sum nonnegative terms: +\[ + -\log P_{n-1} + \le\sum_{h\ge1}\frac1{h(e^{2\gamma h}-1)} + \le\frac1{2\gamma}\sum_{h\ge1}h^{-2} + =\frac{\pi^2}{12\gamma}. +\] +Consequently, for every leading size \(1\le n\le N\), +\[ + \|T_n^{-1}\|_1,\ \|T_n^{-1}\|_\infty + \le nP_{n-1}^{-3}\le N e^{\pi/(4c)}=:Q_{\rm cap}. +\tag{4} +\] +I also have \(02\) proves enclosure. +Otherwise \(f<(J+1)/3\), \(W_X\le2Jh\), and \(X_U\le2J\). +The smallest range-reduction divisor \(a=2^k\) satisfies \(a\le32J\). +Because \(1/8\) lies on the working grid, the rounded interval +\(Y=X/a\) has \(0\le Y_L\le Y_U\le1/8\) and \(W_Y\le W_X/a+2h\). + +The positive Taylor loop stops by \(\lceil J/3\rceil\) terms. +Its term upper endpoints are at most one. Their widths satisfy +\(d_n\le d_{n-1}/8+W_Y+4h\), hence \(d_n\le2W_Y+5h\). +Interval addition is exact; the source adds one grid unit for the tail. +Thus \(W_{\rm sum}\le J(2W_Y+5h)+h\). +The sum has lower endpoint at least one. After reciprocal enclosure, +the width is at most \(W_{\rm sum}+2h\), with endpoints in \([0,1]\). +Each subsequent squaring multiplies the width by at most two and adds +at most \(2h\). Therefore +\[ + W_E\le2JW_X+a(9J+5)h + \le(292J^2+160J)h\le512J^2h, +\] +which proves (5). For unrestricted \(f\), I retain the \(f\)-dependence +of the intermediate \(X\) width before the shortcut. + +For a requested coefficient error \(2^{-b}\), use +\[ + p=b+1,\qquad J=2p+32. +\tag{6} +\] +Equation (5) is at most \(2^{-p}\): check the inequality at \(p=1\), +then note that the ratio decreases as \(p\) increases. +Inward quantization on the \(p\)-grid adds at most \(2^{-p}\). +The returned exact rational receipt therefore bounds the coefficient +error by \(2^{-b}\). I store an exact one whenever \(f=0\). + +To compute a rational conditioning allowance, let +\[ + a_Q=\lceil2/c\rceil,\quad J_Q=2a_Q+36,\quad + [L,U]\ni e^{-\pi/(4c)},\quad \overline Q=N/L. +\tag{7} +\] +From \(\pi<4\) and \(e<4\), \(e^{-\pi/(4c)}>2^{-a_Q}\). +Equation (5) at \(J_Q\) gives width at most \(2^{-(a_Q+2)}\). +It follows that \(L>0\), \(L\ge(3/4)e^{-\pi/(4c)}\), and +\[ + Q_{\rm cap}\le\overline Q\le(4/3)Q_{\rm cap}. +\tag{8} +\] +The working precision \(J_Q=O(1+1/c)\) tracks the size of +\(\log Q_{\rm cap}\). I check positivity and the width bound before +using the reciprocal. + +## 4. Truncation and the same rounded root + +For integer \(W\ge0\), both signs satisfy +\(k(\pm n)\le e^{-\pi c n(n-1)}\). The ratio of successive terms for +\(n\ge W+1\) is at most \(e^{-2\pi c(W+1)}\). Hence +\[ + \sum_{|u|>W}k(u) + \le\frac{2e^{-\pi cW(W+1)}}{1-e^{-2\pi c(W+1)}}. +\tag{9} +\] +I use outward upper endpoints in the numerator and ratio, and require +a positive lower denominator. For \(W=N-1\), the finite matrix has +no omitted offsets and I set its tail allowance to zero. + +Choose +\[ + h_0=t+\left\lceil\log_2(8(R+1)\overline Q^2)\right\rceil,\quad + \varepsilon=2^{-h_0},\quad + b=h_0+\left\lceil\log_2(4(2N-1))\right\rceil. +\tag{10} +\] +I choose the first \(0\le W0\), \(|s|\le1/2\), and +\(T_{ij}=\exp[-\pi c((j-i)^2-2s(j-i))]\), I prove error below \(2^{-t}\) +with online widths +\[ +O(F_0+t+\log N+\log R+1/c). +\] +I derive leading-section conditioning, pay omitted offsets and coefficient +quantization, and pass the same rational root to the full compiler. +I preserve the diagonal ones and both closed half-shift endpoints. +[GAUSSIAN-PROOF.md](GAUSSIAN-PROOF.md) gives the interval termination, +conditioning, perturbation and composed error arguments. +Raw Gaussian gauges, common-reference assembly and physical delivery +require further work. I execute the retained full Toeplitz recursion and its stored inverse, using the rational generator, triangular and FFT circuits below. @@ -104,6 +121,16 @@ python3.12 -B scripts/research_package.py \ I declare eight normal/optimized commands, with limits of 60, 60, 90, 90, 150, 150, 300 and 300 seconds, totaling 1,200 seconds. +In each final slot I run the full verifier followed by the Gaussian +verifier in the same process. I preserve the full receipt and compare +the separate [GAUSSIAN-RESULT.json](GAUSSIAN-RESULT.json) receipt. +I check nine Gaussian cases, 4,527 gates and 38 rounded forward/inverse +states, including both half shifts, paid truncation and a 512-bit +incoming vector. I compare finer scalar interval references and exact +leading-section inverses, and reject 16 controls. The finer references +share the declared interval backend; I claim no independent transcendental +implementation. The complete Gaussian receipt is +`0a0548963b8132b430f43b8534abbd4220b93a138eb9e0a28a9cfd7432922928`. I authenticate the source before parsing its schedule or executing its interval oracle. The adapter preserves bit reversal, gate ownership, disjoint layers, shear orientation and chronological calls. It replaces @@ -223,6 +250,22 @@ near-identity stability argument. I add the explicit global coefficient perturbation, inverse, rounding and intermediate-width contract, with source-bound recursive execution. +For the normalized Gaussian extension, I retain Mason Hale/PR2313's +interval arithmetic and existing dense Gaussian/integer execution, +Jeremy Kothe/IdeaOuter's shifted `Tsec` source, and Vlado's q-LDU factors. +I rederive our PR4193 finite-cap implication in GAUSSIAN-PROOF.md rather +than require an unmerged sibling proof as an unchecked hypothesis. +I preserve full revision `7ab1bc8973cc9e606053c65f094ca3abc4a42fee` +and its complete receipt +`b1bdf09eb959cc4821c5515fa865c06541215224174bcdfc667e6515210bfbb6`. +Its [hosted replay](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39463338955) +remains evidence for that revision. +Before the extension I refreshed unchanged main, inspected 20 recent +open PR records and 8/776 hub comments, and checked our canonical thread. +I found no competing normalized coefficient-to-circuit claim in that +coverage. I compared PR4212's P112 audit and PR4208's CRT stencil work; +they address different native/physical obligations. + I preserve the first completed revision `f02a4ab5ab70a8af80415ac35c5b48cd5a93a475` and its FFT receipt `2e203e42d3d617e828e3341ffeb44254c931db2aebb8a040d04d67338b5b4016`. @@ -250,8 +293,8 @@ Apache-2.0. ## Open next questions -In [TASK.json](TASK.json) I specify the next steps and blockers: bind the -full circuit's conditioning and precision to the actual Gaussian block -producer, then connect its coefficients and data widths to physical -delivery. Complete setup, fixed-tape lowering and independent acceptance -of the complete multiplication theorem remain open. +In [TASK.json](TASK.json) I specify the next steps and blockers: incorporate +the raw Gaussian gauges and scalar factors, compare with the common rounded +reference, and connect the coefficients and data widths to physical delivery. +Reviewers still need to assess the universal proofs and complete setup, +fixed-tape lowering and the full multiplication construction. diff --git a/research/rational-unitary-fft/SOURCES.json b/research/rational-unitary-fft/SOURCES.json index 23ccb215f5..32540a4003 100644 --- a/research/rational-unitary-fft/SOURCES.json +++ b/research/rational-unitary-fft/SOURCES.json @@ -15,6 +15,11 @@ {"path":"research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py","usage":"input","git_blob":"4f0180c88f6ea25b85814883ef7cf2ad41e67110","sha256":"5ff7f25c55b370a1d0cda6c84ab9cdf2c7b0de3266d3f2e7a43280a6491074ab"}, {"path":"research/rational-unitary-fft/vendor/stein_rational.py","usage":"input","pin":"head:0e181260108dd940a594fad6b912588bdee62522:research/rational-unitary-fft/vendor/stein_rational.py","git_blob":"71b240a460a185ee2fe926cd189b56aeac6f997f","sha256":"3065fc5d6d22ec44de533b08fb0c639603449456383668e561dc46629e641502"}, {"path":"research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json","usage":"input","pin":"head:0e181260108dd940a594fad6b912588bdee62522:research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json","git_blob":"c602fb5bb332dbe098950b14d55da05ce2cf4312","sha256":"e7f0e8dc517387e302b1e8bf5dff7e12b29da8aa91abc6ea597f33c1ac08ba62"}, - {"path":"research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md","usage":"input","git_blob":"20585e44fa4e49b13b5d8495bbf6ed254459a362","sha256":"d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116"} + {"path":"research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md","usage":"input","git_blob":"20585e44fa4e49b13b5d8495bbf6ed254459a362","sha256":"d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116"}, + {"path":"research/certified-gaussian-recovery/intervals.py","usage":"input","git_blob":"91d458865aa026bdf0d59653a4ecbfe3a178ecbc","sha256":"159d2a7d32a8cc7c8f56632a8ef127308b984a0a6753c9d6bdccacc6e7d6d4df"}, + {"path":"research/certified-gaussian-recovery/NOTICE.md","usage":"input","git_blob":"5e3269b5afe5e11f044ffae45216c4731b42c7af","sha256":"10acae9e78402ca70c6afd0e8d6da8fefc0da519c646cf43d93f923aba85b86d"}, + {"path":"research/certified-gaussian-recovery/PROOF.md","usage":"input","git_blob":"9dace009004a4336d6ce57764c255190fbf87e64","sha256":"65480dd23fb1aae314365aeea4aacbc146e160e5c8a7a4ac4f602f22ae4581de"}, + {"path":"research/capped-fusion-t74-k554/outer/check_gaugefree.py","usage":"input","git_blob":"3fbb4a1b2acc3d5120d93d2ffd7cd5007d4f7f0e","sha256":"5274b2838f322dc56a88570e4f9262fcd3857cd387160ad138e083ccdfbe9761"}, + {"path":"research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md","usage":"input","git_blob":"59d9128d75c57b790e6e602da8b2ab9c2cf7fa57","sha256":"c6a032ed495325806ddf491c5c85bc822a7fdd613aff906a7be553960d0f85a3"} ] } diff --git a/research/rational-unitary-fft/TASK.json b/research/rational-unitary-fft/TASK.json index e3600a8e5e..cc9318ffb6 100644 --- a/research/rational-unitary-fft/TASK.json +++ b/research/rational-unitary-fft/TASK.json @@ -38,21 +38,21 @@ ] }, "scope_status": "refined", - "actual_scope": "Execute the retained full Toeplitz recursion and stored inverse with exact rational reference coefficients, certified generator/FFT/triangle gates, a polynomial completed-downset/frontier bound, one global grid and explicit online widths.", - "scope_change_reason": "The completed generator contract supports the full recursive caller. I preserve the three earlier receipts and add a global LDU/downset proof that avoids multiplying conditioning losses through recursion depth, including unequal sibling schedules and active fragments.", + "actual_scope": "Execute the retained normalized Gaussian Tsec/full_plan caller with requested-precision rational coefficients, closed-half-shift leading-section conditioning and complete forward/inverse ideal-kernel error and online-width bounds.", + "scope_change_reason": "The completed full circuit requires conditioning for the same rational root it receives. I connect the retained Gaussian evaluator to that boundary, rederive the finite-cap bound, and pay truncation, quantization and circuit errors while preserving four prior receipts.", "bottleneck_link": "The Gaussian caller needs full circuit execution and a quantified conditioning/rounding contract before physical delivery and the complete numerical composition can be admitted.", "budget_minutes": 120, "stop_rule": "Stop at an invalid phase enclosure, nonunitary adapted gate, missing multiplier/twist error, understated word width or source-schedule mismatch. Do not equate coefficient-level unitarity with exact rounded restoration or physical execution.", "outcome": "completed", - "evidence": "I preserve the completed FFT, triangle and generator proofs/receipts. FULL-PROOF.md gives the leading-section, polynomial downset/frontier, perturbation, scalar and width arguments. FULL-RESULT.json records 11 cases, 3659 gates, 68 rounded states, 151 completed-call sets and 12 physical cuts. Schedule checks cover every cut; operator factorizations cover the selected cuts. Nontrivial fixtures reach size8; size16 is an identity/zero-call case. Scientific acceptance remains separate.", + "evidence": "I preserve the completed FFT, triangle, generator and full proofs/receipts. GAUSSIAN-PROOF.md derives the interval width/termination, finite-cap conditioning and complete normalized-kernel error bounds. GAUSSIAN-RESULT.json records 9 cases, 4527 gates, 38 rounded states and 16 controls. I execute Tsec and its adjacent full_plan call, compare finer scalar interval references sharing the declared backend, and compute exact leading inverses. Scientific acceptance remains separate.", "author": "Jose Manuel (jmbarrancoml)", "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, source analysis, implementation and writing, including source-analysis agents. No independent human or formal-kernel acceptance.", "open_questions": [ { - "question": "Can the actual Gaussian block producer supply the full circuit's leading-section conditioning and reference-error contract?", - "why_it_matters": "The full circuit theorem assumes a rational Toeplitz root and bounded leading inverses. The physical Gaussian caller must supply the same rounded/truncated root and pay its approximation error.", - "first_step": "Connect PR4193's conditioning estimates and the retained Gaussian coefficient producer to exact rational roots, then allocate truncation/coefficient/circuit errors in the actual block call.", - "blocked_by": "A source-bound Gaussian-to-full-circuit interface and complete preparation/error ledger." + "question": "Can the normalized circuit contract cover the raw Gaussian gauges and common rounded reference?", + "why_it_matters": "The actual broad and narrow callers include diagonal and scalar factors outside Tsec. Those factors change input radii, inverse sensitivity and error propagation.", + "first_step": "Bind one retained raw broad or narrow block formula to certified gauge coefficients, compose its rounded forward/inverse calls and compare with the same common Gaussian reference.", + "blocked_by": "A complete raw-factor preparation and error allocation, including boundary factors that the caller applies without inversion." }, { "question": "Can the coefficient preparation and canonical data widths be joined to the physical Gaussian caller?", diff --git a/research/rational-unitary-fft/VERIFY.json b/research/rational-unitary-fft/VERIFY.json index bb1d4242be..f0d5c70c62 100644 --- a/research/rational-unitary-fft/VERIFY.json +++ b/research/rational-unitary-fft/VERIFY.json @@ -15,7 +15,12 @@ "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py", "research/rational-unitary-fft/vendor/stein_rational.py", "research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json", - "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md" + "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md", + "research/certified-gaussian-recovery/intervals.py", + "research/certified-gaussian-recovery/NOTICE.md", + "research/certified-gaussian-recovery/PROOF.md", + "research/capped-fusion-t74-k554/outer/check_gaugefree.py", + "research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md" ], "commands": [ { @@ -49,14 +54,14 @@ "checks": "Repeat the complete generator program with source assertions enabled under optimized Python and compare its full typed receipt." }, { - "argv": ["{python}", "-I", "-B", "{package}/verify_full.py"], + "argv": ["{python}", "-I", "-B", "{package}/verify_full.py", "--with-gaussian"], "timeout_seconds": 300, - "checks": "Execute the retained full Toeplitz recursion and stored inverse, check leading-section admission, legal downsets/frontiers and active fragments, and compare exact endpoints plus rounded global error/width budgets." + "checks": "Execute the retained full Toeplitz recursion and stored inverse, check leading-section admission, legal downsets/frontiers and active fragments, and compare exact endpoints plus rounded global error/width budgets. Then execute the retained normalized Gaussian Tsec/full_plan caller with certified coefficients and truncation, compare ideal forward/inverse references and preserve both complete receipts." }, { - "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_full.py"], + "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_full.py", "--with-gaussian"], "timeout_seconds": 300, - "checks": "Repeat full recursive execution and discriminating conditioning/schedule controls under optimized Python with authenticated source assertions and a complete typed receipt." + "checks": "Repeat full recursive and normalized Gaussian execution sequentially under optimized Python with authenticated source assertions, conditioning/schedule/orientation/stale-certificate controls and both complete typed receipts." } ] } diff --git a/research/rational-unitary-fft/gaussian.py b/research/rational-unitary-fft/gaussian.py new file mode 100644 index 0000000000..52ad2e1d60 --- /dev/null +++ b/research/rational-unitary-fft/gaussian.py @@ -0,0 +1,274 @@ +"""Certified normalized Gaussian coefficients at the retained full-plan caller. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. Interval arithmetic: Mason Hale; Tsec: Jeremy Kothe/IdeaOuter; +full circuit and q-LDU identities: Vlado Vojdanovski. +""" +import ast +import copy +from dataclasses import dataclass +from fractions import Fraction as Q +from hashlib import sha256 +import sys +from types import ModuleType, SimpleNamespace + +from exact import C, ceil_log2, need, vector_hash +from full import Full, Matrix, allocation + +INTERVALS = "research/certified-gaussian-recovery/intervals.py" +EVALUATOR = "research/capped-fusion-t74-k554/outer/check_gaugefree.py" + + +def rational(value): + need(type(value) in (int, Q), "exact Gaussian parameter") + return Q(value) + + +def load_intervals(content): + name = "_retained_gaussian_intervals_" + sha256(content).hexdigest() + if name not in sys.modules: + module = ModuleType(name) + sys.modules[name] = module + try: + exec(compile(content, "", "exec", + optimize=0), module.__dict__) + except BaseException: + del sys.modules[name] + raise + return sys.modules[name] + + +@dataclass(frozen=True) +class PiMultiple: + coefficient: Q + + def __neg__(self): + return PiMultiple(-self.coefficient) + + def __add__(self, other): + if isinstance(other, Grid): + return other + self + need(isinstance(other, PiMultiple), "one symbolic pi factor") + return PiMultiple(self.coefficient + other.coefficient) + + def __mul__(self, other): + if isinstance(other, Grid): + return other * self + return PiMultiple(self.coefficient * rational(other)) + + __rmul__ = __mul__ + + +class Grid: + """Only exact scalar arithmetic and the two source broadcast axes.""" + def __init__(self, rows): + self.rows = tuple(tuple(row) for row in rows) + need(self.rows and self.rows[0] + and all(len(row) == len(self.rows[0]) for row in self.rows), "grid shape") + self.shape = len(self.rows), len(self.rows[0]) + + def binary(self, other, operation): + if not isinstance(other, Grid): + return Grid([[operation(x, other) for x in row] for row in self.rows]) + m, n = max(self.shape[0], other.shape[0]), max(self.shape[1], other.shape[1]) + need(all(a in (1, b) for a, b in zip(self.shape, (m, n))) + and all(a in (1, b) for a, b in zip(other.shape, (m, n))), "source broadcasting") + return Grid([[operation(self.rows[i % self.shape[0]][j % self.shape[1]], + other.rows[i % other.shape[0]][j % other.shape[1]]) + for j in range(n)] for i in range(m)]) + + def __sub__(self, other): + return self.binary(other, lambda a, b: a - b) + + def __add__(self, other): + return self.binary(other, lambda a, b: a + b) + + def __mul__(self, other): + return self.binary(other, lambda a, b: a * b) + + def __pow__(self, exponent): + need(exponent == 2, "source square only") + return Grid([[x * x for x in row] for row in self.rows]) + + +class Range: + def __init__(self, n): + self.n = n + + def __getitem__(self, key): + if key == (None, slice(None)): + return Grid([range(self.n)]) + need(key == (slice(None), None), "source arange axes") + return Grid([[i] for i in range(self.n)]) + + +class GaussianMatrix(Matrix): + def astype(self, tag): + need(tag is complex, "retained complex tag without floating conversion") + return GaussianMatrix(self) + + +@dataclass(frozen=True) +class Specification: + n: int + c: Q + shift: Q + radius: Q + target: int + incoming: int + + +class Gaussian: + def __init__(self, raw, n, c, shift, radius, target, incoming=0, window=None): + c, shift, radius = rational(c), rational(shift), rational(radius) + need(type(n) is int and n > 0 and n & (n - 1) == 0, "dyadic Gaussian dimension") + need(c > 0 and abs(shift) <= Q(1, 2) and radius >= 1, "Gaussian domain") + need(type(target) is int and target >= 1 and type(incoming) is int and incoming >= 0, + "Gaussian target and incoming grid") + need(window is None or (type(window) is int and 0 <= window < n), "Gaussian cutoff") + self.spec = Specification(n, c, shift, radius, target, incoming) + self.raw, self.oracle = raw, load_intervals(raw[INTERVALS]) + a = -(-(2 / c).numerator // (2 / c).denominator) + self.conditioning_bits = 2 * a + 36 + box = self.exponential(1 / (4 * c), self.conditioning_bits) + lower = Q(box.lo, box.scale) + need(lower > 0 and Q(box.hi - box.lo, box.scale) <= Q(1, 1 << (a + 2)), + "positive conditioning enclosure with certified width") + self.conditioning_box = box + self.qbar = n / lower + self.error_bits = target + ceil_log2(8 * (radius + 1) * self.qbar ** 2) + self.epsilon = Q(1, 1 << self.error_bits) + self.coefficient_error_bits = self.error_bits + ceil_log2(4 * (2 * n - 1)) + self.coefficient_bits = self.coefficient_error_bits + 1 + self.working_bits = 2 * self.coefficient_bits + 32 + if window is None: + for window in range(n): + tail = self.tail(window) + if tail is not None and tail <= self.epsilon / 2: + break + self.window, self.tail_bound = window, self.tail(window) + need(self.tail_bound is not None and self.tail_bound <= self.epsilon / 2, + "requested cutoff has a paid Gaussian tail") + self.offsets, self.builder, self.root, self.plan = {}, None, None, None + tree = ast.parse(raw[EVALUATOR]) + definitions = [copy.deepcopy(node) for node in tree.body + if isinstance(node, ast.FunctionDef) and node.name == "Tsec"] + need(len(definitions) == 1, "one retained Tsec definition") + function = ast.Module(body=definitions, type_ignores=[]) + pairs = [] + for parent in ast.walk(tree): + body = getattr(parent, "body", None) + if not isinstance(body, list): + continue + for left, right in zip(body, body[1:]): + if (isinstance(left, ast.Assign) and isinstance(right, ast.Assign) + and ast.unparse(left) == "A = Tsec(n, g, s).astype(complex)" + and ast.unparse(right) == "plan = M.full_plan(list(range(n)), A)"): + pairs.append([copy.deepcopy(left), copy.deepcopy(right)]) + need(len(pairs) == 1, "one retained adjacent Tsec/full_plan call") + self.callsite = ast.Module(body=pairs[0], type_ignores=[]) + self.source_ast = sha256(ast.dump(function).encode()).hexdigest() + self.callsite_ast = sha256(ast.dump(self.callsite).encode()).hexdigest() + self.namespace = dict(np=SimpleNamespace(arange=Range, exp=self.prepare_exponent_grid), + n=n, g=PiMultiple(c), s=shift, + M=SimpleNamespace(full_plan=self.compile_root)) + exec(compile(function, "", "exec", optimize=0), self.namespace) + + def exponential(self, factor, bits): + factor = rational(factor) + need(factor >= 0 and bits >= 16, "nonnegative exponent and working precision") + if factor == 0: + return self.oracle.Interval.rational(1, bits) + result = self.oracle.exp_negative(self.oracle.pi_interval(bits) * factor) + need(Q(result.hi - result.lo, result.scale) <= 512 * bits ** 2 * Q(1, 1 << bits), + "retained exponential width allowance") + return result + + def tail(self, window): + if window == self.spec.n - 1: + return Q(0) + numerator = self.exponential(self.spec.c * window * (window + 1), self.working_bits) + ratio = self.exponential(2 * self.spec.c * (window + 1), self.working_bits) + denominator = 1 - Q(ratio.hi, ratio.scale) + return 2 * Q(numerator.hi, numerator.scale) / denominator if denominator > 0 else None + + def coefficient(self, offset): + if offset not in self.offsets: + factor = self.spec.c * (offset * offset - 2 * self.spec.shift * offset) + box = self.exponential(factor, self.working_bits) + zero = self.oracle.Interval.rational(0, self.working_bits) + numerators, error = self.oracle.inward_quantize((box, zero), self.coefficient_bits) + value = Q(numerators[0], 1 << self.coefficient_bits) + need(numerators[1] == 0 and 0 <= value <= 1 + and error <= Q(1, 1 << self.coefficient_error_bits), "Gaussian coefficient receipt") + need(factor != 0 or value == 1, "exact zero-exponent coefficient") + self.offsets[offset] = dict(factor=factor, value=value, error=error, + lower=Q(box.lo, box.scale), upper=Q(box.hi, box.scale)) + return self.offsets[offset]["value"] + + def prepare_exponent_grid(self, grid): + n, c, shift = self.spec.n, self.spec.c, self.spec.shift + need(isinstance(grid, Grid) and grid.shape == (n, n), "Tsec exponent shape") + rows = [] + for i, row in enumerate(grid.rows): + output = [] + for j, exponent in enumerate(row): + u = j - i + need(isinstance(exponent, PiMultiple) + and exponent.coefficient == -c * (u * u - 2 * shift * u), + "retained normalized Tsec orientation") + output.append(self.coefficient(u) if abs(u) <= self.window else Q(0)) + rows.append(output) + return GaussianMatrix(rows) + + def compile_root(self, indices, matrix): + n = self.spec.n + need(self.builder is None and indices == list(range(n)), "one exact Gaussian caller") + need(set(self.offsets) == set(range(-self.window, self.window + 1)), + "shared complete retained offset table") + expected = GaussianMatrix([[self.offsets[j - i]["value"] if abs(j - i) <= self.window else 0 + for j in range(n)] for i in range(n)]) + need(matrix == expected, "same prepared root at full-plan boundary") + self.root = Matrix(matrix) + self.coefficient_error = sum((row["error"] for row in self.offsets.values()), Q(0)) + self.kernel_error = self.tail_bound + self.coefficient_error + need(self.coefficient_error <= self.epsilon / 2 and self.kernel_error <= self.epsilon + and self.epsilon * self.qbar <= Q(1, 2), "Gaussian perturbation admission") + bound = allocation(n, n, 2 * self.qbar, self.spec.radius, self.spec.target + 1, + self.spec.incoming) + self.builder = Full(bound, self.raw) + self.plan = self.builder.plan(matrix, indices) + self.forward_error = bound["total_error"] + self.spec.radius * self.kernel_error + self.inverse_error = bound["total_error"] + 2 * self.qbar ** 2 * self.spec.radius * self.kernel_error + need(max(self.forward_error, self.inverse_error) < Q(1, 1 << self.spec.target), + "complete ideal Gaussian budget") + self.root_hash = vector_hash([z for row in self.root for z in row]) + return self.plan + + def compile(self): + need(self.builder is None, "Gaussian caller executes once") + need(self.namespace["n"] == self.spec.n and self.namespace["g"] == PiMultiple(self.spec.c) + and self.namespace["s"] == self.spec.shift, "same prepared caller parameters") + exec(compile(self.callsite, "", "exec", optimize=0), + self.namespace) + need(self.namespace["plan"] is self.plan and self.namespace["A"] == self.root, + "retained caller returns the certified root and plan") + return self.plan + + def record(self): + need(self.builder is not None and self.plan is not None, "compiled Gaussian receipt") + return dict(n=self.spec.n, c=str(self.spec.c), shift=str(self.spec.shift), + radius=str(self.spec.radius), target_bits=self.spec.target, + incoming_bits=self.spec.incoming, window=self.window, + conditioning_bits=self.conditioning_bits, + conditioning_interval=[str(Q(self.conditioning_box.lo, self.conditioning_box.scale)), + str(Q(self.conditioning_box.hi, self.conditioning_box.scale))], + ideal_inverse_bound=str(self.qbar), error_bits=self.error_bits, + coefficient_bits=self.coefficient_bits, working_bits=self.working_bits, + epsilon=str(self.epsilon), tail_bound=str(self.tail_bound), + coefficient_error=str(self.coefficient_error), kernel_error=str(self.kernel_error), + forward_error=str(self.forward_error), inverse_error=str(self.inverse_error), + root_sha256=self.root_hash, source_ast_sha256=self.source_ast, + callsite_ast_sha256=self.callsite_ast, + offsets=[dict(offset=u, **{key: str(value) for key, value in self.offsets[u].items()}) + for u in sorted(self.offsets)]) diff --git a/research/rational-unitary-fft/verify.py b/research/rational-unitary-fft/verify.py index 60ecb552d8..70d98f5514 100644 --- a/research/rational-unitary-fft/verify.py +++ b/research/rational-unitary-fft/verify.py @@ -33,7 +33,7 @@ def rejected(operation, label): raise ValueError("control unexpectedly passed: " + label) -def authenticate(include_generator=False, include_full=False): +def authenticate(include_generator=False, include_full=False, include_gaussian=False): rows = [r for r in json.loads((HERE / "SOURCES.json").read_text())["files"] if r["usage"] == "input"] raw = {} for row in rows: @@ -50,9 +50,17 @@ def authenticate(include_generator=False, include_full=False): "research/rational-unitary-fft/vendor/STEIN-PROVENANCE.json", } full = {"research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md"} + gaussian = { + "research/certified-gaussian-recovery/intervals.py", + "research/certified-gaussian-recovery/NOTICE.md", + "research/certified-gaussian-recovery/PROOF.md", + "research/capped-fusion-t74-k554/outer/check_gaugefree.py", + "research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md", + } return raw, [{"path": r["path"], "sha256": r["sha256"]} for r in rows if (include_generator or r["path"] not in added) - and (include_full or r["path"] not in full)] + and (include_full or r["path"] not in full) + and (include_gaussian or r["path"] not in gaussian)] def operator(plan, n): diff --git a/research/rational-unitary-fft/verify_full.py b/research/rational-unitary-fft/verify_full.py index ecee33aa15..882c068000 100644 --- a/research/rational-unitary-fft/verify_full.py +++ b/research/rational-unitary-fft/verify_full.py @@ -255,6 +255,7 @@ def run(): def main(): parser = argparse.ArgumentParser(description=__doc__) parser.add_argument("--write-result", action="store_true") + parser.add_argument("--with-gaussian", action="store_true") args = parser.parse_args() result = run() path = HERE / "FULL-RESULT.json" @@ -268,6 +269,9 @@ def main(): downsets=sum(len(x["downsets"]) for x in result["cases"]), physical_cuts=sum(len(x["physical_cuts"]) for x in result["cases"]), result_sha256=sha256(canonical(result)).hexdigest()), sort_keys=True)) + if args.with_gaussian: + from verify_gaussian import verify_result + verify_result(args.write_result) if __name__ == "__main__": diff --git a/research/rational-unitary-fft/verify_gaussian.py b/research/rational-unitary-fft/verify_gaussian.py new file mode 100644 index 0000000000..cf53315e22 --- /dev/null +++ b/research/rational-unitary-fft/verify_gaussian.py @@ -0,0 +1,209 @@ +"""Check the retained normalized Gaussian coefficient-to-circuit boundary. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +from exact import (C, Mat, Vec, apply, ceil_log2, component_l1, eye, multiply, + need, norm2, upper, vector_hash) +from full import admission +from gaussian import EVALUATOR, Gaussian, GaussianMatrix +from verify import authenticate, canonical, mv, operator, plan_hash, rejected +from verify_triangle import eliminate, frobenius_error, structure + + +def finer_reference(producer): + """Dense scalar reference; shares the declared interval backend, not Tsec.""" + spec = producer.spec + precision = producer.error_bits + 16 + ceil_log2(2 * spec.n) + bits = 2 * precision + 32 + table, errors = {}, [] + for u in range(1 - spec.n, spec.n): + factor = spec.c * ((Q(u) - spec.shift) ** 2 - spec.shift ** 2) + box = producer.exponential(factor, bits) + value = Q(box.lo, box.scale) + table[u] = value + errors.append(Q(box.hi - box.lo, box.scale)) + epsilon = sum(errors, Q(0)) + need(epsilon * producer.qbar <= Q(1, 2), "fine reference inverse admission") + matrix = Mat([[table[j - i] for j in range(spec.n)] for i in range(spec.n)]) + return matrix, epsilon, bits + + +def one_case(label, raw, n, c, shift, target=20, incoming=3, window=None): + vectors = [Vec(C(Q((3 * i + 1) % 7 - 3, 8), Q((5 * i + 2) % 9 - 4, 8)) + for i in range(n)), + Vec(C((-1) ** i * 32, i % 2) for i in range(n))] + if incoming > 3: + vectors.append(Vec([C(Q(1, 1 << incoming))] + [C()] * (n - 1))) + radius = max(Q(1), *(component_l1(v) for v in vectors)) + producer = Gaussian(raw, n, c, shift, radius, target, incoming, window) + plan = producer.compile() + inverse = producer.builder.inverse(plan) + bound = producer.builder.bound + lam, q, leading = admission(producer.root) + need(lam <= n and q <= 2 * producer.qbar, "separate exact leading-section admission") + need(all(producer.root[i][i] == 1 for i in range(n)), "exact normalized diagonal") + fine, fine_error, fine_bits = finer_reference(producer) + fine_inverse = eliminate(fine) + actual, actual_inverse = operator(plan, n), operator(inverse, n) + rational_inverse = eliminate(producer.root) + need(multiply(actual, actual_inverse) == eye(n), "Gaussian stored inverse identity") + forward = frobenius_error(actual, producer.root) + backward = frobenius_error(actual_inverse, rational_inverse) + need(max(forward, backward) <= n * bound["coefficient_error"] ** 2, + "Gaussian full circuit rational reference endpoints") + height = max(structure(plan, bound), structure(inverse, bound)) + states = [] + for direction, program, stored, root, reference, allowance, reference_error in ( + ("forward", plan, actual, producer.root, fine, producer.forward_error, fine_error), + ("inverse", inverse, actual_inverse, rational_inverse, fine_inverse, + producer.inverse_error, 2 * producer.qbar ** 2 * fine_error)): + for vector in vectors: + need(norm2(vector) <= radius ** 2, "Gaussian input radius") + need(all((part * (1 << incoming)).denominator == 1 + for z in vector for part in (z.re, z.im)), "declared Gaussian incoming grid") + output, stats = apply(program, vector, bound["data_bits"]) + root_error = norm2(output - mv(root, vector)) + rounding_error = norm2(output - mv(stored, vector)) + need(root_error <= bound["total_error"] ** 2 + and rounding_error <= bound["rounding_error"] ** 2, "Gaussian rational circuit errors") + # The component-l1 distance bounds the vector 2-norm. + ideal_error = component_l1(output - mv(reference, vector)) + reference_error * radius + need(ideal_error <= allowance and allowance < Q(1, 1 << target), + "complete rounded error against ideal Gaussian kernel") + need(stats["max_observed_height"] <= bound["temporary_bits"] + and stats["max_grid_numerator_bits"] + 1 <= bound["data_bits"] + bound["integer_bits"] + and stats["max_reduced_fractional_bits"] <= bound["data_bits"], + "Gaussian online width and temporary allowances") + states.append(dict(direction=direction, input_sha256=vector_hash(vector), + output_sha256=vector_hash(output), statistics=stats, + root_error_squared_upper=str(upper(root_error)), + rounding_error_squared_upper=str(upper(rounding_error)), + ideal_error_upper=str(upper(ideal_error)))) + record = producer.record() + record.update(label=label, layers=len(plan), gates=sum(map(len, plan)), + plan_sha256=plan_hash(plan), inverse_sha256=plan_hash(inverse), + full_source_ast_sha256=producer.builder.source_ast, + exact_leading_bounds=[dict(n=b, bound=str(value)) for b, value in leading], + fine_reference_bits=fine_bits, fine_reference_error=str(fine_error), + fine_reference_sha256=vector_hash([z for row in fine for z in row]), + budget={key: str(value) if isinstance(value, Q) else value + for key, value in bound.items()}, + coefficient_height=height, states=states) + return record + + +def controls(raw): + results = [] + for label, kwargs in ( + ("floating-gaussian-scale", dict(c=0.5)), + ("nonpositive-gaussian-scale", dict(c=0)), + ("outside-closed-half-shift", dict(shift=Q(5001, 10000))), + ("nondyadic-gaussian-size", dict(n=3)), + ("unpaid-truncation", dict(window=0)), + ("fractional-target-bits", dict(target=Q(3, 2))), + ("negative-incoming-grid", dict(incoming=-1))): + values = dict(n=2, c=Q(1, 2), shift=Q(1, 3), radius=1, target=16, incoming=0) + values.update(kwargs) + results.append(rejected(lambda: Gaussian(raw, **values), label)) + for label, before, after in ( + ("reversed-gaussian-offset", b"np.arange(n)[None,:]-np.arange(n)[:,None]", + b"np.arange(n)[:,None]-np.arange(n)[None,:]"), + ("omitted-normalization", b"-g*(u-s)**2+g*s*s", b"-g*(u-s)**2")): + changed = dict(raw) + need(raw[EVALUATOR].count(before) == 1, "unique Gaussian source mutation") + changed[EVALUATOR] = raw[EVALUATOR].replace(before, after) + results.append(rejected(lambda: Gaussian(changed, 2, 1, Q(1, 3), 1, 16).compile(), label)) + producer = Gaussian(raw, 2, 1, Q(1, 3), 1, 16) + original = producer.namespace["Tsec"] + def changed_root(*args): + matrix = original(*args) + matrix[0][0] = C(2) + return matrix + producer.namespace["Tsec"] = changed_root + results.append(rejected(producer.compile, "altered-root-with-old-coefficient-certificate")) + producer = Gaussian(raw, 2, 1, Q(1, 3), 1, 16) + producer.namespace["g"] = Q(1) + results.append(rejected(producer.compile, "omitted-symbolic-pi")) + producer = Gaussian(raw, 2, 1, Q(1, 3), 1, 16) + producer.coefficient_bits = 1 + results.append(rejected(producer.compile, "underfunded-coefficient-grid")) + producer = Gaussian(raw, 1, 1, 0, 1, 16) + plan = producer.compile() + results.append(rejected(producer.compile, "reused-stateful-caller")) + results.append(rejected(lambda: apply(plan, Vec([Q(1, 1 << 512)]), 20), + "discarded-gaussian-incoming-grid")) + results.append(rejected(lambda: GaussianMatrix([[1]]).astype(float), "floating-cast-tag")) + need(canonical({"window": 1}) != canonical({"window": True}), "typed Gaussian receipt") + return results + ["typed-gaussian-receipt"] + + +def run(): + raw, sources = authenticate(include_generator=True, include_full=True, include_gaussian=True) + fixtures = [ + ("normalized-scalar", 1, Q(1, 3), Q(1, 2), 20, 3, None), + ("broad-half-shift", 2, Q(1, 8), Q(1, 2), 20, 3, None), + ("positive-half-shift", 4, Q(1, 2), Q(1, 2), 20, 3, None), + ("negative-half-shift", 4, Q(1, 2), Q(-1, 2), 20, 3, None), + ("fine-incoming", 4, Q(2), Q(-3, 8), 20, 512, None), + ("truncated-narrow", 8, Q(2), Q(1, 3), 20, 3, None), + ("near-half-shift", 8, Q(1, 2), Q(499, 1000), 20, 3, None), + ("low-requested-precision", 2, Q(1), Q(1, 3), 8, 3, 1), + ("high-requested-precision", 2, Q(1), Q(1, 3), 36, 3, 1), + ] + records = [] + for label, *args in fixtures: + records.append(one_case(label, raw, *args)) + print("CHECK_GAUSSIAN_CASE", label, flush=True) + need(any(row["window"] < row["n"] - 1 and Q(row["tail_bound"]) > 0 for row in records), + "nonzero paid truncation witness") + need(any(Q(row["kernel_error"]) > 0 for row in records), "nonzero coefficient approximation") + need(records[-1]["coefficient_bits"] > records[-2]["coefficient_bits"], + "requested precision changes preparation") + probe = Gaussian(raw, 1, 1, 0, 1, 16) + enclosures = [] + for bits in (16, 38, 70): + for factor in (Q(0), Q(1, 10 ** 8), Q(1, 4), Q(3), Q(10 ** 6)): + box = probe.exponential(factor, bits) + need(0 <= box.lo <= box.hi <= box.scale, "exponential range") + if factor == 0: + need(box.lo == box.hi == box.scale, "exact-one exponential") + enclosures.append(dict(bits=bits, factor=str(factor), lo=box.lo, hi=box.hi)) + return dict(schema_version=1, scope="retained normalized Gaussian coefficients and full logical circuit", + sources=sources, cases=records, enclosures=enclosures, controls=controls(raw), + normalized_gaussian_gate_execution=True, complete_gaussian_execution=False, + raw_gauges=False, physical_tape_execution=False, universal_proof_kernel=False, + independent_transcendental_backend=False) + + +def verify_result(write=False): + result = run() + path = HERE / "GAUSSIAN-RESULT.json" + if write: + path.write_text(json.dumps(result, indent=2) + "\n") + else: + need(canonical(result) == canonical(json.loads(path.read_text())), "complete typed Gaussian receipt") + print(json.dumps(dict(status="PASS_NORMALIZED_GAUSSIAN_CIRCUIT", cases=len(result["cases"]), + gates=sum(row["gates"] for row in result["cases"]), + rounded_states=sum(len(row["states"]) for row in result["cases"]), + controls=len(result["controls"]), + result_sha256=sha256(canonical(result)).hexdigest()), sort_keys=True)) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-result", action="store_true") + verify_result(parser.parse_args().write_result) + + +if __name__ == "__main__": + main() From c9db4069d59bfad26a15fe000b91cc3a058684e5 Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 20:44:20 +0200 Subject: [PATCH 7/9] Certify raw Gaussian interiors against shared rounded references --- .../RAW-GAUSSIAN-PROOF.md | 470 ++++ .../RAW-GAUSSIAN-RESULT.json | 2443 +++++++++++++++++ research/rational-unitary-fft/README.md | 75 +- research/rational-unitary-fft/SOURCES.json | 8 +- research/rational-unitary-fft/TASK.json | 14 +- research/rational-unitary-fft/VERIFY.json | 16 +- .../gaussian_reference.py | 321 +++ research/rational-unitary-fft/raw_gaussian.py | 163 ++ research/rational-unitary-fft/verify.py | 13 +- research/rational-unitary-fft/verify_full.py | 5 + .../verify_raw_gaussian.py | 263 ++ 11 files changed, 3766 insertions(+), 25 deletions(-) create mode 100644 research/rational-unitary-fft/RAW-GAUSSIAN-PROOF.md create mode 100644 research/rational-unitary-fft/RAW-GAUSSIAN-RESULT.json create mode 100644 research/rational-unitary-fft/gaussian_reference.py create mode 100644 research/rational-unitary-fft/raw_gaussian.py create mode 100644 research/rational-unitary-fft/verify_raw_gaussian.py diff --git a/research/rational-unitary-fft/RAW-GAUSSIAN-PROOF.md b/research/rational-unitary-fft/RAW-GAUSSIAN-PROOF.md new file mode 100644 index 0000000000..1fb549d24a --- /dev/null +++ b/research/rational-unitary-fft/RAW-GAUSSIAN-PROOF.md @@ -0,0 +1,470 @@ +# A common Gaussian reference and anchored raw interior circuits + +Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance. +Apache-2.0. + +## 1. Domain, source geometry and scope + +Choose integers \(00. +\tag{4} +\] +The stored numerator is at least \(2^{b+2}\). I store the exact reciprocal +when needed; its relative error is at most \(2^{-b-1}\). +Both the value and reciprocal have numerator/denominator heights at most +\(P+1\). I use certified lower approximations rather than assume that +I can determine an irrational exponential's exact downward rounding. + +For an absolute error \(2^{-b}\), I use the earlier producer with output +grid \(b+1\) and working grid \(2(b+1)+32\). That producer may return zero +for a sufficiently small kernel entry. I use (3) for weights and gauges +that require a positive reciprocal. + +## 3. One fixed rounded reference + +Choose source halfwidth \(w\) with \(2w0\), the actual rational narrow reference has diagonal one, +row gap at least \(d\), and row norm at most \(2-d\). Hole identity +coordinates preserve this statement. If (13) is nonpositive, I return +no global conditioning certificate; the reference itself remains defined. + +All principal restrictions inherit the gap and the original circular +coordinates. Write such a restriction as \(I-E\). +Then \(\|E\|_\infty\le1-d\), and each column of \(E^k\) can reach at +most \(\min(T,2kW_N+1)\) physical positions. The Neumann series gives +\[ + \|(I-E)^{-1}\|_\infty\le d^{-1},\qquad + \|(I-E)^{-1}\|_1 + \le d^{-1}+2W_N(1-d)d^{-2}. +\tag{14} +\] +I keep the circular metric in this support argument; an arbitrary +linear reindexing would not preserve the same bandwidth. +Equation (14) concerns the narrow reference. I claim no arbitrary +principal inverse bound for the global broad reference. + +## 6. Anchored raw factors + +For an admitted interval \(I=[L,L+n)\), let +\(m=L+(n-1)/2\), \(\beta_m=\theta m-c_L\), and \(t_i=i-m\). +The narrow ideal restriction is +\[ + N_I=D T D^{-1},\quad + D_i=e^{\pi\alpha^2\theta t_i^2},\quad + c_T=\alpha^2\rho,\quad s_T=-\beta_m/\rho. +\tag{15} +\] +Here \(T_{ij}=e^{-\pi c_T[(j-i)^2-2s_T(j-i)]}\). + +For broad selected rows/columns, the target center is \(m+c_L\); +its centered offsets still equal \(t_i\). +Set \(\delta=\beta_m/\rho\), \(v=\lambda\theta/\alpha^2\). +The exact ideal block is \(A T B\), where +\[ + \begin{split} + c_T&=\lambda/\alpha^2,\quad s_T=-\delta(1-\theta),\\ + A_i&=\alpha^{-1} + e^{\pi[-\delta^2/\alpha^2+v(\lambda t_i^2+2\delta t_i)]},\\ + B_i&=e^{-\pi v(t_i^2+4\delta t_i)}. + \end{split} +\tag{16} +\] +Expanding the exponents gives +\(\alpha^{-1}e^{-(\pi/\alpha^2)(j-\lambda q_i)^2}\). +I verify that rational identity before compiling. +The normalized kernel already absorbs its shift normalization; +I add no further \(e^{-\gamma s_T^2}\) scalar. + +Both shifts lie in \([-1/2,1/2]\). +Let \(F_g\) bound the absolute rational coefficients multiplying \(\pi\) +in the two diagonals. The exact conservative allowance +\[ + G=\alpha\,2^{\lceil8F_g\rceil} +\tag{17} +\] +bounds \(A,B,A^{-1},B^{-1}\). +Use the normalized theorem's \(\overline Q\) and set +\(H=\max(n,\overline Q)\), \(Q_{\rm raw}=G^2H\). +Thus \(H\) bounds \(T,T^{-1}\), and \(Q_{\rm raw}\) bounds the inverse +of the ideal raw block. + +## 7. One stored forward/inverse program and one grid + +For target \(\tau=2^{-t}\), radius \(R\ge1\), choose +\[ + \delta_g=\frac{\tau}{24GH(R+1)},\quad + b_g=\left\lceil\log_2(4G/\delta_g)\right\rceil,\quad + R_*=2G(R+1),\quad + t_*=t+\lceil\log_2(16G)\rceil. +\tag{18} +\] +Generate each decay coefficient through (3) at relative precision \(b_g\), +and use its exact reciprocal for growth. Multiply exact rational +prefactors separately. By (4) and (17), both stored diagonal errors +and the errors of their exact reciprocals are at most \(\delta_g\). +For the narrow similarity I store one chirp and its reciprocal. + +Compile the normalized circuit at radius \(R_*\), target \(t_*\), and +one data grid +\[ + F\ge F_0,\quad + F\ge t+\lceil\log_2(24nGH)\rceil, +\tag{19} +\] +also satisfying the child's own required grid. Its incoming-grid +dependence is a maximum, so I can compute (19), declare it as the +child's incoming precision, and retain that same grid throughout. + +The complete forward program is +\(\widehat B;\mathcal C;\widehat A\). +For its inverse I invert the stored gates and reverse their order: +\(\widehat A^{-1};\mathcal C^{-1};\widehat B^{-1}\). +Each diagonal layer has rounding norm at most \(r=n2^{-F}\). +The rounded input diagonal fits radius +\((G+\delta_g)R+r\le R_*\). +For either direction, telescoping the three stages gives +\[ + E_{\rm raw} + \le2G E_{\rm norm}+3GH\delta_g R+3GHr<3\tau/8, +\tag{20} +\] +where \(E_{\rm norm}\le\tau/(16G)\). +This includes the input diagonal's error before the inverse amplifies it. + +Let \(E_{\rm ref}\) be the appropriate allowance in (11). Require +\[ + E_{\rm ref}\le + \frac{\tau}{8(R+1)Q_{\rm raw}^2}. +\tag{21} +\] +Otherwise the raw caller rejects that fixed reference. Equations (11), +(16), and the Neumann resolvent identity give +\(\|M_{\rm ref}^{-1}\|\le2Q_{\rm raw}\) and inverse comparison error +at most \(2Q_{\rm raw}^2E_{\rm ref}\). +Thus the complete forward/inverse errors against the same rational +reference restriction are bounded by +\[ + E_{\rm raw}+R E_{\rm ref}<\tau,\qquad + E_{\rm raw}+2R Q_{\rm raw}^2E_{\rm ref}<\tau. +\tag{22} +\] +I check the reference description and exact restriction again before +execution. A changed page cannot reuse the earlier preparation snapshot. + +### Uniform construction for a requested cap and accuracy + +I also supply `Reference.for_cap`, which chooses one reference before +any block requests. Given a dyadic cap \(B\), \(2B\le S\), radius \(R\) +and target \(t\), set +\[ + F_*=1+\alpha^2\theta(B^2+B+1),\quad + G_*=\alpha2^{\lceil8F_*\rceil},\quad + H_*=2\overline Q(B,1/(\alpha^2\rho)),\quad Q_*=G_*^2H_*. +\] +Equations (15)–(16), \(|t_i|\le B/2\) and \(|\beta_m|\le1/2\) show +that \(F_*\) bounds both raw exponent ranges. The factor-two margin +in \(H_*\) covers the normalized allowance for either family and any +\(n\le B\): each local allowance is at most \(4/3\) of its exact +finite-cap upper bound. Thus \(G_*,H_*,Q_*\) dominate the local budgets. + +Let +\[ + e=t+\left\lceil\log_2(8(R+1)Q_*^2)\right\rceil,\qquad + A=\alpha^2\rho^2. +\] +Choose integers +\[ + w=\left\lceil\sqrt{(e+5)/(3\alpha^2)}\right\rceil,\qquad + W=\max\{\lceil A\rceil,\lceil\sqrt{A(e+5)/3}\rceil,2w+2\}, +\] +and \(b_e=e+3+\lceil\log_2(2\max(w,W)+1)\rceil\). +I compute the square-root ceilings through integer arithmetic. +If \(2w\ge S\) or \(2W\ge T\), I reject the period/cap/accuracy request. + +For legal cuts, both geometric ratios are below \(\exp(-6)<1/64\). +The true tail numerators are at most \(2^{-e-5}\), since \(\pi>3\) +and \(\exp(1)>2\). The working-grid width is at most +\(2^{-b_e-23}\). Therefore each rational tail allowance is less than +\(2^{-e}/4\); each coefficient row/column allowance is at most +\(2^{-e}/8\). Consequently both errors in (11) are below \(2^{-e}\), +which satisfies (21) for all admitted blocks of size at most \(B\) +and radius at most \(R\). I check the actual rational inequalities +before returning the reference. + +The chosen broad width is +\(O(\alpha^2\rho^2+\alpha\rho\sqrt e+w)\). +In the later caller's regime, \(t+\log R=O(p)\), \(\alpha^2=O(\log p)\) +and \(\theta B^2\log p=o(1)\), so \(e=O(p)\), +\(w=O(\sqrt{p/\log p})\) and +\(W=O(\sqrt{p\log p}+\log p)\). +These widths fit the retained bandwidth order. The original axis +lengths must still pass the explicit period checks. + +## 8. Setup, online widths and remaining work + +Full-cell weights use +\[ + P_{\rm weight}=\lceil2\alpha^2/\theta\rceil+b_e+7 + =O(b_e+\alpha^2/\theta). +\tag{23} +\] +Under the later caller's parameters this is \(O(b_e+p^2\log p)\). +If \(a\le16\lceil\log_2p\rceil\), \(\theta\ge1/(4p^2)\), \(p\ge2\), +and \(b_e=Cp\), its retained setup allowance +\(Cp+64p^2\lceil\log_2p\rceil\) exceeds (23). +The kernel grid (6) adds \(O(\alpha^2w)\), which is \(o(p)\) at the +intended bandwidth. Increasing the fixed precision constant absorbs +that allowance. + +I keep full-cell weights and reference ratios in offline preparation. +Online gauges use only the anchored exponent range \(F_g\). +Combining (17)–(19) with the completed normalized theorem gives +\[ + O(F_0+t+\log n+\log R+1/c_T+F_g+\log\alpha) +\tag{24} +\] +bits for coefficients, data records and gate temporaries. +I add two parallel scalar layers and \(2n\) gates. +For internal headroom I use the child's bound at radius \(R_*\) and +the final bound \((G+1)(HR_*+E_{\rm norm})+r\). +The verifier checks the resulting signed-grid and temporary allowances. + +For capped blocks in the later parameter range, \(F_g\) is bounded by +the anchored quadratic/linear range, while \(1/c_T=O(\log p)\) on the +broad side. These are local online bounds. I leave the carve, complete +Schur/boundary program, coefficient delivery, simultaneous tensor layout +and tape costs outside this receipt. The broad boundary Schur factor +requires application; the narrow inverse caller needs its separate solve. + +## 9. Verification and attribution + +I check a complete finite periodic reference against three explicit +windings plus analytic bounds for omitted windings, in both induced norms. +I check actual row gaps, a principal inverse crossing a phase boundary +and a hole, retained cuts, physical support and overlapping pages. +The exponential comparisons share Mason's declared interval backend. +The written series bounds cover the infinite tails. + +I execute broad/narrow raw blocks with positive and negative center +phases, reverse the same stored program, and compare rounded outputs +with exact rational reference matrices and separate inverse elimination. +Controls cover underfunded tails/coefficients, zero setup weights, +stale pages, hole completion, normalization and inverse wrapper order. +Increasing reference setup precision changes the reference while +leaving the independently allocated online circuit unchanged. + +I credit Vlado Vojdanovski for phase-cell references, q-LDU and circuits; +Jeremy Kothe/IdeaOuter for shifted raw identities; Mason Hale for interval +arithmetic and earlier Gaussian execution; Chafik Boukhalfa for phases +and fine-prime setup work; Cinna Davis for the later caller and comparison +contract; and classical Neumann-series and diagonal-scaling methods. +Internal agents belong to our author process. Independent reviewers +must assess the universal proof separately from the finite receipts. +I retain ToikY's separate scalar-backend audit in +[PR4220](https://github.com/CrocSwap/integer-mult-bounds/pull/4220), +head `5cf23ba47a28b78a98eb0665177b5030e65c01b9`, at our completed +normalized source `66b95cc916552f487203be14fd99a7aea0bdd78e`. +That audit excludes the raw/reference argument presented here. diff --git a/research/rational-unitary-fft/RAW-GAUSSIAN-RESULT.json b/research/rational-unitary-fft/RAW-GAUSSIAN-RESULT.json new file mode 100644 index 0000000000..c937b461e9 --- /dev/null +++ b/research/rational-unitary-fft/RAW-GAUSSIAN-RESULT.json @@ -0,0 +1,2443 @@ +{ + "schema_version": 1, + "scope": "common periodic Gaussian reference and raw capped interior circuits", + "sources": [ + { + "path": "research/unconditional-batched-crt/GAUSSIAN.md", + "sha256": "2f63c91c58f2d1cf31294812c75970ea390c61866a8da7ed700d4316a033a242" + }, + { + "path": 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"error_squared_upper": "362320712784518463787/1461501637330902918203684832716283019655932542976" + }, + { + "direction": "inverse", + "input_sha256": "2c8dad3fec566a91b5122bc30bab9dd4c6cc69b5ac54d5431141a718eabe4526", + "output_sha256": "3983898c161418659a67fa51e34f88696c915c7393c7dd7b04cc3e7839f9422d", + "statistics": { + "max_observed_height": 1158, + "max_grid_numerator_bits": 516, + "max_reduced_fractional_bits": 512, + "rounded_outputs": 248 + }, + "error_squared_upper": "86279865314717762568795/730750818665451459101842416358141509827966271488" + }, + { + "direction": "inverse", + "input_sha256": "39adbc13169e5aecb8c72f64164ecd93942813ae52b72061a5df6221c78a14b4", + "output_sha256": "543aa7003af732cba87fbc2a5888aae4de2eb4b8cb21db0bc33df3faff516011", + "statistics": { + "max_observed_height": 1155, + "max_grid_numerator_bits": 2, + "max_reduced_fractional_bits": 512, + "rounded_outputs": 248 + }, + "error_squared_upper": "1/1461501637330902918203684832716283019655932542976" + } + ] + } + ], + "controls": [ + "interior-crosses-period-cut", + "interior-crosses-original-cell", + "ambiguous-periodic-lift", + "floating-reference-parameter", + "period-too-short-for-uniform-reference-accuracy", + "unpaid-common-reference-tail", + "underfunded-common-reference-coefficients", + "changed-reference-after-preparation", + "zero-weight-at-online-precision", + "underfunded-positive-weight-grid", + "uncertified-global-row-gap", + "erased-broad-hole-output", + "forced-unit-broad-diagonal", + "reference-setup-precision-separate-from-online-gates", + "wrong-inverse-diagonal-order", + "wrong-raw-exponential-constant", + "typed-raw-receipt" + ], + "raw_interior_gate_execution": true, + "complete_gaussian_execution": false, + "physical_tape_execution": false, + "boundary_schur_execution": false, + "independent_transcendental_backend": false, + "universal_proof_kernel": false +} diff --git a/research/rational-unitary-fft/README.md b/research/rational-unitary-fft/README.md index cfee76200f..664a630d90 100644 --- a/research/rational-unitary-fft/README.md +++ b/research/rational-unitary-fft/README.md @@ -1,4 +1,29 @@ -# Certified normalized Gaussian calls through full Toeplitz circuits +# Raw Gaussian interiors against a shared rounded reference + +I construct fixed rational narrow and broad references for a periodic +source/target axis, retaining original phase cells, the period cut and +their different hole-completion rules. I execute anchored raw +constant-phase block restrictions through the normalized circuit below, +including the inverse of the same stored program. +I pay positive-weight setup, kernel tails, diagonal coefficients, circuit +rounding and comparison with that same reference. +[RAW-GAUSSIAN-PROOF.md](RAW-GAUSSIAN-PROOF.md) gives the all-size bounds. + +For target \(2^{-t}\), radius \(R\), incoming grid \(F_0\), normalized +scale \(c\), and anchored exponent range \(F_g\), I obtain online widths +\[ +O(F_0+t+\log n+\log R+1/c+F_g+\log\alpha). +\] +I keep full-cell weights in offline setup at +\(O(b_e+\alpha^2/\theta)\) bits. I also compute a row-gap certificate for +the same narrow reference; a positive gap gives dimension-independent +row/column inverse bounds for its principal restrictions in the original +circular coordinates. I leave the carve, boundary program, physical +delivery and complete tensor/tape execution open. +`Reference.for_cap` chooses one global cutoff and coefficient precision +for a declared block cap, radius and accuracy. I reject periods too short +for those cuts. Coordinate and coefficient queries avoid enumerating the +whole axis; the verifier includes a window on an axis of size \(10^{20}+1\). I connect the retained normalized Gaussian `Tsec` function and its `full_plan` call to requested-precision rational coefficients and the @@ -14,8 +39,8 @@ quantization, and pass the same rational root to the full compiler. I preserve the diagonal ones and both closed half-shift endpoints. [GAUSSIAN-PROOF.md](GAUSSIAN-PROOF.md) gives the interval termination, conditioning, perturbation and composed error arguments. -Raw Gaussian gauges, common-reference assembly and physical delivery -require further work. +The raw extension above composes this normalized contract with its +diagonal/scalar factors and reference comparison. I execute the retained full Toeplitz recursion and its stored inverse, using the rational generator, triangular and FFT circuits below. @@ -121,9 +146,17 @@ python3.12 -B scripts/research_package.py \ I declare eight normal/optimized commands, with limits of 60, 60, 90, 90, 150, 150, 300 and 300 seconds, totaling 1,200 seconds. -In each final slot I run the full verifier followed by the Gaussian -verifier in the same process. I preserve the full receipt and compare -the separate [GAUSSIAN-RESULT.json](GAUSSIAN-RESULT.json) receipt. +In each final slot I run the full, normalized Gaussian and raw Gaussian +verifiers sequentially in the same process. I preserve the full and +[GAUSSIAN-RESULT.json](GAUSSIAN-RESULT.json) receipts and compare the +separate [RAW-GAUSSIAN-RESULT.json](RAW-GAUSSIAN-RESULT.json) receipt. +For the raw extension I check 2,178 entries across two complete +33-by-33 references, with three explicit windings and analytic tails. +I check row gaps, a principal inverse across a cell boundary and a hole, +shared overlapping pages and period cuts. Six raw-circuit fixtures +exercise 1,086 gates and 26 rounded states; 17 controls pass. +The raw scientific receipt is +`f9c2bfc6b61ca8a52c403d912d2f687beb3641955cec71d3879ab017b067d6f7`. I check nine Gaussian cases, 4,527 gates and 38 rounded forward/inverse states, including both half shifts, paid truncation and a 512-bit incoming vector. I compare finer scalar interval references and exact @@ -266,6 +299,28 @@ I found no competing normalized coefficient-to-circuit claim in that coverage. I compared PR4212's P112 audit and PR4208's CRT stencil work; they address different native/physical obligations. +I preserve normalized revision `66b95cc916552f487203be14fd99a7aea0bdd78e` +and its [hosted evidence](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39470174495). +ToikY's [PR4220](https://github.com/CrocSwap/integer-mult-bounds/pull/4220) +at `5cf23ba47a28b78a98eb0665177b5030e65c01b9` supplies a separate +implementation and written audit of that revision's scalar enclosure, +width and precision rules. The author reports 128 independent intervals +covering 186 source observations. I retain that scoped credit; its +coverage excludes the full circuit, matrix-inverse theorem and this +raw/reference extension. Our verifier's own comparisons still share +Mason's backend. +For the raw extension I use the retained narrow weight-ratio reference, +while I choose a fixed entrywise broad reference and prove its comparison. +I do not identify it with the older factorwise-rounded broad matrix. +The block admission checks constant phase and the original cut; it does +not implement the outer carving algorithm. +Before this extension I refreshed unchanged main, checked 15 recent PR +records and 6/790 hub comments, and compared PR4214's GL3 audit and +PR4208's CRT stencil work. I found no overlapping raw-reference claim +in that coverage. Broader history and unpublished work remain outside it. +Before publication I checked 12 newer PR records, 5/815 hub comments +and the new scalar-audit handoff while canonical main remained unchanged. + I preserve the first completed revision `f02a4ab5ab70a8af80415ac35c5b48cd5a93a475` and its FFT receipt `2e203e42d3d617e828e3341ffeb44254c931db2aebb8a040d04d67338b5b4016`. @@ -293,8 +348,8 @@ Apache-2.0. ## Open next questions -In [TASK.json](TASK.json) I specify the next steps and blockers: incorporate -the raw Gaussian gauges and scalar factors, compare with the common rounded -reference, and connect the coefficients and data widths to physical delivery. -Reviewers still need to assess the universal proofs and complete setup, +In [TASK.json](TASK.json) I specify the next steps and blockers: bind the +actual interior/boundary carve and Schur pages to these shared references, +then connect the coefficients and data widths to physical delivery. +Reviewers still need to assess the universal proofs, complete setup, fixed-tape lowering and the full multiplication construction. diff --git a/research/rational-unitary-fft/SOURCES.json b/research/rational-unitary-fft/SOURCES.json index 32540a4003..0fe6eb0e50 100644 --- a/research/rational-unitary-fft/SOURCES.json +++ b/research/rational-unitary-fft/SOURCES.json @@ -20,6 +20,12 @@ {"path":"research/certified-gaussian-recovery/NOTICE.md","usage":"input","git_blob":"5e3269b5afe5e11f044ffae45216c4731b42c7af","sha256":"10acae9e78402ca70c6afd0e8d6da8fefc0da519c646cf43d93f923aba85b86d"}, {"path":"research/certified-gaussian-recovery/PROOF.md","usage":"input","git_blob":"9dace009004a4336d6ce57764c255190fbf87e64","sha256":"65480dd23fb1aae314365aeea4aacbc146e160e5c8a7a4ac4f602f22ae4581de"}, {"path":"research/capped-fusion-t74-k554/outer/check_gaugefree.py","usage":"input","git_blob":"3fbb4a1b2acc3d5120d93d2ffd7cd5007d4f7f0e","sha256":"5274b2838f322dc56a88570e4f9262fcd3857cd387160ad138e083ccdfbe9761"}, - {"path":"research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md","usage":"input","git_blob":"59d9128d75c57b790e6e602da8b2ab9c2cf7fa57","sha256":"c6a032ed495325806ddf491c5c85bc822a7fdd613aff906a7be553960d0f85a3"} + {"path":"research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md","usage":"input","git_blob":"59d9128d75c57b790e6e602da8b2ab9c2cf7fa57","sha256":"c6a032ed495325806ddf491c5c85bc822a7fdd613aff906a7be553960d0f85a3"}, + {"path":"research/unconditional-batched-crt/PROOF.md","usage":"input","git_blob":"406eb561cc8d6041ea3fba1a849653b654f2198c","sha256":"689fc7b01c5f04f77019ff02d1b79cb4f7020bd98e05b1052020bad295dec060"}, + {"path":"research/unconditional-batched-crt/vendor/FINE-PRIME.md","usage":"input","git_blob":"bf9899ee21be4058e4aac0b64bfa67d01ee8ddd3","sha256":"47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce"}, + {"path":"research/capped-phase-fusion/BILATERAL-LDU.md","usage":"input","git_blob":"5cad6ee19f60d626beb9f6d7be78cbc22fdd08c6","sha256":"0d65e66d227464dda1a19f781ae0a0be671cbfd347c41e37ec954f3d629d8a77"}, + {"path":"research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md","usage":"input","git_blob":"3bcf490a8c164d44aff5b7e2ec2e0f4bf07d7a73","sha256":"e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272"}, + {"path":"research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md","usage":"input","git_blob":"bbeeee7431720817f031360f67b2186abac5cd23","sha256":"54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a"}, + {"path":"research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md","usage":"input","git_blob":"0f7f80b3f74d469c2f72f3b9f338d32568b4eae3","sha256":"e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0"} ] } diff --git a/research/rational-unitary-fft/TASK.json b/research/rational-unitary-fft/TASK.json index cc9318ffb6..02be43ed22 100644 --- a/research/rational-unitary-fft/TASK.json +++ b/research/rational-unitary-fft/TASK.json @@ -38,21 +38,21 @@ ] }, "scope_status": "refined", - "actual_scope": "Execute the retained normalized Gaussian Tsec/full_plan caller with requested-precision rational coefficients, closed-half-shift leading-section conditioning and complete forward/inverse ideal-kernel error and online-width bounds.", - "scope_change_reason": "The completed full circuit requires conditioning for the same rational root it receives. I connect the retained Gaussian evaluator to that boundary, rederive the finite-cap bound, and pay truncation, quantization and circuit errors while preserving four prior receipts.", + "actual_scope": "Construct shared rounded periodic narrow/broad references and execute anchored raw constant-phase interior restrictions and their stored inverses, with positive-weight setup, global tail/row-gap bounds and complete reference-error/online-width budgets.", + "scope_change_reason": "The normalized circuit omits raw diagonal/scalar factors and the fixed-reference comparison. I preserve five earlier receipts and supply those factors and their complete error composition, with source geometry, distinct hole rules, original period cuts and shared pages.", "bottleneck_link": "The Gaussian caller needs full circuit execution and a quantified conditioning/rounding contract before physical delivery and the complete numerical composition can be admitted.", "budget_minutes": 120, "stop_rule": "Stop at an invalid phase enclosure, nonunitary adapted gate, missing multiplier/twist error, understated word width or source-schedule mismatch. Do not equate coefficient-level unitarity with exact rounded restoration or physical execution.", "outcome": "completed", - "evidence": "I preserve the completed FFT, triangle, generator and full proofs/receipts. GAUSSIAN-PROOF.md derives the interval width/termination, finite-cap conditioning and complete normalized-kernel error bounds. GAUSSIAN-RESULT.json records 9 cases, 4527 gates, 38 rounded states and 16 controls. I execute Tsec and its adjacent full_plan call, compare finer scalar interval references sharing the declared backend, and compute exact leading inverses. Scientific acceptance remains separate.", + "evidence": "I preserve the five completed scientific receipts. RAW-GAUSSIAN-PROOF.md derives positive relative coefficients, uniformly allocated common periodic references, both tail norms, a narrow row-gap certificate and complete raw forward/inverse reference-error bounds. RAW-GAUSSIAN-RESULT.json records two complete 33x33 references, 6 circuit fixtures, 1086 gates, 26 rounded states and 17 controls, plus a local query on a source axis of size10^20+1 without axis enumeration. Comparisons share the declared interval backend; scientific acceptance remains separate.", "author": "Jose Manuel (jmbarrancoml)", "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, source analysis, implementation and writing, including source-analysis agents. No independent human or formal-kernel acceptance.", "open_questions": [ { - "question": "Can the normalized circuit contract cover the raw Gaussian gauges and common rounded reference?", - "why_it_matters": "The actual broad and narrow callers include diagonal and scalar factors outside Tsec. Those factors change input radii, inverse sensitivity and error propagation.", - "first_step": "Bind one retained raw broad or narrow block formula to certified gauge coefficients, compose its rounded forward/inverse calls and compare with the same common Gaussian reference.", - "blocked_by": "A complete raw-factor preparation and error allocation, including boundary factors that the caller applies without inversion." + "question": "Can the actual carve and five-factor boundary word use the same common reference and raw interior programs?", + "why_it_matters": "Correct constant-phase restrictions do not establish separated interiors, shared boundary pages or the complete Gaussian map. Broad boundary Schur factors require application; narrow inverse execution needs a solve.", + "first_step": "Construct the retained separated dyadic carve with original cuts and holes, derive its exact Schur and coupling pages from Reference, and bind the applied broad and solved narrow boundary stages to the same forward/inverse error ledger.", + "blocked_by": "Carve and boundary-program integration, followed by physical setup, routing and tape costs." }, { "question": "Can the coefficient preparation and canonical data widths be joined to the physical Gaussian caller?", diff --git a/research/rational-unitary-fft/VERIFY.json b/research/rational-unitary-fft/VERIFY.json index f0d5c70c62..672ac0c05f 100644 --- a/research/rational-unitary-fft/VERIFY.json +++ b/research/rational-unitary-fft/VERIFY.json @@ -20,7 +20,13 @@ "research/certified-gaussian-recovery/NOTICE.md", "research/certified-gaussian-recovery/PROOF.md", "research/capped-fusion-t74-k554/outer/check_gaugefree.py", - "research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md" + "research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md", + "research/unconditional-batched-crt/PROOF.md", + "research/unconditional-batched-crt/vendor/FINE-PRIME.md", + "research/capped-phase-fusion/BILATERAL-LDU.md", + "research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md", + "research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md", + "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md" ], "commands": [ { @@ -54,14 +60,14 @@ "checks": "Repeat the complete generator program with source assertions enabled under optimized Python and compare its full typed receipt." }, { - "argv": ["{python}", "-I", "-B", "{package}/verify_full.py", "--with-gaussian"], + "argv": ["{python}", "-I", "-B", "{package}/verify_full.py", "--with-gaussian", "--with-raw"], "timeout_seconds": 300, - "checks": "Execute the retained full Toeplitz recursion and stored inverse, check leading-section admission, legal downsets/frontiers and active fragments, and compare exact endpoints plus rounded global error/width budgets. Then execute the retained normalized Gaussian Tsec/full_plan caller with certified coefficients and truncation, compare ideal forward/inverse references and preserve both complete receipts." + "checks": "Execute the full Toeplitz verifier, then normalized Gaussian verification, preserving both receipts. Then check one periodic rounded reference, holes/cuts/overlaps and row-gap admission, execute anchored raw broad/narrow programs and their stored inverses, and compare rounded errors and widths against the same reference restrictions." }, { - "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_full.py", "--with-gaussian"], + "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_full.py", "--with-gaussian", "--with-raw"], "timeout_seconds": 300, - "checks": "Repeat full recursive and normalized Gaussian execution sequentially under optimized Python with authenticated source assertions, conditioning/schedule/orientation/stale-certificate controls and both complete typed receipts." + "checks": "Repeat full, normalized and raw Gaussian verification sequentially under optimized Python. Preserve source assertions, compare three complete typed receipts, and reject precision, inverse-wrapper, stale-reference, hole-completion and period/cell controls." } ] } diff --git a/research/rational-unitary-fft/gaussian_reference.py b/research/rational-unitary-fft/gaussian_reference.py new file mode 100644 index 0000000000..92addfabde --- /dev/null +++ b/research/rational-unitary-fft/gaussian_reference.py @@ -0,0 +1,321 @@ +"""One rounded periodic Gaussian reference, with original cells and hole rules. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. Geometry/reference: Vlado, Jeremy Kothe and Cinna Davis. +Outward interval arithmetic: Mason Hale. +""" +from dataclasses import dataclass +from fractions import Fraction as Q +from hashlib import sha256 +import json +from math import isqrt + +from exact import Mat, ceil_log2, need +from gaussian import INTERVALS, load_intervals, rational + + +def ceiling(x): + x = rational(x) + return -(-x.numerator // x.denominator) + + +def nearest_up(x): + x = rational(x) + Q(1, 2) + return x.numerator // x.denominator + + +def ceiling_sqrt(x): + x = rational(x) + need(x >= 0, "nonnegative exact square root bound") + root = isqrt(x.numerator // x.denominator) + return root + int(root * root < x) + + +class Exponentials: + def __init__(self, raw): + self.oracle = load_intervals(raw[INTERVALS]) + + def decay_box(self, factor, bits): + factor = rational(factor) + need(factor >= 0 and type(bits) is int and bits >= 16, "decay domain") + if factor == 0: + return self.oracle.Interval.rational(1, bits) + box = self.oracle.exp_negative(self.oracle.pi_interval(bits) * factor) + need(Q(box.hi - box.lo, box.scale) <= 512 * bits ** 2 * Q(1, 1 << bits), + "retained decay width") + return box + + def absolute(self, factor, error_bits): + need(type(error_bits) is int and error_bits >= 1, "absolute exponential precision") + p = error_bits + 1 + box = self.decay_box(factor, 2 * p + 32) + result, error = self.oracle.inward_quantize( + (box, self.oracle.Interval.rational(0, box.bits)), p) + value = Q(result[0], 1 << p) + need(result[1] == 0 and 0 <= value <= 1 and error <= Q(1, 1 << error_bits), + "absolute decay receipt") + return value, error + + def relative(self, factor, exponent_bound, relative_bits, precision=None): + factor, exponent_bound = rational(factor), rational(exponent_bound) + need(0 <= factor <= exponent_bound and type(relative_bits) is int and relative_bits >= 1, + "relative positive exponential domain") + length = ceiling(8 * exponent_bound) + required = length + relative_bits + 3 + p = required if precision is None else precision + need(type(p) is int and p >= required, "positive weight setup precision") + box = self.decay_box(factor, 2 * p + 32) + step = 1 << (box.bits - p) + value = Q(box.lo // step, 1 << p) + absolute_error = max(abs(value - Q(box.lo, box.scale)), abs(value - Q(box.hi, box.scale))) + relative_error = absolute_error * (1 << length) + need(value > 0 and relative_error <= Q(1, 1 << (relative_bits + 2)), + "positive relative weight receipt") + return dict(value=value, absolute_error=absolute_error, relative_error=relative_error, + precision=p, working_bits=box.bits, exponent=factor, + lower=Q(box.lo, box.scale), upper=Q(box.hi, box.scale)) + + def geometric_tail(self, first, step, scale, bits): + numerator, ratio = self.decay_box(first, bits), self.decay_box(step, bits) + denominator = 1 - Q(ratio.hi, ratio.scale) + need(denominator > 0, "positive Gaussian tail denominator") + return rational(scale) * Q(numerator.hi, numerator.scale) / denominator + + +@dataclass(frozen=True) +class ReferenceSpec: + source_size: int + excess: int + alpha: int + source_halfwidth: int + target_halfwidth: int + entry_bits: int + + +@dataclass(frozen=True) +class Entry: + value: Q + kind: str + lift: int + retained_error: Q + + +class Coordinates: + def __init__(self, size, function): + self.size, self.function = size, function + + def __getitem__(self, key): + if isinstance(key, slice): + return tuple(self.function(j) for j in range(*key.indices(self.size))) + need(type(key) is int, "integer coordinate index") + key = key + self.size if key < 0 else key + if not 0 <= key < self.size: + raise IndexError(key) + return self.function(key) + + def __iter__(self): + return (self.function(j) for j in range(self.size)) + + def __len__(self): + return self.size + + +class SelectedAddresses: + def __init__(self, selected, source_size, target_size): + self.selected, self.source_size, self.target_size = selected, source_size, target_size + + def get(self, coordinate): + candidate = nearest_up(Q(self.source_size * coordinate, self.target_size)) + if 0 <= candidate < self.source_size and self.selected[candidate] == coordinate: + return candidate + return None + + +class Reference: + def __init__(self, raw, source_size, excess, alpha, source_halfwidth, + target_halfwidth, entry_bits): + need(all(type(x) is int for x in (source_size, excess, alpha, source_halfwidth, + target_halfwidth, entry_bits)), "integer reference parameters") + need(0 < excess < source_size and alpha >= 1 and entry_bits >= 8, + "Gaussian reference geometry") + target_size = source_size + excess + need(0 <= 2 * source_halfwidth < source_size and 0 <= 2 * target_halfwidth < target_size, + "unique retained periodic lifts") + self.spec = ReferenceSpec(source_size, excess, alpha, source_halfwidth, + target_halfwidth, entry_bits) + self.allocation = None + self.theta, self.rho, self.lam = Q(excess, source_size), Q(target_size, source_size), Q(source_size, target_size) + self.target_size = target_size + self.phases = Coordinates(source_size, lambda j: nearest_up(self.theta * j)) + self.selected = Coordinates(source_size, lambda j: j + self.phases[j]) + need(self.selected[0] == 0 and self.selected[-1] < target_size, "selected endpoint range") + self.source_at = SelectedAddresses(self.selected, source_size, target_size) + self._holes = None + self.betas = Coordinates(source_size, lambda j: self.theta * j - self.phases[j]) + self.exp = Exponentials(raw) + self.weight_exponent_bound = Q(alpha ** 2, 4) / self.theta + self.weight_relative_bits = entry_bits + 4 + self.weight_precision = ceiling(8 * self.weight_exponent_bound) + self.weight_relative_bits + 3 + self.ratio_log_bound = ceiling(8 * alpha ** 2 * source_halfwidth) + self.kernel_error_bits = entry_bits + self.ratio_log_bound + 4 + self.weights, self.kernel, self.entries = {}, {}, {} + work = 2 * (entry_bits + ceiling(8 * alpha ** 2 * self.rho ** 2) + 16) + 32 + w, width = source_halfwidth, target_halfwidth + self.narrow_tail = self.exp.geometric_tail(alpha ** 2 * w * (w + 1), + 2 * alpha ** 2 * (w + 1), 2, work) + broad_c = 1 / (alpha ** 2 * self.rho ** 2) + self.broad_tail = self.exp.geometric_tail(broad_c * (width + Q(1, 2)) ** 2, + 2 * broad_c * (width + 1), Q(2, alpha), work) + self.entry_error = Q(1, 1 << entry_bits) + self.errors = { + "narrow": self.narrow_tail + (2 * w + 1) * self.entry_error, + "broad": self.broad_tail + (2 * width + 1) * self.entry_error, + } + adjacent = self.exp.decay_box(2 * alpha ** 2 * self.theta, work) + distant = self.exp.decay_box(2 * alpha ** 2, work) + self.narrow_gap = (1 - Q(adjacent.hi, adjacent.scale) + - 5 * Q(distant.hi, distant.scale) - self.errors["narrow"]) + self.narrow_inverse_bounds = None + if self.narrow_gap > 0: + width = 2 * source_halfwidth + 2 + self.narrow_inverse_bounds = ( + 1 / self.narrow_gap, + 1 / self.narrow_gap + 2 * width * (1 - self.narrow_gap) / self.narrow_gap ** 2) + description = {key: getattr(self.spec, key) for key in self.spec.__dataclass_fields__} + self.description_hash = sha256(json.dumps(description, sort_keys=True).encode()).hexdigest() + + @property + def holes(self): + if self._holes is None: + self._holes = tuple(k for k in range(self.target_size) if self.source_at.get(k) is None) + return self._holes + + @classmethod + def for_cap(cls, raw, source_size, excess, alpha, cap, radius, target): + need(all(type(x) is int for x in (source_size, excess, alpha, cap, target)) + and 0 < excess < source_size and alpha >= 1 and cap > 0 + and cap & (cap - 1) == 0 and 2 * cap <= source_size and target >= 1, + "uniform reference cap and geometry") + radius = rational(radius) + need(radius >= 1, "uniform reference radius") + theta, rho = Q(excess, source_size), Q(source_size + excess, source_size) + from gaussian import Gaussian + exponent = 1 + alpha ** 2 * theta * (cap ** 2 + cap + 1) + gauge = alpha * (1 << ceiling(8 * exponent)) + middle = 2 * Gaussian(raw, cap, 1 / (alpha ** 2 * rho), 0, 1, 1).qbar + inverse = gauge ** 2 * middle + error_bits = target + ceil_log2(8 * (radius + 1) * inverse ** 2) + source_width = ceiling_sqrt(Q(error_bits + 5, 3 * alpha ** 2)) + a = alpha ** 2 * rho ** 2 + target_width = max(ceiling(a), ceiling_sqrt(a * (error_bits + 5) / 3), + 2 * source_width + 2) + entry_bits = error_bits + 3 + ceil_log2(2 * max(source_width, target_width) + 1) + result = cls(raw, source_size, excess, alpha, source_width, target_width, entry_bits) + need(max(result.errors.values()) <= Q(1, 1 << error_bits), "uniform cap reference error") + result.allocation = dict(cap=cap, radius=str(radius), target_bits=target, + gauge_exponent_bound=str(exponent), gauge_bound=str(gauge), + normalized_bound=str(middle), inverse_bound=str(inverse), + error_bits=error_bits) + return result + + def weight(self, j): + need(type(j) is int and 0 <= j < self.spec.source_size, "global weight index") + if j not in self.weights: + factor = self.spec.alpha ** 2 * (Q(1, 4) - self.betas[j] ** 2) / self.theta + self.weights[j] = self.exp.relative(factor, self.weight_exponent_bound, + self.weight_relative_bits, self.weight_precision) + return self.weights[j]["value"] + + def kernel_value(self, h): + h = abs(h) + need(h <= self.spec.source_halfwidth, "retained unshifted source offset") + if h not in self.kernel: + self.kernel[h] = self.exp.absolute(self.spec.alpha ** 2 * self.rho * h ** 2, + self.kernel_error_bits) + return self.kernel[h][0] + + def _narrow(self, row, column): + i, j = self.source_at.get(row), self.source_at.get(column) + if i is None or j is None: + return Entry(Q(int(row == column)), "hole-identity", 0, Q(0)) + if i == j: + return Entry(Q(1), "unit-diagonal", 0, Q(0)) + source_size = self.spec.source_size + lift = -nearest_up(Q(j - i, source_size)) + h = j + lift * source_size - i + if abs(h) > self.spec.source_halfwidth: + return Entry(Q(0), "omitted", lift, Q(0)) + if lift == 0 and self.phases[i] == self.phases[j]: + value = self.weight(i) * self.kernel_value(h) / self.weight(j) + return Entry(value, "shared-weight-ratio", lift, self.entry_error) + factor = self.spec.alpha ** 2 * ((self.rho * h + self.betas[i]) ** 2 - self.betas[j] ** 2) + value, error = self.exp.absolute(factor, self.spec.entry_bits + 2) + return Entry(value, "cross-period" if lift else "cross-cell", lift, error) + + def _broad(self, row, column): + j = self.source_at.get(column) + if j is None: + return Entry(Q(int(row == column)), "hole-column", 0, Q(0)) + lift = -nearest_up(Q(column - row, self.target_size)) + distance = column + lift * self.target_size - row + if abs(distance) > self.spec.target_halfwidth: + return Entry(Q(0), "omitted", lift, Q(0)) + factor = (j + lift * self.spec.source_size - self.lam * row) ** 2 / self.spec.alpha ** 2 + value, error = self.exp.absolute(factor, self.spec.entry_bits + 2) + return Entry(value / self.spec.alpha, "broad-gaussian", lift, error / self.spec.alpha) + + def entry(self, kind, row, column): + need(kind in ("narrow", "broad") and type(row) is int and type(column) is int + and 0 <= row < self.target_size and 0 <= column < self.target_size, "reference entry domain") + key = kind, row, column + if key not in self.entries: + self.entries[key] = (self._narrow if kind == "narrow" else self._broad)(row, column) + return self.entries[key] + + def window(self, kind, indices): + need(len(set(indices)) == len(indices), "distinct reference window coordinates") + return Mat([[self.entry(kind, i, j).value for j in indices] for i in indices]) + + def interior(self, start, n): + size = self.spec.source_size + need(type(start) is int and type(n) is int and n > 0 and n & (n - 1) == 0 + and 0 <= start and start + n <= size and 2 * n <= size, + "dyadic interior inside original period cut") + phase = self.phases[start] + need(all(self.phases[j] == phase for j in range(start, start + n)), + "interior stays inside one original phase cell") + center = Q(2 * start + n - 1, 2) + beta = self.theta * center - phase + coordinates = self.selected[start:start + n] + need(coordinates == tuple(range(coordinates[0], coordinates[0] + n)), + "unit target gaps in the interior") + return coordinates, center, beta + + def require_narrow_gap(self): + need(self.narrow_gap > 0, "positive common narrow reference row gap") + return self.narrow_inverse_bounds + + def record(self): + materialize = self.spec.source_size <= 1024 + return dict(description_sha256=self.description_hash, + allocation=self.allocation, + parameters={key: getattr(self.spec, key) for key in self.spec.__dataclass_fields__}, + target_size=self.target_size, geometry_materialized=materialize, + selected=list(self.selected) if materialize else None, + holes=list(self.holes) if materialize else None, + phases=list(self.phases) if materialize else None, + weight_precision=self.weight_precision, + weight_relative_bits=self.weight_relative_bits, kernel_error_bits=self.kernel_error_bits, + narrow_tail=str(self.narrow_tail), broad_tail=str(self.broad_tail), + narrow_error=str(self.errors["narrow"]), broad_error=str(self.errors["broad"]), + narrow_gap_lower=str(self.narrow_gap), + narrow_inverse_bounds=None if self.narrow_inverse_bounds is None + else list(map(str, self.narrow_inverse_bounds)), + narrow_physical_bandwidth=2 * self.spec.source_halfwidth + 2, + broad_physical_bandwidth=self.spec.target_halfwidth, + weight_receipts=[dict(index=j, **{key: str(value) if isinstance(value, Q) else value + for key, value in self.weights[j].items()}) + for j in sorted(self.weights)], + kernel_receipts=[dict(offset=h, value=str(self.kernel[h][0]), + error=str(self.kernel[h][1])) + for h in sorted(self.kernel)]) diff --git a/research/rational-unitary-fft/raw_gaussian.py b/research/rational-unitary-fft/raw_gaussian.py new file mode 100644 index 0000000000..db7040f282 --- /dev/null +++ b/research/rational-unitary-fft/raw_gaussian.py @@ -0,0 +1,163 @@ +"""Anchored raw Gaussian circuits compared with one rounded reference. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. Raw identities: Jeremy Kothe/IdeaOuter and retained Gaussian +lineage; circuit: Vlado; interval arithmetic: Mason Hale. +""" +from fractions import Fraction as Q + +from exact import C, Mat, ceil_log2, need, vector_hash +from full import allocation +from gaussian import Gaussian, rational +from gaussian_reference import ceiling + + +class RawGaussian: + def __init__(self, raw, reference, kind, start, n, radius, target, incoming=0): + need(kind in ("narrow", "broad"), "raw Gaussian direction family") + radius = rational(radius) + need(radius >= 1 and type(target) is int and target >= 1 + and type(incoming) is int and incoming >= 0, "raw radius and precision") + coordinates, center, beta = reference.interior(start, n) + self.raw, self.reference, self.kind = raw, reference, kind + self.start, self.n, self.radius, self.target, self.incoming = start, n, radius, target, incoming + self.coordinates, self.center, self.beta = coordinates, center, beta + alpha, theta, rho, lam = reference.spec.alpha, reference.theta, reference.rho, reference.lam + offsets = [Q(j) - center for j in range(start, start + n)] + if kind == "narrow": + self.c, self.shift = alpha ** 2 * rho, -beta / rho + self.a_factors = [alpha ** 2 * theta * x ** 2 for x in offsets] + self.b_factors = [-x for x in self.a_factors] + self.a_scale = Q(1) + else: + delta, rate = beta / rho, lam * theta / alpha ** 2 + self.c, self.shift = lam / alpha ** 2, -delta * (1 - theta) + self.a_factors = [-delta ** 2 / alpha ** 2 + rate * (lam * y ** 2 + 2 * delta * y) + for y in offsets] + self.b_factors = [-rate * (x ** 2 + 4 * delta * x) for x in offsets] + self.a_scale = Q(1, alpha) + self.gauge_exponent_bound = max(abs(x) for x in self.a_factors + self.b_factors) + self.gauge_bound = alpha * (1 << ceiling(8 * self.gauge_exponent_bound)) + probe = Gaussian(raw, n, self.c, self.shift, 1, 1) + self.qbar, self.middle_bound = probe.qbar, max(Q(n), probe.qbar) + g, h = self.gauge_bound, self.middle_bound + self.raw_inverse_bound = g ** 2 * h + tau = Q(1, 1 << target) + self.reference_error = reference.errors[kind] + need(self.reference_error <= tau / (8 * (radius + 1) * self.raw_inverse_bound ** 2), + "common reference accuracy at this raw inverse caller") + self.diagonal_error = tau / (24 * g * h * (radius + 1)) + self.relative_bits = max(1, ceil_log2(4 * g / self.diagonal_error)) + self.middle_radius = 2 * g * (radius + 1) + self.middle_target = target + ceil_log2(16 * g) + provisional = allocation(n, n, 2 * self.qbar, self.middle_radius, + self.middle_target + 1, incoming) + self.data_bits = max(incoming, target + ceil_log2(24 * n * g * h), + provisional["data_bits"]) + self.child = Gaussian(raw, n, self.c, self.shift, self.middle_radius, + self.middle_target, self.data_bits) + need(self.child.qbar == self.qbar, "same normalized conditioning allowance") + self.decays, self.diagonal_records = {}, [] + self.a_values = [self.coefficient(f, self.a_scale) for f in self.a_factors] + self.b_values = [self.coefficient(f, Q(1)) for f in self.b_factors] + if kind == "narrow": + need(all(a * b == 1 for a, b in zip(self.a_values, self.b_values)), + "one stored narrow chirp and its exact reciprocal") + self.check_identity() + self.reference_matrix = reference.window(kind, coordinates) + self.reference_description = reference.description_hash + self.reference_hash = vector_hash([z for row in self.reference_matrix for z in row]) + self.plan, self.inverse_plan = None, None + + def coefficient(self, factor, scale): + absolute = abs(factor) + if absolute not in self.decays: + self.decays[absolute] = self.reference.exp.relative( + absolute, self.gauge_exponent_bound, self.relative_bits) + row = self.decays[absolute] + if factor >= 0: + value, lower, upper = 1 / row["value"], 1 / row["upper"], 1 / row["lower"] + else: + value, lower, upper = row["value"], row["lower"], row["upper"] + value, lower, upper = scale * value, scale * lower, scale * upper + error = max(abs(value - lower), abs(value - upper)) + inverse_error = max(abs(1 / value - 1 / lower), abs(1 / value - 1 / upper)) + need(value > 0 and max(error, inverse_error) <= self.diagonal_error, + "direct and exact stored reciprocal diagonal errors") + self.diagonal_records.append(dict(factor=str(factor), scale=str(scale), value=str(value), + direct_error=str(error), inverse_error=str(inverse_error), + precision=row["precision"])) + return value + + def ideal_factor(self, i, j): + ref, alpha = self.reference, self.reference.spec.alpha + row, column = self.start + i, self.start + j + if self.kind == "narrow": + return -alpha ** 2 * ((ref.rho * (column - row) + ref.betas[row]) ** 2 + - ref.betas[column] ** 2) + return -(Q(column) - ref.lam * ref.selected[row]) ** 2 / alpha ** 2 + + def check_identity(self): + for i in range(self.n): + for j in range(self.n): + u = j - i + total = (self.a_factors[i] - self.c * (u ** 2 - 2 * self.shift * u) + + self.b_factors[j]) + need(total == self.ideal_factor(i, j), "exact anchored raw source identity") + + def compile(self): + need(self.plan is None, "raw caller executes once") + need(self.reference.description_hash == self.reference_description + and self.reference.window(self.kind, self.coordinates) == self.reference_matrix, + "same common reference at raw execution boundary") + middle = self.child.compile() + self.plan = ([[(tuple([i]), Mat([[value]])) for i, value in enumerate(self.b_values)]] + + list(middle) + + [[(tuple([i]), Mat([[value]])) for i, value in enumerate(self.a_values)]]) + self.inverse_plan = self.child.builder.inverse(self.plan) + cb = self.child.builder.bound + need(cb["data_bits"] == self.data_bits, "one raw and normalized data grid") + g, h, radius = self.gauge_bound, self.middle_bound, self.radius + self.rounding_error = self.n * Q(1, 1 << self.data_bits) + need((g + self.diagonal_error) * radius + self.rounding_error <= self.middle_radius, + "rounded input diagonal fits common normalized radius") + self.middle_error = max(self.child.forward_error, self.child.inverse_error) + self.raw_error = (2 * g * self.middle_error + 3 * g * h * self.diagonal_error * radius + + 3 * g * h * self.rounding_error) + self.forward_error = self.raw_error + self.reference_error * radius + self.inverse_error = self.raw_error + 2 * self.raw_inverse_bound ** 2 * self.reference_error * radius + tau = Q(1, 1 << self.target) + need(self.raw_error < 3 * tau / 8 + and max(self.forward_error, self.inverse_error) < tau, "complete raw reference error budget") + amplitude = max(radius, cb["amplitude_bound"], + (g + 1) * (h * self.middle_radius + self.middle_error) + self.rounding_error) + integer_bits = max(1, ceil_log2(max(Q(1), amplitude))) + 2 + gauge_height = max(row["precision"] for row in self.diagonal_records) + self.reference.spec.alpha.bit_length() + 2 + height = max(gauge_height, cb["coefficient_height_bound"]) + self.bound = dict(n=self.n, data_bits=self.data_bits, integer_bits=integer_bits, + coefficient_height_bound=height, + temporary_bits=2 * self.data_bits + 8 * height + 2 * integer_bits + 32, + amplitude_bound=amplitude, + layer_bound=cb["layer_bound"] + 2, gate_bound=cb["gate_bound"] + 2 * self.n) + need(len(self.plan) <= self.bound["layer_bound"] + and sum(map(len, self.plan)) <= self.bound["gate_bound"], "raw circuit size") + return self.plan + + def record(self): + need(self.plan is not None, "compiled raw receipt") + return dict(kind=self.kind, start=self.start, n=self.n, coordinates=list(self.coordinates), + center=str(self.center), beta=str(self.beta), + reference_description_sha256=self.reference.description_hash, + reference_matrix_sha256=self.reference_hash, + radius=str(self.radius), target_bits=self.target, incoming_bits=self.incoming, + gauge_exponent_bound=str(self.gauge_exponent_bound), + gauge_bound=str(self.gauge_bound), normalized_bound=str(self.middle_bound), + raw_inverse_bound=str(self.raw_inverse_bound), + reference_error=str(self.reference_error), diagonal_error=str(self.diagonal_error), + diagonal_relative_bits=self.relative_bits, diagonal_coefficients=self.diagonal_records, + middle_radius=str(self.middle_radius), middle_error=str(self.middle_error), + raw_error=str(self.raw_error), forward_error=str(self.forward_error), + inverse_error=str(self.inverse_error), + budget={key: str(value) if isinstance(value, Q) else value + for key, value in self.bound.items()}, + child=self.child.record()) diff --git a/research/rational-unitary-fft/verify.py b/research/rational-unitary-fft/verify.py index 70d98f5514..c0320350b1 100644 --- a/research/rational-unitary-fft/verify.py +++ b/research/rational-unitary-fft/verify.py @@ -33,7 +33,7 @@ def rejected(operation, label): raise ValueError("control unexpectedly passed: " + label) -def authenticate(include_generator=False, include_full=False, include_gaussian=False): +def authenticate(include_generator=False, include_full=False, include_gaussian=False, include_raw=False): rows = [r for r in json.loads((HERE / "SOURCES.json").read_text())["files"] if r["usage"] == "input"] raw = {} for row in rows: @@ -57,10 +57,19 @@ def authenticate(include_generator=False, include_full=False, include_gaussian=F "research/capped-fusion-t74-k554/outer/check_gaugefree.py", "research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md", } + raw_gaussian = { + "research/unconditional-batched-crt/PROOF.md", + "research/unconditional-batched-crt/vendor/FINE-PRIME.md", + "research/capped-phase-fusion/BILATERAL-LDU.md", + "research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md", + "research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md", + "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md", + } return raw, [{"path": r["path"], "sha256": r["sha256"]} for r in rows if (include_generator or r["path"] not in added) and (include_full or r["path"] not in full) - and (include_gaussian or r["path"] not in gaussian)] + and (include_gaussian or r["path"] not in gaussian) + and (include_raw or r["path"] not in raw_gaussian)] def operator(plan, n): diff --git a/research/rational-unitary-fft/verify_full.py b/research/rational-unitary-fft/verify_full.py index 882c068000..0720dda749 100644 --- a/research/rational-unitary-fft/verify_full.py +++ b/research/rational-unitary-fft/verify_full.py @@ -256,7 +256,9 @@ def main(): parser = argparse.ArgumentParser(description=__doc__) parser.add_argument("--write-result", action="store_true") parser.add_argument("--with-gaussian", action="store_true") + parser.add_argument("--with-raw", action="store_true") args = parser.parse_args() + need(not args.with_raw or args.with_gaussian, "raw verification follows normalized verification") result = run() path = HERE / "FULL-RESULT.json" if args.write_result: @@ -272,6 +274,9 @@ def main(): if args.with_gaussian: from verify_gaussian import verify_result verify_result(args.write_result) + if args.with_raw: + from verify_raw_gaussian import verify_result as verify_raw_result + verify_raw_result(args.write_result) if __name__ == "__main__": diff --git a/research/rational-unitary-fft/verify_raw_gaussian.py b/research/rational-unitary-fft/verify_raw_gaussian.py new file mode 100644 index 0000000000..5a068070d9 --- /dev/null +++ b/research/rational-unitary-fft/verify_raw_gaussian.py @@ -0,0 +1,263 @@ +"""Verify common Gaussian pages, raw factors and complete stored inverses. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +from exact import (C, Mat, Vec, apply, component_l1, eye, multiply, need, + norm2, upper, vector_hash) +from gaussian_reference import Entry, Reference +from raw_gaussian import RawGaussian +from verify import authenticate, canonical, mv, operator, plan_hash, rejected +from verify_triangle import eliminate, frobenius_error, structure + + +def reference_check(ref): + """Three explicit windings plus analytic tails; shared interval backend.""" + size, alpha, source_size = ref.target_size, ref.spec.alpha, ref.spec.source_size + need(len(set(ref.selected)) == source_size and len(ref.holes) == ref.spec.excess + and all(Q(-1, 2) <= beta < Q(1, 2) for beta in ref.betas), "complete finite geometry") + bits = 2 * (ref.spec.entry_bits + 24) + 32 + records = [] + for kind in ("narrow", "broad"): + matrix, errors = [], [] + for row in range(size): + values, differences = [], [] + for column in range(size): + entry = ref.entry(kind, row, column) + values.append(C(entry.value)) + i, j = ref.source_at.get(row), ref.source_at.get(column) + if j is None or (kind == "narrow" and i is None): + need(entry.value == int(row == column), "exact hole completion") + differences.append(Q(0)) + continue + lo = hi = Q(0) + for winding in (-1, 0, 1): + if kind == "narrow": + factor = alpha ** 2 * ((ref.rho * (j + winding * source_size) + - ref.selected[i]) ** 2 - ref.betas[j] ** 2) + scale = Q(1) + else: + factor = (j + winding * source_size - ref.lam * row) ** 2 / alpha ** 2 + scale = Q(1, alpha) + box = ref.exp.decay_box(factor, bits) + lo += scale * Q(box.lo, box.scale) + hi += scale * Q(box.hi, box.scale) + differences.append(max(abs(entry.value - lo), abs(entry.value - hi))) + matrix.append(values) + errors.append(differences) + if kind == "narrow": + remainder = ref.exp.geometric_tail(alpha ** 2 * source_size * (source_size + 1), + 2 * alpha ** 2 * (source_size + 1), 2, bits) + else: + c = 1 / (alpha ** 2 * ref.rho ** 2) + remainder = ref.exp.geometric_tail(c * (size + Q(1, 2)) ** 2, + 2 * c * (size + 1), Q(2, alpha), bits) + measured = max(max(map(sum, errors)), max(map(sum, zip(*errors)))) + remainder + need(measured <= ref.errors[kind], "periodic reference row and column error") + if kind == "narrow": + ref.require_narrow_gap() + gap = min(1 - sum((matrix[i][j].re for j in range(size) if i != j), Q(0)) + for i in range(size)) + need(gap >= ref.narrow_gap, "actual common reference row gap") + for i in range(size): + for j in range(size): + distance = min(abs(i - j), size - abs(i - j)) + width = 2 * ref.spec.source_halfwidth + 2 if kind == "narrow" else ref.spec.target_halfwidth + need(distance <= width or matrix[i][j] == 0, "physical reference bandwidth") + records.append(dict(kind=kind, entries=size ** 2, + winding_terms=3 * source_size * (source_size if kind == "narrow" else size), + matrix_sha256=vector_hash([z for row in matrix for z in row]), + measured_error_upper=str(upper(measured)), allowance=str(ref.errors[kind]))) + need(any(ref.entry("broad", h, q).value != 0 for h in ref.holes for q in ref.selected), + "broad Gaussian outputs at hole rows") + need(all(ref.entry("narrow", h, q).value == 0 for h in ref.holes for q in ref.selected), + "narrow direct-sum hole completion") + wrap = ref.entry("narrow", ref.selected[-1], ref.selected[0]) + need(wrap.kind == "cross-period" and wrap.lift == 1, "original period cut retained") + first = ref.window("narrow", ref.selected[1:5]) + second = ref.window("narrow", ref.selected[3:7]) + need(all(first[i + 2][j + 2] == second[i][j] for i in range(2) for j in range(2)), + "overlapping pages use identical global entries") + principal = ref.window("narrow", list(ref.selected[13:17]) + list(ref.holes[:1])) + inverse = eliminate(principal) + row_bound = max(component_l1(row) for row in inverse) + column_bound = max(component_l1(column) for column in zip(*inverse)) + need(row_bound <= ref.narrow_inverse_bounds[0] and column_bound <= ref.narrow_inverse_bounds[1], + "principal inverse across a cell boundary and a hole") + return dict(reference=ref.record(), matrices=records, overlap_entries=4, + wrap_kind=wrap.kind, weights=len(ref.weights), kernels=len(ref.kernel), + principal_inverse_bounds=[str(row_bound), str(column_bound)]) + + +def one_case(label, raw, ref, kind, start, n, fine=False): + vectors = [Vec(C(Q((3 * i + 1) % 7 - 3, 8), Q((5 * i + 2) % 9 - 4, 8)) + for i in range(n)), + Vec(C((-1) ** i * 8, i % 2) for i in range(n))] + incoming = 512 if fine else 3 + if fine: + vectors.append(Vec([C(Q(1, 1 << incoming))] + [C()] * (n - 1))) + radius = max(Q(1), *(component_l1(v) for v in vectors)) + caller = RawGaussian(raw, ref, kind, start, n, radius, 16, incoming) + if ref.allocation is not None: + need(n <= ref.allocation["cap"] and radius <= Q(ref.allocation["radius"]) + and caller.raw_inverse_bound <= Q(ref.allocation["inverse_bound"]), + "uniform reference allocation covers this caller") + plan = caller.compile() + inverse = caller.inverse_plan + actual, actual_inverse = operator(plan, n), operator(inverse, n) + need(multiply(actual, actual_inverse) == eye(n), "exact inverse of stored raw program") + reference, reference_inverse = caller.reference_matrix, eliminate(caller.reference_matrix) + height = max(structure(plan, caller.bound), structure(inverse, caller.bound)) + states = [] + for direction, program, wanted, allowance in ( + ("forward", plan, reference, caller.forward_error), + ("inverse", inverse, reference_inverse, caller.inverse_error)): + for vector in vectors: + need(norm2(vector) <= radius ** 2 and all((part * (1 << incoming)).denominator == 1 + for z in vector for part in (z.re, z.im)), "raw incoming contract") + output, stats = apply(program, vector, caller.data_bits) + error = norm2(output - mv(wanted, vector)) + need(error <= allowance ** 2 and allowance < Q(1, 1 << caller.target), + "rounded raw forward/inverse error against common reference") + need(stats["max_observed_height"] <= caller.bound["temporary_bits"] + and stats["max_grid_numerator_bits"] + 1 <= caller.data_bits + caller.bound["integer_bits"] + and stats["max_reduced_fractional_bits"] <= caller.data_bits, "raw online record widths") + states.append(dict(direction=direction, input_sha256=vector_hash(vector), + output_sha256=vector_hash(output), statistics=stats, + error_squared_upper=str(upper(error)))) + record = caller.record() + record.update(label=label, layers=len(plan), gates=sum(map(len, plan)), + plan_sha256=plan_hash(plan), inverse_sha256=plan_hash(inverse), + coefficient_height=height, states=states) + print("CHECK_RAW_GAUSSIAN_CASE", label, flush=True) + return record, caller + + +def controls(raw, ref, caller): + results = [] + results.append(rejected(lambda: ref.interior(ref.spec.source_size - 2, 4), "interior-crosses-period-cut")) + results.append(rejected(lambda: ref.interior(14, 4), "interior-crosses-original-cell")) + results.append(rejected(lambda: Reference(raw, 32, 1, 3, 16, 14, 96), "ambiguous-periodic-lift")) + results.append(rejected(lambda: Reference(raw, 32, 1, 3.0, 3, 14, 96), "floating-reference-parameter")) + results.append(rejected(lambda: Reference.for_cap(raw, 32, 1, 3, 4, 64, 64), + "period-too-short-for-uniform-reference-accuracy")) + poor = Reference(raw, 32, 1, 3, 3, 2, 96) + results.append(rejected(lambda: RawGaussian(raw, poor, "broad", 2, 2, 1, 16), + "unpaid-common-reference-tail")) + coarse = Reference(raw, 32, 1, 3, 3, 14, 8) + results.append(rejected(lambda: RawGaussian(raw, coarse, "broad", 2, 2, 1, 16), + "underfunded-common-reference-coefficients")) + fresh = Reference(raw, 32, 1, 3, 3, 14, 96) + pending = RawGaussian(raw, fresh, "narrow", 2, 2, 1, 16) + key = "narrow", fresh.selected[2], fresh.selected[2] + fresh.entries[key] = Entry(Q(2), "unit-diagonal", 0, Q(0)) + results.append(rejected(pending.compile, "changed-reference-after-preparation")) + weight = ref.weight(0) + wrong, _ = ref.exp.absolute(ref.weight_exponent_bound, 16) + need(wrong == 0 and weight > 0, "online precision can zero a setup weight") + results.append("zero-weight-at-online-precision") + results.append(rejected(lambda: ref.exp.relative(ref.weight_exponent_bound, + ref.weight_exponent_bound, 16, 16), "underfunded-positive-weight-grid")) + weak = Reference(raw, 32, 1, 1, 0, 2, 32) + results.append(rejected(weak.require_narrow_gap, "uncertified-global-row-gap")) + need(any(ref.entry("broad", h, q).value > 0 for h in ref.holes for q in ref.selected), + "broad hole rows cannot use narrow direct sum") + results.append("erased-broad-hole-output") + need(any(ref.entry("broad", q, q).value != 1 for q in ref.selected), + "raw broad diagonal retains alpha and displacement") + results.append("forced-unit-broad-diagonal") + finer = Reference(raw, 32, 1, 3, 3, 14, 112) + original_caller = RawGaussian(raw, ref, "broad", 2, 2, 1, 16) + finer_caller = RawGaussian(raw, finer, "broad", 2, 2, 1, 16) + original_caller.compile() + finer_caller.compile() + need(original_caller.reference_hash != finer_caller.reference_hash + and plan_hash(original_caller.plan) == plan_hash(finer_caller.plan), + "full reference setup precision stays off online coefficient wires") + results.append("reference-setup-precision-separate-from-online-gates") + # Swap only the wrapper factors around the unchanged stored middle inverse. + need(caller.kind == "narrow" and caller.n >= 4, "noncommuting wrapper control fixture") + middle_inverse = caller.child.builder.inverse(caller.child.plan) + wrong_inverse = ([[((i,), Mat([[1 / a]])) for i, a in enumerate(caller.b_values)]] + + list(middle_inverse) + + [[((i,), Mat([[1 / a]])) for i, a in enumerate(caller.a_values)]]) + need(frobenius_error(operator(wrong_inverse, caller.n), eliminate(caller.reference_matrix)) + > caller.n * caller.inverse_error ** 2, "wrong anchored inverse wrapper") + results.append("wrong-inverse-diagonal-order") + original = caller.a_factors[0] + caller.a_factors[0] += 1 + results.append(rejected(caller.check_identity, "wrong-raw-exponential-constant")) + caller.a_factors[0] = original + need(canonical({"excess": 1}) != canonical({"excess": True}), "typed raw receipt") + return results + ["typed-raw-receipt"] + + +def run(): + raw, sources = authenticate(include_generator=True, include_full=True, + include_gaussian=True, include_raw=True) + small = Reference(raw, 32, 1, 3, 3, 14, 96) + reference = reference_check(small) + print("CHECK_COMMON_GAUSSIAN_REFERENCE", flush=True) + large = Reference.for_cap(raw, 128, 1, 3, 4, 64, 16) + sparse = Reference(raw, 10 ** 20 + 1, 10 ** 18, 3, 3, 30, 96) + coordinates, _, _ = sparse.interior(10 ** 10, 4) + sparse_window = sparse.window("narrow", coordinates) + need(all(sparse.source_at.get(coordinates[i]) == 10 ** 10 + i for i in range(4)) + and sparse._holes is None and not sparse.record()["geometry_materialized"], + "large-axis queries avoid global geometry enumeration") + fixtures = [ + ("broad-small", small, "broad", 2, 2, False), + ("narrow-small", small, "narrow", 2, 2, False), + ("broad-positive-cell", large, "broad", 60, 4, False), + ("broad-negative-cell", large, "broad", 65, 4, False), + ("narrow-positive-cell", large, "narrow", 60, 4, False), + ("narrow-fine-negative-cell", large, "narrow", 65, 4, True), + ] + records, callers = [], [] + for label, ref, kind, start, n, fine in fixtures: + record, caller = one_case(label, raw, ref, kind, start, n, fine) + records.append(record) + callers.append(caller) + need(all(row["budget"]["data_bits"] >= row["incoming_bits"] for row in records), + "preserved raw incoming grids") + return dict(schema_version=1, scope="common periodic Gaussian reference and raw capped interior circuits", + sources=sources, global_reference=reference, large_reference=large.record(), + large_address_query=dict(reference=sparse.record(), coordinates=list(coordinates), + window_sha256=vector_hash([z for row in sparse_window for z in row])), + cases=records, controls=controls(raw, small, callers[-1]), + raw_interior_gate_execution=True, complete_gaussian_execution=False, + physical_tape_execution=False, boundary_schur_execution=False, + independent_transcendental_backend=False, universal_proof_kernel=False) + + +def verify_result(write=False): + result = run() + path = HERE / "RAW-GAUSSIAN-RESULT.json" + if write: + path.write_text(json.dumps(result, indent=2) + "\n") + else: + need(canonical(result) == canonical(json.loads(path.read_text())), "complete typed raw Gaussian receipt") + print(json.dumps(dict(status="PASS_RAW_GAUSSIAN_REFERENCE_CIRCUIT", cases=len(result["cases"]), + gates=sum(row["gates"] for row in result["cases"]), + rounded_states=sum(len(row["states"]) for row in result["cases"]), + controls=len(result["controls"]), + result_sha256=sha256(canonical(result)).hexdigest()), sort_keys=True)) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-result", action="store_true") + verify_result(parser.parse_args().write_result) + + +if __name__ == "__main__": + main() From c13c08ba4f957af70cfc2560ea43d837fe730d21 Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 21:16:58 +0200 Subject: [PATCH 8/9] Retain the rounded boundary adapter for core integration --- .../vendor/BOUNDARY-PROVENANCE.json | 11 + .../vendor/boundary_adapter.py | 400 ++++++++++++++++++ 2 files changed, 411 insertions(+) create mode 100644 research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json create mode 100644 research/rational-unitary-fft/vendor/boundary_adapter.py diff --git a/research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json b/research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json new file mode 100644 index 0000000000..7286022fdd --- /dev/null +++ b/research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json @@ -0,0 +1,11 @@ +{ + "schema_version": 1, + "original_pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/4186", + "original_commit": "a20501df8fc07f5ceb32e6fd64bce9ad8fe594bf", + "original_path": "research/rounded-boundary-blocks/verify.py", + "original_git_blob": "cc9c6815791a8290537d56bb69320201173e1978", + "sha256": "6738df51cdf7ecb55d1521b9f8d38e627c8feb4c4afe242ba2344ffc024c772e", + "retained_path": "research/rational-unitary-fft/vendor/boundary_adapter.py", + "attribution": "Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. Retained exact block/Strassen program: Vlado Vojdanovski and source lineage. Apache-2.0.", + "scope": "Unchanged source archive for exact definition extraction. Its standalone CLI and old package-relative authentication are not invoked here. The new loader authenticates this archive and the actual block source before extracting the adapter." +} diff --git a/research/rational-unitary-fft/vendor/boundary_adapter.py b/research/rational-unitary-fft/vendor/boundary_adapter.py new file mode 100644 index 0000000000..cc9c681579 --- /dev/null +++ b/research/rational-unitary-fft/vendor/boundary_adapter.py @@ -0,0 +1,400 @@ +"""Explicit rounded block-boundary audit; Apache-2.0. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +The retained exact Strassen/block program keeps its original attribution. +This is a numerical/interface model, not a fixed-tape execution. +""" +from __future__ import annotations + +import argparse +from fractions import Fraction as Q +import hashlib +import json +from pathlib import Path + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +SUBJECT = "research/capped-phase-fusion/evidence/radical_reduction_ultra/check_constructions.py" + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def sha(data): + return hashlib.sha256(data).hexdigest() + + +def encoded(value): + return json.dumps(value, sort_keys=True, separators=(",", ":")).encode() + + +def matrix_digest(matrix): + return sha(encoded([[str(x) for x in row] for row in matrix])) + + +def authenticate(override=None): + pins = json.loads((HERE / "SOURCES.json").read_text()) + result = {} + for row in pins["files"]: + if row["usage"] == "input": + name = row["path"] + data = override[name] if override and name in override else (ROOT / name).read_bytes() + need(sha(data) == row["sha256"], "source bytes: " + name) + result[name] = data + return result + + +def maximum(matrix): + return max((abs(x) for row in matrix for x in row), default=Q(0)) + + +def norm(matrix): + return max((sum(abs(x) for x in row) for row in matrix), default=Q(0)) + + +def identity(n): + return [[Q(i == j) for j in range(n)] for i in range(n)] + + +def multiply(a, b): + return [[sum(a[i][k] * b[k][j] for k in range(len(b))) + for j in range(len(b[0]))] for i in range(len(a))] + + +def subtract(a, b): + return [[x - y for x, y in zip(ar, br)] for ar, br in zip(a, b)] + + +def solve(a, rhs): + """Independent exact reference; ordinary elimination on the augmented matrix.""" + n = len(a) + augmented = [list(a[i]) + list(rhs[i]) for i in range(n)] + for k in range(n): + pivot = next((i for i in range(k, n) if augmented[i][k]), None) + need(pivot is not None, "invertible reference") + augmented[k], augmented[pivot] = augmented[pivot], augmented[k] + divisor = augmented[k][k] + augmented[k] = [x / divisor for x in augmented[k]] + for i in range(n): + if i != k: + factor = augmented[i][k] + augmented[i] = [x - factor * y for x, y in zip(augmented[i], augmented[k])] + return [row[n:] for row in augmented] + + +def lu(a): + n = len(a) + lower, upper = identity(n), [[Q(0)] * n for _ in range(n)] + for i in range(n): + for j in range(i, n): + upper[i][j] = a[i][j] - sum(lower[i][k] * upper[k][j] for k in range(i)) + need(upper[i][i] != 0, "no-pivot LU") + for j in range(i + 1, n): + lower[j][i] = (a[j][i] - sum(lower[j][k] * upper[k][i] for k in range(i))) / upper[i][i] + need(multiply(lower, upper) == a, "exact factor identity") + return lower, upper + + +def reverse_both(a): + return [list(reversed(row)) for row in reversed(a)] + + +def rounded(a, bits): + unit = 1 << bits + return [[Q(int(x * unit), unit) for x in row] for row in a] + + +def on_grid(a, bits): + unit = 1 << bits + return all((x * unit).denominator == 1 for row in a for x in row) + + +def reject(call, label): + try: + call() + except (ValueError, AssertionError): + return + raise ValueError("accepted control: " + label) + + +class RoundedSubject: + """Keep the source block order; intercept only pages and declared rounding sites.""" + + def __init__(self, source, bits, width, y_bound, inverse_bound, sub_bound, fault=None): + need(fault in (None, "drop_previous", "skip_solved_round"), "known controlled fault") + self.fault = fault + self.bits, self.width = bits, width + self.eta = Q(2, 1 << bits) + self.y_bound, self.inverse_bound, self.sub_bound = y_bound, inverse_bound, sub_bound + self.temporary_bound = 128 * width ** 4 * inverse_bound ** 2 * (sub_bound + 1) * y_bound + self.depth = self.products = self.updates = self.inverse_pages = 0 + self.max_value, self.max_fractional = Q(0), 0 + self.inverse_ids, self.lower_pages = set(), {} + self.ns = {"__file__": SUBJECT, "__name__": "retained_boundary_subject"} + exec(compile(source, SUBJECT, "exec", optimize=0), self.ns) + self.exact_mul, self.exact_add, self.exact_inverse = ( + self.ns["mul"], self.ns["add"], self.ns["inverse"]) + self.ns.update(mul=self.mul, add=self.add, inverse=self.inverse) + + def observe(self, a): + self.max_value = max(self.max_value, maximum(a)) + need(self.max_value <= self.temporary_bound, "exact Strassen temporary magnitude") + for row in a: + for x in row: + d = x.denominator + need(d & (d - 1) == 0, "dyadic live arithmetic") + self.max_fractional = max(self.max_fractional, d.bit_length() - 1) + need(self.max_fractional <= 2 * self.bits, "double-fraction temporary width") + + def inverse(self, a): + reference = self.exact_inverse(a) + need(norm(reference) <= self.inverse_bound, "consecutive inverse-block bound") + page = rounded(reference, self.bits) + need(norm(subtract(page, reference)) <= self.width * self.eta, "inverse-page error") + self.inverse_pages += 1 + self.inverse_ids.add(id(page)) + self.observe(page) + return page + + def add(self, a, b, sign=1): + result = self.exact_add(a, b, sign) + self.observe(result) + if self.depth == 0: + need(sign == -1, "declared completed RHS update") + self.updates += 1 + result = rounded(result, self.bits) + return result + + def mul(self, a, b): + root = self.depth == 0 + inverse_product = root and id(a) in self.inverse_ids + if root: + self.products += 1 + need(on_grid(b, self.bits), "completed block input grid") + if not inverse_product: + key = tuple(map(tuple, a)) + if key not in self.lower_pages: + page = rounded(a, self.bits) + need(norm(subtract(page, a)) <= self.width * self.eta, "subdiagonal-page error") + self.lower_pages[key] = page + a = self.lower_pages[key] + if self.fault == "drop_previous": + b = [[Q(0) for _ in row] for row in b] + need(on_grid(a, self.bits), "factor-page grid") + self.observe(a) + self.observe(b) + self.depth += 1 + try: + result = self.exact_mul(a, b) + finally: + self.depth -= 1 + self.observe(result) + if inverse_product and self.fault != "skip_solved_round": + result = rounded(result, self.bits) + return result + + def run(self, a, rhs): + n, columns = len(a), len(rhs[0]) + w = self.width + need(w > 0 and w & (w - 1) == 0, "dyadic block width") + need(all(len(row) == n for row in a) and all(len(row) == columns for row in rhs), + "matrix and RHS shapes") + need(all(not a[i][j] for i in range(n) for j in range(n) + if j > i or i - j > w), "natural lower band") + need(on_grid(rhs, self.bits), "RHS input grid") + need(maximum(rhs) <= self.y_bound and self.y_bound >= 1, "RHS magnitude") + before = matrix_digest(a), matrix_digest(rhs) + result = self.ns["block_solve"](a, rhs, w) + need(before == (matrix_digest(a), matrix_digest(rhs)), "borrowed source preservation") + need(on_grid(result, self.bits), "completed solved blocks") + blocks, batches = (n + w - 1) // w, (columns + w - 1) // w + need(self.products == (2 * blocks - 1) * batches, "complete two-product schedule") + need(self.updates == (blocks - 1) * batches, "complete RHS rounding schedule") + need(self.inverse_pages == blocks, "one reference inverse page per block") + need(self.ns["OPS"] == self.products * 7 ** (w.bit_length() - 1), + "actual seven-product multiplication count") + return result + + +def pass_constants(a, sub, q, width): + return a * width + 2 * a + 1, width * (a * (sub + 1) + 1), q + + +def check_pass(source, matrix, rhs, p, bits, width, upper=False, fault=None): + a, sub, q = (Q(2), Q(2), Q(1040 * p ** 5)) if upper else ( + Q(65 * p ** 3), Q(65 * p ** 3), Q(32 * p ** 2)) + eta, y = Q(2, 1 << bits), max(Q(1), maximum(rhs)) + c0, c1, _ = pass_constants(a, sub, q, width) + need(width * eta <= 1 and eta * c0 <= 1 and q * eta * c1 <= Q(1, 2), + "residual absorption") + need(norm(matrix) <= max(a, sub), "triangular norm contract") + need(norm(solve(matrix, identity(len(matrix)))) <= q, "whole triangular inverse bound") + worker = RoundedSubject(source, bits, width, y, q, sub, fault=fault) + answer = worker.run(matrix, rhs) + residual = maximum(subtract(multiply(matrix, answer), rhs)) + magnitude = maximum(answer) + residual_bound = eta * (c0 * y + c1 * magnitude) + need(residual <= residual_bound, "actual complete residual") + need(magnitude <= 4 * q * y, "absorbed output magnitude") + error = maximum(subtract(answer, solve(matrix, rhs))) + error_bound = q * eta * (c0 + 4 * q * c1) * y + need(error <= error_bound, "triangular solution error") + return answer, { + "residual": str(residual), "residual_bound": str(residual_bound), + "error": str(error), "error_bound": str(error_bound), + "maximum": str(magnitude), "temporary_maximum": str(worker.max_value), + "temporary_bound": str(worker.temporary_bound), + "temporary_fractional_bits": worker.max_fractional, + "root_products": worker.products, "scalar_products": worker.ns["OPS"], + "rhs_roundings": worker.updates, "inverse_pages": worker.inverse_pages, + "batches": (len(rhs[0]) + width - 1) // width, + } + + +def ceil_log2(q): + q = Q(q) + need(q > 0, "positive logarithm") + e = q.numerator.bit_length() - q.denominator.bit_length() + power = Q(1 << e) if e >= 0 else Q(1, 1 << -e) + return e if q <= power else e + 1 + + +def schur(a, eliminated): + kept = [i for i in range(len(a)) if i not in eliminated] + aa = [[a[i][j] for j in eliminated] for i in eliminated] + bb = [[a[i][j] for j in kept] for i in eliminated] + cc = [[a[i][j] for j in eliminated] for i in kept] + dd = [[a[i][j] for j in kept] for i in kept] + return subtract(dd, multiply(cc, solve(aa, bb))) + + +def cases(): + for p in (1, 2, 4): + rho = 1 - Q(1, 16 * p * p) + for n, band, width, columns in ((3, 1, 1, 3), (5, 1, 2, 3), + (9, 2, 4, 5), (12, 3, 4, 7)): + if band > 2 * p: + continue + a = identity(n) + for i in range(n): + neighbors = [j for j in range(n) if i != j and abs(i - j) <= band] + for j in neighbors: + a[i][j] = rho * (-1 if (i + 2 * j) % 3 else 1) / len(neighbors) + yield f"band-p{p}-n{n}-w{width}-r{columns}", p, width, columns, a + # A pair reaches the inverse gap bound and needs non-dyadic inverse pages. + yield f"near-gap-p{p}", p, 2, 3, [[Q(1), -rho], [-rho, Q(1)]] + a = identity(12) + for i in range(12): + for j in range(12): + if 0 < abs(i - j) <= 2: + a[i][j] = Q(-1, 8) + yield "finite-section-schur", 2, 4, 5, schur(a, [3, 4, 8, 9]) + + +def audit(source): + result = {} + for name, p, width, columns, a in cases(): + n = len(a) + delta = Q(1, 16 * p * p) + gap = min(abs(a[i][i]) - sum(abs(a[i][j]) for j in range(n) if j != i) + for i in range(n)) + band = max((abs(i - j) for i in range(n) for j in range(n) if a[i][j]), default=0) + need(norm(a) <= 2 and gap >= delta and band <= 2 * p, + "fine-prime narrow boundary contract") + need(band <= width <= 4 * p, "consecutive blocks cover scalar bandwidth") + lower, upper = lu(a) + real = [[Q(((i + 2 * j) % 7) - 3, 8) for j in range(columns)] for i in range(n)] + imag = [[Q(((3 * i + j) % 5) - 2, 8) for j in range(columns)] for i in range(n)] + if name == "near-gap-p4": + real = [[17 * x for x in row] for row in real] + y0 = max(Q(1), max(abs(x) + abs(y) for ar, ai in zip(real, imag) for x, y in zip(ar, ai))) + bits = 8 + 41 + 16 * ceil_log2(p) + ceil_log2(y0) + eta = Q(2, 1 << bits) + need(eta <= Q(1, (1 << 21) * p ** 9), "caller absorption guard") + parts, records = [], [] + for rhs in (real, imag): + forward, first = check_pass(source, lower, rhs, p, bits, width) + backward, second = check_pass(source, reverse_both(upper), list(reversed(forward)), + p, bits, width, upper=True) + answer = list(reversed(backward)) + parts.append(answer) + records.append({"lower": first, "upper": second}) + reference = [solve(a, real), solve(a, imag)] + error = max(abs(parts[0][i][j] - reference[0][i][j]) + + abs(parts[1][i][j] - reference[1][i][j]) + for i in range(n) for j in range(columns)) + # Each real pass uses eta/2 or less; the shared complex-l1 allowance + # above already bounds the joint rounding by eta. + bound = (1 << 40) * p ** 16 * eta * y0 + need(error <= bound <= Q(1, 256), "two-pass complex solution error") + magnitude = max(abs(parts[0][i][j]) + abs(parts[1][i][j]) + for i in range(n) for j in range(columns)) + need(magnitude <= (1 << 20) * p ** 7 * y0, "two-pass magnitude") + need(all(Q(r[side]["temporary_maximum"]) <= (1 << 46) * p ** 16 * y0 + for r in records for side in ("lower", "upper")), "two-pass temporary bound") + result[name] = { + "p": p, "rows": n, "columns": columns, "width": width, "bits": bits, + "gap": str(gap), "band": band, "Y0": str(y0), "error": str(error), + "error_bound": str(bound), "magnitude": str(magnitude), + "production_batch_domain": columns >= width, + "passes": records, "answer_sha256": [matrix_digest(x) for x in parts], + } + need(any(Q(case["error"]) > 0 for case in result.values()), "nonzero rounded solve error") + # Inverting an already rounded block violates the page's required error. + raw = [[Q(1, 3)]] + correct = solve(raw, identity(1)) + wrong = rounded(solve(rounded(raw, 4), identity(1)), 4) + need(norm(subtract(wrong, correct)) > Q(2, 16), "rounded-input inverse-page control") + # A nonconsecutive triangular inverse is not the corresponding full block. + tri = [[Q(1), Q(0), Q(0)], [Q(1, 2), Q(1), Q(0)], [Q(0), Q(1, 2), Q(1)]] + full = solve(tri, identity(3)) + need(full[2][0] == Q(1, 4) and solve([[tri[i][j] for j in (0, 2)] for i in (0, 2)], + identity(2))[1][0] == 0, "consecutive-block control") + reject(lambda: check_pass(source, identity(2), [[Q(1)], [Q(1)]], 1, 1, 1), + "insufficient absorption precision") + reject(lambda: check_pass(source, [[Q(1), Q(0)], [Q(-1, 2), Q(1)]], + [[Q(1)], [Q(0)]], 1, 49, 1, fault="drop_previous"), + "omitted previous block") + reject(lambda: check_pass(source, [[Q(3)]], [[Q(1, 8)]], 1, 49, 1, + fault="skip_solved_round"), "omitted solved-block rounding") + bad = [[Q(1), Q(1, 4)], [Q(0), Q(1)]] + reject(lambda: RoundedSubject(source, 64, 1, Q(1), Q(32), Q(65)).run(bad, [[Q(1)], [Q(1)]]), + "unreversed upper direction") + # A cyclic corner is outside the admitted ordinary lower band. + corner = identity(4) + corner[3][0] = Q(1, 4) + reject(lambda: RoundedSubject(source, 64, 1, Q(1), Q(32), Q(65)).run( + corner, [[Q(1)]] * 4), "unhandled corner") + return { + "status": "PASS_ROUNDED_BOUNDARY_BLOCKS", "cases": result, + "case_count": len(result), "real_triangular_passes": 4 * len(result), + "controls": ["rounded-input inverse page", "nonconsecutive inverse block", + "insufficient absorption precision", "unreversed upper direction", + "unhandled cyclic corner", "omitted previous block", + "omitted solved-block rounding"], + "scope": "Source-bound rounded two-product solves and explicit residual/headroom proof for the finite-section narrow boundary; no physical tape execution, broad inverse or full multiplication theorem.", + } + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--emit", type=Path) + args = parser.parse_args() + sources = authenticate() + reject(lambda: authenticate({SUBJECT: sources[SUBJECT] + b"\n"}), "source corruption") + result = audit(sources[SUBJECT]) + if args.emit: + with args.emit.open("x") as handle: + json.dump(result, handle, sort_keys=True, indent=2) + handle.write("\n") + else: + need(encoded(result) == encoded(json.loads((HERE / "RESULT.json").read_text())), + "complete typed retained receipt") + print(json.dumps({"status": result["status"], "cases": result["case_count"], + "receipt_sha256": sha(encoded(result)), "scope": result["scope"]}, sort_keys=True)) + + +if __name__ == "__main__": + main() From 78c72e563173f3822fcb4efed9ceeccf05630258 Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 22:23:15 +0200 Subject: [PATCH 9/9] Execute rounded Gaussian factor words on shared finite-core references --- research/rational-unitary-fft/CORE-PROOF.md | 354 ++ .../rational-unitary-fft/CORE-RESULT.json | 5389 +++++++++++++++++ research/rational-unitary-fft/README.md | 78 +- research/rational-unitary-fft/SOURCES.json | 7 +- research/rational-unitary-fft/TASK.json | 14 +- research/rational-unitary-fft/VERIFY.json | 31 +- research/rational-unitary-fft/core.py | 629 ++ research/rational-unitary-fft/verify.py | 13 +- research/rational-unitary-fft/verify_core.py | 229 + research/rational-unitary-fft/verify_full.py | 5 + 10 files changed, 6712 insertions(+), 37 deletions(-) create mode 100644 research/rational-unitary-fft/CORE-PROOF.md create mode 100644 research/rational-unitary-fft/CORE-RESULT.json create mode 100644 research/rational-unitary-fft/core.py create mode 100644 research/rational-unitary-fft/verify_core.py diff --git a/research/rational-unitary-fft/CORE-PROOF.md b/research/rational-unitary-fft/CORE-PROOF.md new file mode 100644 index 0000000000..a491eb52da --- /dev/null +++ b/research/rational-unitary-fft/CORE-PROOF.md @@ -0,0 +1,354 @@ +# Rounded Gaussian factor words on ordinary finite cores + +I join the shared references and raw interior circuits of +[RAW-GAUSSIAN-PROOF.md](RAW-GAUSSIAN-PROOF.md) to the retained carve and +boundary program. I prove a maximum-norm error bound for a complete +finite narrow inverse or broad forward word. I use exact rational +preparation and one online data grid. + +## Domain and claim + +Fix the `Reference` geometry: integers \(S>K>0,\alpha\ge1\), +\(\theta=K/S,\rho=1+\theta,\lambda=\rho^{-1}\), source half-cut \(w\) +and target half-cut \(W_b\). Use its immutable rational narrow or broad +reference. Choose an ordinary half-open target window \([a,b)\) +with complementary length at least \(W\), where +\[ + W\ge2w+2\quad\hbox{(narrow)},\qquad + W\ge W_b\quad\hbox{(broad)}. +\] +I retain the original period cut, phase labels and hole conventions. +The shortest connection across the omitted complement has distance at +least \(W+1\); consequently no periodic corner survives inside this +window. I also check its ordinary bandwidth against \(W\). +Write \(Q\) for the principal reference matrix on this finite window. + +Choose a positive dyadic cap \(B\) with \(2B\le S\), a rational +input bound \(Y_0\ge1\), target \(t\ge1\), and incoming grid \(F_0\ge0\). +For complex data I use +\[ + \|x\|_*=\max_{i,j}(|\Re x_{ij}|+|\Im x_{ij}|). +\] +The column index ranges over the declared RHS columns. Real +absolute-row-sum bounds act on this norm without a dimension factor. + +For narrow execution require the same reference's certified gap \(d>0\). +For broad execution require \(E_M Q_{\rm ideal}\le1/2\), with +\(Q_{\rm ideal}\) defined below. For either direction require the +reference accuracy needed by each raw child at the amplified radius +and target below. `Core` rejects an underfunded fixed reference before +preparing its Schur pages. `allocated_reference` supplies a sufficient +allocation before the user chooses a window. + +Under these hypotheses I execute the retained six-call word and obtain +\[ + \|\widehat x-Q^{-1}x\|_* \le 2^{-t-1} + \quad\hbox{(narrow)},\qquad + \|\widehat x-Qx\|_* \le 2^{-t-1} + \quad\hbox{(broad)}. \tag{1} +\] +I state these bounds against the finite rational reference. I do not +include a core/halo approximation in (1). + +## Carve and exact reference pages + +I extract the unchanged `carve` function from the pinned +`check_capped_gauges.py`. I apply it to each clipped run of consecutive +selected coordinates with one original phase. I retain holes as +boundary coordinates. For a run of length \(L\le6W\), the source +chooses only boundary coordinates. Otherwise it keeps the first \(W\), +sets \(R=L-2W\), and while \(R>4W\) chooses an interior length +\[ + \min(B,2^{\lfloor\log_2(R-W)\rfloor}), +\] +followed by \(W\) boundary coordinates. It finishes with its residual +and final boundary strips. I retain this order, including a possible +coalesced terminal boundary group of \(6W\). + +Each interior has a positive dyadic length at most \(B\). Consecutive +interiors have more than \(W\) coordinate distance, so I obtain +\[ + Q=\begin{pmatrix}A&B_0\\ C_0&D\end{pmatrix}, + \qquad A=\bigoplus_i A_i . +\] +Here \(B_0,C_0\) are the original reference couplings; the cap \(B\) +has no relation to the coupling symbol \(B_0\). I prepare +\[ + F=D-C_0A^{-1}B_0 \tag{2} +\] +from exact reference entries and exact interior inverses. +I never substitute rounded online inverse applications into (2). +An eliminated interval touches at most \(W\) retained neighbors on +each side. Their compressed separation is at most \(2W-1\). +Original couplings cannot increase their separation under deletion. +Thus the compressed boundary half-bandwidth obeys \(b_F\le2W-1\). + +The source carve permits \(B\le W\). I require \(B>W\) only for the +extra claim that each chosen interior exceeds \(W\). The finite +fixtures exercise the former domain and record `cap_exceeds_band=false`. + +## Exact chronology and factor norms + +I extract the source `seq` assignment from `check_union.py`. Its +narrow inverse chronology is +\[ + A^{-1}\text{ on }I;\quad E_{-C_0};\quad + F^{-1}\text{ on }J;\quad A\text{ on }I;\quad + E_{-B_0};\quad A^{-1}\text{ on }I. \tag{3} +\] +I use the original couplings in the two shears. I reverse the source +labels in (3) and exchange forward/inverse labels to obtain the broad +forward chronology +\[ + A\text{ on }I;\quad E_{B_0};\quad + A^{-1}\text{ on }I;\quad F\text{ on }J;\quad + E_{C_0};\quad A\text{ on }I. \tag{4} +\] +Direct multiplication with (2) proves both endpoints without assuming +that the broad factors in (4) have inverses. In particular, +I apply the broad \(F\); I require no broad \(F^{-1}\). +The two middle calls act on disjoint banks and form one direct-sum +factor, giving five factors in either word. + +For the narrow reference, diagonal one and row gap \(d>0\) give row +norm at most two. During elimination, triangle inequalities give +\[ + \operatorname{gap}'_i\ge\operatorname{gap}_i+ + |a_{ik}/a_{kk}|\operatorname{gap}_k,\qquad + \operatorname{sum}'_i\le\operatorname{sum}_i- + |a_{ik}/a_{kk}|\operatorname{gap}_k . +\] +Therefore \(A_i,F\) retain norm at most two and gap at least \(d\). +At a largest component of \(x\), the gap inequality gives +\(\|Fx\|_*\ge d\|x\|_*\), and likewise for \(A_i\). +I may take +\[ + \mathcal K_N=\max(3,d^{-1}) +\] +for the five exact factor norms. + +For the broad reference, inward coefficient rounding and the Gaussian +lattice-sum bound \(1+\sigma\) give both row and column sums at most +three. This includes the identity on hole columns. To bound the +ideal interior inverse, put \(r=(B-1)/2\), \(v=\lambda\theta/\alpha^2\) +and +\[ + F_g=\max\left\{ + \alpha^2\theta r^2,\ + {1\over4\alpha^2\rho^2}+v(\lambda r^2+r/\rho),\ + v(r^2+2r/\rho)\right\}. +\] +The anchored identities in RAW-GAUSSIAN-PROOF.md and +\(|\delta|\le1/(2\rho)\) bound the absolute exponents of both diagonal +families by \(F_g\). Choose a certified positive +\(q\le e^{-\pi F_g}\), put \(G=\alpha/q\), and use the earlier +cap-wide conditioning certificate +\[ + H=\overline Q(B,\lambda/\alpha^2),\qquad Q_{\rm ideal}=G^2H . +\] +It bounds the actual ideal kernels and their inverses for either +family, including smaller interiors and closed half shifts. It need +not dominate each independently computed certificate's upper endpoint. +Neumann perturbation at \(E_MQ_{\rm ideal}\le1/2\) gives +\(\|A^{-1}\|_\infty\le2Q_{\rm ideal}\). From (2), +\[ + \|F\|_\infty\le3+18Q_{\rm ideal},\qquad + \mathcal K_M=3+18Q_{\rm ideal}. +\] +Since \(Q_{\rm ideal}\ge1\), this also bounds both shears, \(A\) and +\(A^{-1}\). + +## One radius, reference allocation and data grid + +Write \(\mathcal K\) for the appropriate factor bound and set +\[ + Y_*=6\mathcal K^5Y_0,\quad + e={2^{-t}\over10\mathcal K^4},\quad + R_D=\lceil\sqrt B\rceil Y_*,\quad + t_D=t+\lceil\log_2(20\mathcal K^4)\rceil. \tag{5} +\] +Each interior vector at radius \(Y_*\) has Euclidean norm at most +\(R_D\). The raw contract gives Euclidean error below +\(2^{-t_D}\le e/2\); conversion to maximum complex-l1 error costs at +most \(\sqrt2\), hence remains below \(e\). + +For a noncircular reference allocation, I first enclose +\[ + d_0=1-\operatorname{upper}(e^{-2\pi\alpha^2\theta}) + -5\operatorname{upper}(e^{-2\pi\alpha^2}) +\] +and require \(d_0>0\). Before preparing a reference, I choose +\(\mathcal K_*=\max(3,2/d_0,4+32Q_{\rm ideal})\) and compute (5) +with this bound. I call `Reference.for_cap` at target at least +\[ + \max(t_D,\lceil\log_2(2/d_0)\rceil, + \lceil\log_2(2Q_{\rm ideal})\rceil,1). +\] +I check the resulting \(E_N\le d_0/2\), actual certified \(d\ge d_0/2\), +and \(E_MQ_{\rm ideal}\le1/2\). The reference allocator's conservative +cap-wide gauge also dominates the raw children's requirements. +Thus \(\mathcal K_*\) funds both later directions without selecting +the window or computing its Schur complement. I reject periods too +short for the allocated cuts, as in the earlier reference contract. + +For a fixed reference, `Core` constructs each raw probe at \((R_D,t_D)\) +and rejects inadequate reference precision. It then sets one grid +\(P\) to the maximum of \(F_0\), the raw children's grids and the +sparse/boundary requirements below. Reinstantiation at incoming grid +\(P\) preserves that same \(P\), because the child allocations use a +maximum with their incoming precision. + +## Prepared boundary execution + +For narrow \(F\), choose the auxiliary integer +\[ + p=\max(W+1,\lceil\sqrt{1/(16d)}\rceil). +\] +This \(p\) is an envelope parameter for the boundary theorem, not an +assertion about the source construction's original precision parameter. +Then \(d\ge1/(16p^2)\) and \(b_F<2p\). Choose a dyadic block width +\(\max(1,b_F)\le w_B\le4p\). + +I prepare exact ordinary banded \(F=LU\). Gap and row-norm preservation +give positive pivots at least \(d\), \(\|U\|_\infty\le2\), and +\[ + \|L\|_\infty\le1+2b_F/d\le65p^3,\quad + \|L^{-1}\|_\infty\le32p^2,\quad + \|U^{-1}\|_\infty\le1040p^5. \tag{6} +\] +The last two follow from \(L^{-1}=UF^{-1}\) and \(U^{-1}=F^{-1}L\). +Consecutive triangular inverse pages equal principal blocks of the +full triangular inverse. I pad the final page by identity. + +I retain PR4186's rounding adapter and the underlying block/Strassen +program byte for byte. I prepare rounded factor pages and rounded +inverses of the **exact** consecutive diagonal blocks. During execution +I replace the source inverse callback with a sequential reader of +these prepared inverse pages. I check the next rounded diagonal page +against its expected value. The callback performs no inverse. +The retained adapter's rounding of an already dyadic subdiagonal page +preserves its value; its multiplication, addition and output-rounding +sites remain unchanged. This proves value equivalence with the retained +rounding adapter evaluated on exact reference factors. In the finite +comparison I cache exact inverse evaluations for that baseline. +I replace only its diagnostic matrix digest with binary rational hashing. + +I restate the needed residual argument to avoid an unchecked dependency +on the sibling PR's proof. Let \(\eta=2^{1-P}\). For exact diagonal +block \(A_0\) and subdiagonal \(B_1\), the prepared pages obey +\(\|\Delta D\|,\|\Delta B\|\le w_B\eta\). The source executes +\[ + v=\operatorname{rnd}_P(y-\widehat B_1x_{\rm prev}),\qquad + x=\operatorname{rnd}_P(\widehat Dv). +\] +With rounding errors \(u,f\), the exact residual is +\[ + A_0x+B_1x_{\rm prev}-y + =A_0\Delta Dv-\Delta Bx_{\rm prev}+u+A_0f. \tag{7} +\] +For \(a\ge\|A_0\|,\ b_0\ge\|B_1\|,\ Q_T\ge\|T^{-1}\|\), define +\[ + c_0=aw_B+2a+1,\qquad c_1=w_B[a(b_0+1)+1]. +\] +If \(w_B\eta\le1,\eta c_0\le1,Q_T\eta c_1\le1/2\), (7) gives +\[ + X\le4Q_TY,\qquad + \|\widehat x-T^{-1}y\|_* + \le Q_T\eta(c_0+4Q_Tc_1)Y . \tag{8} +\] +I derive \(X\) by absorption from the completed residual, rather than +assuming the exact-state bound for rounded data. + +Substitution of (6) gives per-pass error constants at most +\(2^{28}p^{11}\) for \(L\) and \(2^{30}p^{11}\) for \(U\). +The guard \(\eta\le2^{-21}p^{-9}\) suffices for both absorptions. +The first output has magnitude at most \(2^7p^2Y_*\). +After reversing both axes for the upper pass and restoring row order, +I obtain +\[ + \|\widehat x-F^{-1}y\|_*<2^{40}p^{16}\eta Y_* . \tag{9} +\] +The two contributions are at most \(2^{39}p^{16}\eta Y_*\) and +\(2^{37}p^{13}\eta Y_*\). +Real/imaginary RHS packing preserves (7)-(9): the pair of component +truncations already costs at most \(\eta\) in complex-l1 norm. + +For \(q_e=t+\lceil\log_2(10\mathcal K^4)\rceil\), I require +\[ + P\ge q_e+41+16\lceil\log_2p\rceil+\lceil\log_2Y_*\rceil . +\] +Then (9) is at most \(2^{-q_e}\le e\), and the absorption guards hold. + +## Sparse stages, total error and finite storage + +For a row with at most \(s\) coefficients, I round the exact real +coefficients once, accumulate products exactly, then truncate both +completed output components. At input radius \(Y_*\), this costs at +most \(\eta(sY_*+1)\). I use \(s=2W+1\) for original coupling rows +and \(s=2b_F+1\) for broad Schur rows. It suffices to require +\[ + P\ge1+\left\lceil\log_2{(sY_*+1)\over e}\right\rceil . +\] +I compute each shear's updated bank from the prior bank values and +commit the completed rows before the next stage. + +Let \(E_j\) denote the discrepancy after \(j\) exact factors. +Equations (5), (9) and the sparse bound give +\(E_{j+1}\le\mathcal K E_j+e\). +The two middle calls update disjoint banks, so I use their maximum +error, with no extra summand. Thus +\[ + E_5\le5\mathcal K^4 e=2^{-t-1}. +\] +The corresponding magnitude recurrence keeps each actual factor input +and the intermediate middle state below \(Y_*=6\mathcal K^5Y_0\). +This induction justifies the input envelopes used for the local bounds. +The number of interiors and RHS columns does not enter this maximum norm. + +For sparse arithmetic, inward coefficient rounding cannot increase an +absolute row sum. The allowance \(16(\mathcal K+1)Y_*\) covers products, +partial accumulators and completed shears. PR4186's exact Strassen +scratch argument bounds both boundary passes by \(2^{46}p^{16}Y_*\): +at depth \(d\) each operand has at most \(2^d\) root terms, while each +return combines at most four child products. The conservative per-pass +allowance is \(128w_B^4Q_T^2(b_0+1)Y\). +Products use the \(2P\)-grid and additions preserve it. +I take the maximum of these bounds and each raw child's temporary +bound, retaining its rational coefficient-height charge. + +For \(n_J\) boundary rows and \(k\) complex RHS columns, each triangular +pass uses +\[ + (2\lceil n_J/w_B\rceil-1)\lceil2k/w_B\rceil + \quad\hbox{products of size }w_B,\qquad + 7^{\log_2w_B}\quad\hbox{scalar products per product}. +\] +I retain the ceiling tariff for partial row and RHS batches. +I keep exact Schur formation, LU, inverse-page generation and checker +oracles outside the online arithmetic counts. I make no fixed-tape or +complete setup-cost claim from their host-language implementation. + +## Evidence, credit and remaining scope + +`verify_core.py` checks exact reference residuals after the narrow +boundary stage and complete inverse word; division by the certified gap +converts those residuals to error bounds. For broad execution it compares +the complete result with the exact reference product. It checks each +stage against its actual current input, chronological continuity, input +grids and measured arithmetic widths. + +I credit Vlado Vojdanovski's retained carve, factor word, block program, +Strassen implementation and gap-preservation argument. I reuse our +PR4186 residual analysis and the earlier PR4205 raw/normalized circuits, +with the predecessor credits in README.md. I add the shared-reference +core integration, prepared-page lowering and one composed error budget. +Internal analysis agents contributed to the author process. + +Reviewers still need to assess this quantified proof. The finite receipt +does not establish arbitrary-size execution. I leave periodic corner +treatment, the core/halo comparison, tensor composition, physical page +delivery and the complete multiplication theorem open. 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false, + "tensor_execution": false, + "stored_global_inverse_pair": false, + "independent_transcendental_backend": false, + "universal_proof_kernel": false +} diff --git a/research/rational-unitary-fft/README.md b/research/rational-unitary-fft/README.md index 664a630d90..33000ee730 100644 --- a/research/rational-unitary-fft/README.md +++ b/research/rational-unitary-fft/README.md @@ -1,4 +1,26 @@ -# Raw Gaussian interiors against a shared rounded reference +# Rounded Gaussian factor words on shared finite-core references + +I execute complete ordinary finite-core narrow inverse and broad forward +words against one shared rational reference. I retain the original +constant-phase carve and six-call order, prepare exact Schur pages from +that reference, and join the raw interior circuits below to PR4186's +rounded boundary program. I supply the boundary program with prepared +dyadic inverse pages and preserve its arithmetic and rounding sites. + +For input maximum complex-l1 norm \(Y_0\), five-factor norm bound +\(\mathcal K\), and target \(2^{-t}\), I allocate +\[ +Y_*=6\mathcal K^5Y_0,\qquad e=2^{-t}/(10\mathcal K^4). +\] +I fund the common reference for the amplified dense inputs, preserve +the incoming grid and obtain complete error at most \(2^{-t-1}\). +[CORE-PROOF.md](CORE-PROOF.md) gives the domain, exact-page identities, +boundary residual argument and composed error/temporary bounds. +I require an ordinary window with enough omitted coordinates to remove +periodic corner couplings. I leave core/halo approximation, tensor +composition and physical page/tape costs open. + +## Retained raw and child results I construct fixed rational narrow and broad references for a periodic source/target axis, retaining original phase cells, the period cut and @@ -18,8 +40,9 @@ I keep full-cell weights in offline setup at \(O(b_e+\alpha^2/\theta)\) bits. I also compute a row-gap certificate for the same narrow reference; a positive gap gives dimension-independent row/column inverse bounds for its principal restrictions in the original -circular coordinates. I leave the carve, boundary program, physical -delivery and complete tensor/tape execution open. +circular coordinates. The finite-core extension above now joins the +carve and boundary program; physical delivery and complete tensor/tape +execution remain open. `Reference.for_cap` chooses one global cutoff and coefficient precision for a declared block cap, radius and accuracy. I reject periods too short for those cuts. Coordinate and coefficient queries avoid enumerating the @@ -144,12 +167,21 @@ python3.12 -B scripts/research_package.py \ --package research/rational-unitary-fft ``` -I declare eight normal/optimized commands, with limits of 60, 60, 90, -90, 150, 150, 300 and 300 seconds, totaling 1,200 seconds. -In each final slot I run the full, normalized Gaussian and raw Gaussian -verifiers sequentially in the same process. I preserve the full and -[GAUSSIAN-RESULT.json](GAUSSIAN-RESULT.json) receipts and compare the -separate [RAW-GAUSSIAN-RESULT.json](RAW-GAUSSIAN-RESULT.json) receipt. +I declare eight normal/optimized commands, with limits of 20, 20, 30, +30, 50, 50, 500 and 500 seconds, totaling 1,200 seconds. +In each final slot I run the full, normalized, raw and finite-core +verifiers sequentially in the same process. I preserve six earlier +scientific receipts and compare [CORE-RESULT.json](CORE-RESULT.json). +Four core fixtures include two 143-row words, each with two nontrivial +16-row interiors, and two 11-row windows crossing a hole. +I execute 88,704 dense gate applications and 19,992 boundary scalar +products. I check all six stages against their actual inputs, complete +reference residuals/products, a 768-bit incoming grid and 13 controls. +The core receipt is +`65f0cefaed8d7b8b21125cde5155401b3880e896b49068e8e8c86895b5004e4d`. +These fixtures use cap 16 and physical half-band 18; I make no +large-cap scaling claim from them. I hash exact child plans and large +checker fractions, and report dyadic upper bounds for their errors. For the raw extension I check 2,178 entries across two complete 33-by-33 references, with three explicit windings and analytic tails. I check row gaps, a principal inverse across a cell boundary and a hole, @@ -312,8 +344,9 @@ Mason's backend. For the raw extension I use the retained narrow weight-ratio reference, while I choose a fixed entrywise broad reference and prove its comparison. I do not identify it with the older factorwise-rounded broad matrix. -The block admission checks constant phase and the original cut; it does -not implement the outer carving algorithm. +The raw block admission checks constant phase and the original cut. +The new core adapter applies the retained carving algorithm to those +same source coordinates. Before this extension I refreshed unchanged main, checked 15 recent PR records and 6/790 hub comments, and compared PR4214's GL3 audit and PR4208's CRT stencil work. I found no overlapping raw-reference claim @@ -321,6 +354,23 @@ in that coverage. Broader history and unpublished work remain outside it. Before publication I checked 12 newer PR records, 5/815 hub comments and the new scalar-audit handoff while canonical main remained unchanged. +I preserve raw head `c9db4069d59bfad26a15fe000b91cc3a058684e5` and its +[hosted replay](https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39477979592). +For the core extension I credit Vlado's retained carve, factor order, +block/Strassen program and Schur gap argument. I archive our PR4186 +adapter unchanged with its original provenance and restate the needed +residual proof in CORE-PROOF.md. I add the exact common-reference pages, +prepared-page execution and one whole-word error budget. +I compared Nick Petalas's [PR4217](https://github.com/CrocSwap/integer-mult-bounds/pull/4217) +HALO locality/error audit, including its factor-two correction. That +work addresses a separate obligation; I do not include a HALO theorem +in this finite-core result. Initial overlap covers 15 recent PR records +and 6/816 hub comments. Broader unpublished work remains outside it. +Before submission I refreshed unchanged canonical main and checked +12 recent PR records, 5/853 hub comments and the canonical thread. +The newer connector, fallback-header and GL3 audits address separate +obligations within that coverage. + I preserve the first completed revision `f02a4ab5ab70a8af80415ac35c5b48cd5a93a475` and its FFT receipt `2e203e42d3d617e828e3341ffeb44254c931db2aebb8a040d04d67338b5b4016`. @@ -348,8 +398,8 @@ Apache-2.0. ## Open next questions -In [TASK.json](TASK.json) I specify the next steps and blockers: bind the -actual interior/boundary carve and Schur pages to these shared references, -then connect the coefficients and data widths to physical delivery. +In [TASK.json](TASK.json) I specify the next steps and blockers: connect +the finite-core result to an explicit core/halo error comparison and +join its prepared coefficient pages and data widths to physical delivery. Reviewers still need to assess the universal proofs, complete setup, fixed-tape lowering and the full multiplication construction. diff --git a/research/rational-unitary-fft/SOURCES.json b/research/rational-unitary-fft/SOURCES.json index 0fe6eb0e50..7ee22e983f 100644 --- a/research/rational-unitary-fft/SOURCES.json +++ b/research/rational-unitary-fft/SOURCES.json @@ -26,6 +26,11 @@ {"path":"research/capped-phase-fusion/BILATERAL-LDU.md","usage":"input","git_blob":"5cad6ee19f60d626beb9f6d7be78cbc22fdd08c6","sha256":"0d65e66d227464dda1a19f781ae0a0be671cbfd347c41e37ec954f3d629d8a77"}, {"path":"research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md","usage":"input","git_blob":"3bcf490a8c164d44aff5b7e2ec2e0f4bf07d7a73","sha256":"e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272"}, {"path":"research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md","usage":"input","git_blob":"bbeeee7431720817f031360f67b2186abac5cd23","sha256":"54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a"}, - {"path":"research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md","usage":"input","git_blob":"0f7f80b3f74d469c2f72f3b9f338d32568b4eae3","sha256":"e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0"} + {"path":"research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md","usage":"input","git_blob":"0f7f80b3f74d469c2f72f3b9f338d32568b4eae3","sha256":"e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0"}, + {"path":"research/rational-unitary-fft/vendor/boundary_adapter.py","usage":"input","pin":"head:c13c08ba4f957af70cfc2560ea43d837fe730d21:research/rational-unitary-fft/vendor/boundary_adapter.py","git_blob":"cc9c6815791a8290537d56bb69320201173e1978","sha256":"6738df51cdf7ecb55d1521b9f8d38e627c8feb4c4afe242ba2344ffc024c772e"}, + {"path":"research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json","usage":"input","pin":"head:c13c08ba4f957af70cfc2560ea43d837fe730d21:research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json","git_blob":"7286022fdd083adea45a2e7151b71ef1ec38381e","sha256":"93a0f0584ae8760abc83752f9f151d62e41bfa7f9ade682edef61ff07dec4fb9"}, + {"path":"research/capped-phase-fusion/evidence/radical_reduction_ultra/check_constructions.py","usage":"input","git_blob":"2090a45566fc0fc7af4e2b448833a9c4de578b96","sha256":"91387c9bd763a5c55c22a9907947bd07b6002f739aa6feedf6d3a02adea1e316"}, + {"path":"research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py","usage":"input","git_blob":"d6f5a14cb26c04ab3feb728430005808e56613fa","sha256":"121279bef47dd490bf29fc72ddbce547176cdaf1154af67dc8b96d7baa8cfc6f"}, + {"path":"research/capped-phase-fusion/evidence/phase_layout_audit_ultra/check_union.py","usage":"input","git_blob":"0b3f0af7014dc23d8d80a82c67f2f7b24d3bd1ab","sha256":"74608380d72567edcd4470c8d226b15848f9d3a4e5d77d652bdd8963480ab9ce"} ] } diff --git a/research/rational-unitary-fft/TASK.json b/research/rational-unitary-fft/TASK.json index 02be43ed22..d2e5d013ab 100644 --- a/research/rational-unitary-fft/TASK.json +++ b/research/rational-unitary-fft/TASK.json @@ -38,21 +38,21 @@ ] }, "scope_status": "refined", - "actual_scope": "Construct shared rounded periodic narrow/broad references and execute anchored raw constant-phase interior restrictions and their stored inverses, with positive-weight setup, global tail/row-gap bounds and complete reference-error/online-width budgets.", - "scope_change_reason": "The normalized circuit omits raw diagonal/scalar factors and the fixed-reference comparison. I preserve five earlier receipts and supply those factors and their complete error composition, with source geometry, distinct hole rules, original period cuts and shared pages.", + "actual_scope": "Execute ordinary finite-core narrow inverse and broad forward Gaussian words with retained carving, shared-reference Schur pages, raw interior circuits, prepared rounded boundary arithmetic and one complete radius/grid/error budget.", + "scope_change_reason": "The completed raw interior contract leaves the actual carve, sparse couplings and boundary program unjoined. I preserve six earlier receipts and connect these stages against the same reference, with exact offline pages and one online grid.", "bottleneck_link": "The Gaussian caller needs full circuit execution and a quantified conditioning/rounding contract before physical delivery and the complete numerical composition can be admitted.", "budget_minutes": 120, "stop_rule": "Stop at an invalid phase enclosure, nonunitary adapted gate, missing multiplier/twist error, understated word width or source-schedule mismatch. Do not equate coefficient-level unitarity with exact rounded restoration or physical execution.", "outcome": "completed", - "evidence": "I preserve the five completed scientific receipts. RAW-GAUSSIAN-PROOF.md derives positive relative coefficients, uniformly allocated common periodic references, both tail norms, a narrow row-gap certificate and complete raw forward/inverse reference-error bounds. RAW-GAUSSIAN-RESULT.json records two complete 33x33 references, 6 circuit fixtures, 1086 gates, 26 rounded states and 17 controls, plus a local query on a source axis of size10^20+1 without axis enumeration. Comparisons share the declared interval backend; scientific acceptance remains separate.", + "evidence": "I preserve six earlier scientific receipts. CORE-PROOF.md derives ordinary carve geometry, same-reference factor norms, noncircular allocation, prepared boundary-page equivalence and complete error at most 2^(-t-1). CORE-RESULT.json records four cases, four nontrivial interiors, 88704 dense gate applications, 19992 boundary scalar products and 13 controls. I check stage inputs, narrow residuals, broad exact products and incoming/temporary widths. The finite cap16 fixtures have physical half-band18. Comparisons share the declared scalar backend; scientific acceptance remains separate.", "author": "Jose Manuel (jmbarrancoml)", "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, source analysis, implementation and writing, including source-analysis agents. No independent human or formal-kernel acceptance.", "open_questions": [ { - "question": "Can the actual carve and five-factor boundary word use the same common reference and raw interior programs?", - "why_it_matters": "Correct constant-phase restrictions do not establish separated interiors, shared boundary pages or the complete Gaussian map. Broad boundary Schur factors require application; narrow inverse execution needs a solve.", - "first_step": "Construct the retained separated dyadic carve with original cuts and holes, derive its exact Schur and coupling pages from Reference, and bind the applied broad and solved narrow boundary stages to the same forward/inverse error ledger.", - "blocked_by": "Carve and boundary-program integration, followed by physical setup, routing and tape costs." + "question": "Can the complete finite-core error join the actual core/halo comparison on the same reference?", + "why_it_matters": "A finite reference word does not establish the analytic error of overlapping global cores or remove cyclic corners.", + "first_step": "Bind the caller's actual windows and overlap rule to this ordinary-core domain, incorporate PR4217's scoped HALO corrections, and pay both analytic and numerical errors.", + "blocked_by": "Caller geometry, analytic comparison and a complete outer error ledger." }, { "question": "Can the coefficient preparation and canonical data widths be joined to the physical Gaussian caller?", diff --git a/research/rational-unitary-fft/VERIFY.json b/research/rational-unitary-fft/VERIFY.json index 672ac0c05f..4d9180b2f9 100644 --- a/research/rational-unitary-fft/VERIFY.json +++ b/research/rational-unitary-fft/VERIFY.json @@ -26,48 +26,53 @@ "research/capped-phase-fusion/BILATERAL-LDU.md", "research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md", "research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md", - "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md" + "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md", + "research/rational-unitary-fft/vendor/boundary_adapter.py", + "research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json", + "research/capped-phase-fusion/evidence/radical_reduction_ultra/check_constructions.py", + "research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py", + "research/capped-phase-fusion/evidence/phase_layout_audit_ultra/check_union.py" ], "commands": [ { "argv": ["{python}", "-I", "-B", "{package}/verify.py"], - "timeout_seconds": 60, + "timeout_seconds": 20, "checks": "Authenticate and adapt the actual FFT/shear schedule, certify rational phases and unitary gates, prepare charged spectral coefficients, execute fixed-grid dirty data, compare independent Fourier/Toeplitz references and reject schedule/precision controls." }, { "argv": ["{python}", "-I", "-O", "-B", "{package}/verify.py"], - "timeout_seconds": 60, + "timeout_seconds": 20, "checks": "Repeat the complete gate/rounding certificate under optimized Python, keeping authenticated source assertions and exact typed receipt comparison." }, { "argv": ["{python}", "-I", "-B", "{package}/verify_triangle.py"], - "timeout_seconds": 90, + "timeout_seconds": 30, "checks": "Execute the retained parity recursion and exact stored inverse on one grid; check macro identities, independent exact endpoint inverses, global error/width budgets, reversed indices, nonunitary leaves and adversarial controls." }, { "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_triangle.py"], - "timeout_seconds": 90, + "timeout_seconds": 30, "checks": "Repeat the complete recursive forward/inverse certificate under optimized Python with authenticated source assertions and complete typed receipt comparison." }, { "argv": ["{python}", "-I", "-B", "{package}/verify_generator.py"], - "timeout_seconds": 150, + "timeout_seconds": 50, "checks": "Execute the exact rational producer and retained compensated generator gate program on actual Schur fixtures; check paired spectral coefficients, additive endpoint defects, dirty-source rounding, intermediate widths and cancellation controls." }, { "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_generator.py"], - "timeout_seconds": 150, + "timeout_seconds": 50, "checks": "Repeat the complete generator program with source assertions enabled under optimized Python and compare its full typed receipt." }, { - "argv": ["{python}", "-I", "-B", "{package}/verify_full.py", "--with-gaussian", "--with-raw"], - "timeout_seconds": 300, - "checks": "Execute the full Toeplitz verifier, then normalized Gaussian verification, preserving both receipts. Then check one periodic rounded reference, holes/cuts/overlaps and row-gap admission, execute anchored raw broad/narrow programs and their stored inverses, and compare rounded errors and widths against the same reference restrictions." + "argv": ["{python}", "-I", "-B", "{package}/verify_full.py", "--with-gaussian", "--with-raw", "--with-core"], + "timeout_seconds": 500, + "checks": "Run full, normalized and raw verifiers sequentially, preserving their receipts. Then execute the retained finite-core carve and factor words with shared-reference Schur pages, raw interiors and prepared boundary pages. Check each actual stage input, complete narrow residual and broad product, incoming grids, arithmetic widths and 13 controls." }, { - "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_full.py", "--with-gaussian", "--with-raw"], - "timeout_seconds": 300, - "checks": "Repeat full, normalized and raw Gaussian verification sequentially under optimized Python. Preserve source assertions, compare three complete typed receipts, and reject precision, inverse-wrapper, stale-reference, hole-completion and period/cell controls." + "argv": ["{python}", "-I", "-O", "-B", "{package}/verify_full.py", "--with-gaussian", "--with-raw", "--with-core"], + "timeout_seconds": 500, + "checks": "Repeat full, normalized, raw and finite-core verification sequentially under optimized Python. Preserve source assertions and compare four complete typed receipts, including exact child-plan hashes, prepared-page equivalence and corrupted-page rejection." } ] } diff --git a/research/rational-unitary-fft/core.py b/research/rational-unitary-fft/core.py new file mode 100644 index 0000000000..218bee551a --- /dev/null +++ b/research/rational-unitary-fft/core.py @@ -0,0 +1,629 @@ +"""Shared-reference finite-core Gaussian words and rounded boundary arithmetic. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. Carve, factor order and block/Strassen program: Vlado's retained +lineage. Rounded boundary adapter: PR4186. Gaussian credits remain in README. +""" +import ast +import copy +from fractions import Fraction as Q +import hashlib +import json +from types import SimpleNamespace + +from exact import C, Mat, Vec, apply, ceil_log2, need, qheight, upper, vector_hash +from gaussian import Gaussian, rational +from gaussian_reference import Exponentials, Reference, ceiling_sqrt +from raw_gaussian import RawGaussian + +ADAPTER = "research/rational-unitary-fft/vendor/boundary_adapter.py" +PROVENANCE = "research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json" +SUBJECT = "research/capped-phase-fusion/evidence/radical_reduction_ultra/check_constructions.py" +CARVE = "research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py" +WORD = "research/capped-phase-fusion/evidence/phase_layout_audit_ultra/check_union.py" + + +def matrix_hash(matrix): + shape = (len(matrix), len(matrix[0]) if matrix else 0) + values = (C.wrap(x) for row in matrix for x in row) + return hashlib.sha256((str(shape) + vector_hash(values)).encode()).hexdigest() + + +def plan_hash(plan): + digest = hashlib.sha256() + for layer in plan: + digest.update(str(len(layer)).encode() + b":") + for wires, gate in layer: + digest.update((str(tuple(wires)) + matrix_hash(gate)).encode()) + return digest.hexdigest() + + +def raw_core_record(program): + """Bind exact child values without decimal-expanding checker fractions.""" + errors = [program.reference_error, program.diagonal_error, program.middle_error, + program.raw_error, program.forward_error, program.inverse_error] + bits = max(160, program.target + 32) + return dict(kind=program.kind, start=program.start, n=program.n, + radius=str(program.radius), target_bits=program.target, + incoming_bits=program.incoming, data_bits=program.data_bits, + reference_matrix_sha256=program.reference_hash, + normalized_root_sha256=program.child.root_hash, + source_ast_sha256=program.child.source_ast, + callsite_ast_sha256=program.child.callsite_ast, + plan_sha256=plan_hash(program.plan), + inverse_plan_sha256=plan_hash(program.inverse_plan), + exact_errors_sha256=vector_hash(C(x) for x in errors), + error_upper=dict(zip(("reference", "diagonal", "middle", "raw", + "forward", "inverse"), + (str(upper(x, bits)) for x in errors))), + budget={key: str(upper(value, bits)) if isinstance(value, Q) else value + for key, value in program.bound.items()}) + + +def max_complex(matrix): + return max((abs(x.re) + abs(x.im) for row in matrix for x in row), default=Q(0)) + + +def real(matrix): + need(all(x.im == 0 for row in matrix for x in row), "real reference pages") + return [[x.re for x in row] for row in matrix] + + +def row_norm(matrix): + return max((sum(abs(x) for x in row) for row in matrix), default=Q(0)) + + +def row_gap(matrix): + return min(abs(row[i]) - sum(abs(x) for j, x in enumerate(row) if i != j) + for i, row in enumerate(matrix)) + + +def bandwidth(matrix): + return max((abs(i - j) for i, row in enumerate(matrix) for j, x in enumerate(row) if x), default=0) + + +def product(a, b): + """Exact sparse setup/checker product, outside online arithmetic counts.""" + if not a: + return [] + columns = len(b[0]) if b else 0 + nonzero = [[(j, x) for j, x in enumerate(row) if x] for row in b] + result = [[Q(0) for _ in range(columns)] for _ in a] + for i, row in enumerate(a): + for k, x in enumerate(row): + if x: + for j, y in nonzero[k]: + result[i][j] += x * y + return result + + +def block(matrix, rows, columns): + return [[matrix[i][j] for j in columns] for i in rows] + + +def band_lu(matrix, band): + """Exact natural-order LU; skip entries that ordinary band geometry makes zero.""" + n = len(matrix) + need(bandwidth(matrix) <= band, "ordinary reference factorization band") + lower = [[Q(i == j) for j in range(n)] for i in range(n)] + upper = [[Q(0) for _ in range(n)] for _ in range(n)] + for i in range(n): + for j in range(i, min(n, i + band + 1)): + upper[i][j] = matrix[i][j] - sum(lower[i][k] * upper[k][j] + for k in range(max(0, i - band), i)) + need(upper[i][i] > 0, "positive no-pivot narrow diagonal") + for j in range(i + 1, min(n, i + band + 1)): + lower[j][i] = (matrix[j][i] - sum(lower[j][k] * upper[k][i] + for k in range(max(0, j - band), i))) / upper[i][i] + need(product(lower, upper) == matrix, "exact band LU identity") + return lower, upper + + +def load_boundary(raw): + provenance = json.loads(raw[PROVENANCE]) + need(provenance["retained_path"] == ADAPTER + and hashlib.sha256(raw[ADAPTER]).hexdigest() == provenance["sha256"], + "unchanged boundary adapter provenance") + names = {"need", "sha", "encoded", "matrix_digest", "maximum", "norm", "identity", + "multiply", "subtract", "solve", "lu", "reverse_both", "rounded", "on_grid", + "RoundedSubject", "pass_constants", "check_pass"} + tree = ast.parse(raw[ADAPTER]) + tree.body = [copy.deepcopy(node) for node in tree.body + if isinstance(node, (ast.FunctionDef, ast.ClassDef)) and node.name in names] + need({node.name for node in tree.body} == names, "complete boundary definition closure") + seal = hashlib.sha256(ast.dump(tree).encode()).hexdigest() + namespace = dict(Q=Q, hashlib=hashlib, json=json, SUBJECT=SUBJECT) + exec(compile(tree, "", "exec", optimize=0), namespace) + # This diagnostic hashes exact values without Python's decimal-string limit. + # Arithmetic, pages, rounding sites and source counters remain unchanged. + namespace["matrix_digest"] = matrix_hash + namespace["source_ast_sha256"] = seal + return SimpleNamespace(**namespace) + + +def geometry_bounds(raw, source_size, excess, alpha, cap): + theta, rho = Q(excess, source_size), Q(source_size + excess, source_size) + lam, radius = 1 / rho, Q(cap - 1, 2) + rate = lam * theta / alpha ** 2 + exponent = max(alpha ** 2 * theta * radius ** 2, + 1 / (4 * alpha ** 2 * rho ** 2) + rate * (lam * radius ** 2 + radius / rho), + rate * (radius ** 2 + 2 * radius / rho)) + exp = Exponentials(raw) + decay = exp.relative(exponent, exponent, 24) + gauge = alpha / decay["value"] + middle = Gaussian(raw, cap, lam / alpha ** 2, 0, 1, 1).qbar + return gauge, middle, gauge ** 2 * middle + + +def allocated_reference(raw, source_size, excess, alpha, cap, y0, target): + """Fund one reference for either finite-core direction before choosing windows.""" + need(all(type(x) is int for x in (source_size, excess, alpha, cap, target)) + and 0 < excess < source_size and alpha >= 1 and cap > 0 + and cap & (cap - 1) == 0 and 2 * cap <= source_size and target >= 1, + "core reference geometry and target") + y0 = rational(y0) + need(y0 >= 1, "core reference input envelope") + theta = Q(excess, source_size) + bits = max(96, 2 * max(0, ceil_log2(1 / theta)) + 64) + exp = Exponentials(raw) + near, far = exp.decay_box(2 * alpha ** 2 * theta, bits), exp.decay_box(2 * alpha ** 2, bits) + d0 = 1 - Q(near.hi, near.scale) - 5 * Q(far.hi, far.scale) + need(d0 > 0, "certified ideal narrow gap before reference allocation") + gauge, middle, inverse = geometry_bounds(raw, source_size, excess, alpha, cap) + k = max(Q(3), 2 / d0, 4 + 32 * inverse) + ystar = 6 * k ** 5 * y0 + radius = ceiling_sqrt(cap) * ystar + dense_target = target + ceil_log2(20 * k ** 4) + reference_target = max(dense_target, ceil_log2(2 / d0), ceil_log2(2 * inverse), 1) + reference = Reference.for_cap(raw, source_size, excess, alpha, cap, radius, reference_target) + need(reference.errors["narrow"] <= d0 / 2 + and reference.narrow_gap >= d0 / 2 + and reference.errors["broad"] * inverse <= Q(1, 2), + "funded core gap and broad inverse perturbation") + return reference, dict(ideal_gap_lower=str(d0), scalar_bits=bits, gauge_bound=str(gauge), + normalized_bound=str(middle), raw_inverse_bound=str(inverse), + factor_bound=str(k), stage_radius=str(ystar), dense_radius=str(radius), + dense_target_bits=dense_target, reference_target_bits=reference_target) + + +class Boundary: + def __init__(self, raw, adapter, matrix, width, p, bits, ystar, gap): + self.raw, self.adapter, self.matrix = raw, adapter, matrix + self.width, self.p, self.bits, self.ystar, self.gap = width, p, bits, ystar, gap + n, band = len(matrix), bandwidth(matrix) + need(row_norm(matrix) <= 2 and row_gap(matrix) >= gap >= Q(1, 16 * p ** 2) + and band <= 2 * p and max(1, band) <= width <= 4 * p, "actual boundary admission") + self.lower, upper = band_lu(matrix, band) + self.upper_reverse = adapter.reverse_both(upper) + need(row_norm(self.lower) <= 65 * p ** 3 and row_norm(upper) <= 2, "reference factor norms") + self.exact_inverses, self.pages = {}, [] + self.factor_pages, self.diagonal_pages, self.inverse_pages = {}, {}, {} + prototype = adapter.RoundedSubject(raw[SUBJECT], bits, width, ystar, Q(32 * p ** 2), + Q(65 * p ** 3)) + self.source_inverse = prototype.exact_inverse + for label, factor, inverse_bound in ( + ("lower", self.lower, Q(32 * p ** 2)), + ("upper", self.upper_reverse, Q(1040 * p ** 5))): + self.factor_pages[label] = adapter.rounded(factor, bits) + need(row_norm(adapter.subtract(self.factor_pages[label], factor)) + <= width * Q(2, 1 << bits), "prepared factor-page error") + self.diagonal_pages[label], self.inverse_pages[label] = [], [] + for start in range(0, n, width): + page = [[factor[start + i][start + j] if start + i < n and start + j < n + else Q(i == j) for j in range(width)] for i in range(width)] + key = tuple(map(tuple, page)) + if key not in self.exact_inverses: + inverse = self.source_inverse(page) + need(product(page, inverse) == adapter.identity(width), "exact consecutive inverse page") + self.exact_inverses[key] = tuple(map(tuple, inverse)) + exact = self.exact_inverses[key] + need(row_norm(exact) <= inverse_bound, "reference inverse-page norm") + rounded = adapter.rounded(exact, bits) + need(row_norm(adapter.subtract(rounded, exact)) <= width * Q(2, 1 << bits), + "prepared inverse-page error") + self.diagonal_pages[label].append(adapter.rounded(page, bits)) + self.inverse_pages[label].append(rounded) + self.pages.append(dict(factor=label, start=start, input_sha256=matrix_hash(page), + exact_inverse_sha256=matrix_hash(exact), + rounded_inverse_sha256=matrix_hash(rounded))) + self.temporary_bound = (1 << 46) * p ** 16 * ystar + self.error_bound = (1 << 40) * p ** 16 * Q(2, 1 << bits) * ystar + self.source_ast = adapter.source_ast_sha256 + + def pass_run(self, rhs, upper=False): + p, bits, width = self.p, self.bits, self.width + a, sub, q = (Q(2), Q(2), Q(1040 * p ** 5)) if upper else ( + Q(65 * p ** 3), Q(65 * p ** 3), Q(32 * p ** 2)) + y = 128 * p ** 2 * self.ystar if upper else self.ystar + eta = Q(2, 1 << bits) + c0, c1, _ = self.adapter.pass_constants(a, sub, q, width) + need(width * eta <= 1 and eta * c0 <= 1 and q * eta * c1 <= Q(1, 2), + "boundary residual absorption") + worker = self.adapter.RoundedSubject(self.raw[SUBJECT], bits, width, y, q, sub) + label, cursor = ("upper" if upper else "lower"), [0] + def prepared_inverse(diagonal): + index = cursor[0] + need(index < len(self.inverse_pages[label]) + and diagonal == self.diagonal_pages[label][index], "sequential prepared diagonal page") + page = [list(row) for row in self.inverse_pages[label][index]] + cursor[0] += 1 + worker.inverse_pages += 1 + worker.inverse_ids.add(id(page)) + worker.observe(page) + return page + worker.ns["inverse"] = prepared_inverse + answer = worker.run(self.factor_pages[label], rhs) + need(cursor[0] == len(self.inverse_pages[label]), "complete prepared inverse-page stream") + magnitude = self.adapter.maximum(answer) + need(magnitude <= 4 * q * y and worker.max_value <= self.temporary_bound, + "boundary pass magnitude and scratch") + return answer, dict(root_products=worker.products, scalar_products=worker.ns["OPS"], + inverse_pages=worker.inverse_pages, rhs_roundings=worker.updates, + fractional_bits=worker.max_fractional, + maximum_upper=str(upper_bound(magnitude)), + temporary_maximum_upper=str(upper_bound(worker.max_value)), + batches=(len(rhs[0]) + width - 1) // width) + + def run(self, rhs): + columns = len(rhs[0]) + packed = [[x.re for x in row] + [x.im for x in row] for row in rhs] + forward, lower = self.pass_run(packed) + reversed_answer, upper = self.pass_run(list(reversed(forward)), upper=True) + answer = list(reversed(reversed_answer)) + result = Mat([[C(row[j], row[j + columns]) for j in range(columns)] for row in answer]) + need(max_complex(result) <= (1 << 20) * self.p ** 7 * self.ystar, + "complex boundary output envelope") + return result, dict(lower=lower, upper=upper, real_rhs_columns=2 * columns) + + def baseline(self, rhs): + """Verifier-only comparison with the unchanged reference-factor adapter.""" + columns = len(rhs[0]) + packed = [[x.re for x in row] + [x.im for x in row] for row in rhs] + for matrix, is_upper in ((self.lower, False), (self.upper_reverse, True)): + p = self.p + q, sub = (Q(1040 * p ** 5), Q(2)) if is_upper else (Q(32 * p ** 2), Q(65 * p ** 3)) + y = 128 * p ** 2 * self.ystar if is_upper else self.ystar + worker = self.adapter.RoundedSubject(self.raw[SUBJECT], self.bits, self.width, y, q, sub) + worker.exact_inverse = lambda page: self.exact_inverses[tuple(map(tuple, page))] + if is_upper: + packed = list(reversed(packed)) + packed = worker.run(matrix, packed) + return Mat([[C(row[j], row[j + columns]) for j in range(columns)] + for row in reversed(packed)]) + + def record(self): + return dict(rows=len(self.matrix), band=bandwidth(self.matrix), width=self.width, + p=self.p, bits=self.bits, gap_lower=str(self.gap), + error_bound=str(self.error_bound), temporary_bound=str(self.temporary_bound), + source_ast_sha256=self.source_ast, lower_sha256=matrix_hash(self.lower), + upper_reverse_sha256=matrix_hash(self.upper_reverse), + unique_exact_inverse_pages=len(self.exact_inverses), pages=self.pages) + + +def upper_bound(value): + return upper(value) + + +class Core: + def __init__(self, raw, reference, kind, first, last, cap, y0, target, incoming=0, + physical_band=None): + need(kind in ("narrow", "broad") and all(type(x) is int for x in + (first, last, cap, target, incoming)) and cap > 0 and cap & (cap - 1) == 0 + and target >= 1 and incoming >= 0, "core kind, cap and precision") + y0 = rational(y0) + need(y0 >= 1 and 2 * cap <= reference.spec.source_size, "core radius and source cap") + minimal = (2 * reference.spec.source_halfwidth + 2 if kind == "narrow" + else reference.spec.target_halfwidth) + width = max(minimal, 1) if physical_band is None else physical_band + need(type(width) is int and width >= max(1, minimal), "physical core band covers the reference") + need(0 <= first < last <= reference.target_size + and reference.target_size - (last - first) >= width, "ordinary core with complementary gap") + self.raw, self.reference, self.kind = raw, reference, kind + self.first, self.last, self.cap, self.width = first, last, cap, width + self.y0, self.target, self.incoming = y0, target, incoming + self.n = last - first + self.coordinates = tuple(range(first, last)) + self.adapter = load_boundary(raw) + self.carve, self.carve_ast = self.load_carve() + self.pieces, self.cells = self.partition() + self.interior = sorted(i for cell in self.cells for i in cell["indices"]) + interior_set = set(self.interior) + self.boundary = [i for i in range(self.n) if i not in interior_set] + need(self.boundary, "nonempty retained core boundary") + self.gauge_bound, self.middle_bound, self.raw_inverse_bound = geometry_bounds( + raw, reference.spec.source_size, reference.spec.excess, reference.spec.alpha, cap) + if kind == "narrow": + reference.require_narrow_gap() + self.gap = reference.narrow_gap + self.factor_bound = max(Q(3), 1 / self.gap) + else: + self.gap = None + need(reference.errors["broad"] * self.raw_inverse_bound <= Q(1, 2), + "broad reference interior inverse margin") + self.factor_bound = 3 + 18 * self.raw_inverse_bound + k = self.factor_bound + self.stage_radius = 6 * k ** 5 * y0 + self.stage_error = Q(1, 1 << target) / (10 * k ** 4) + self.dense_radius = ceiling_sqrt(cap) * self.stage_radius + self.dense_target = target + ceil_log2(20 * k ** 4) + for cell in self.cells: + cell["probe"] = RawGaussian(raw, reference, kind, cell["source_start"], len(cell["indices"]), + self.dense_radius, self.dense_target, 0) + self.matrix = real(reference.window(kind, self.coordinates)) + self.reference_hash = matrix_hash(self.matrix) + need(bandwidth(self.matrix) <= width, "no unhandled cyclic core corner") + need(row_norm(self.matrix) <= (2 if kind == "narrow" else 3), "core reference row norm") + if kind == "narrow": + need(row_gap(self.matrix) >= self.gap, "core inherits the common reference gap") + self.prepare_schur() + self.schur_band = bandwidth(self.schur) + need(self.schur_band <= 2 * width - 1, "compressed ordinary Schur bandwidth") + if kind == "narrow": + need(row_norm(self.schur) <= 2 and row_gap(self.schur) >= self.gap, + "same-reference narrow Schur norm and gap") + else: + need(row_norm(self.schur) <= self.factor_bound, "applied broad Schur norm") + self.p = max(width + 1, ceiling_sqrt(1 / (16 * self.gap))) if self.gap else width + 1 + self.block_width = 1 << max(0, (max(1, self.schur_band) - 1).bit_length()) + qe = target + ceil_log2(10 * k ** 4) + boundary_bits = qe + 41 + 16 * ceil_log2(self.p) + ceil_log2(self.stage_radius) + sparse_bits = 1 + ceil_log2(((2 * width + 1) * self.stage_radius + 1) / self.stage_error) + apply_bits = 1 + ceil_log2(((2 * self.schur_band + 1) * self.stage_radius + 1) / self.stage_error) + self.bits = max(incoming, sparse_bits, boundary_bits if kind == "narrow" else apply_bits, + *(cell["probe"].data_bits for cell in self.cells)) + self.upper_page = self.adapter.rounded(self.upper_coupling, self.bits) + self.lower_page = self.adapter.rounded(self.lower_coupling, self.bits) + self.schur_page = self.adapter.rounded(self.schur, self.bits) if kind == "broad" else None + self.solver = (Boundary(raw, self.adapter, self.schur, self.block_width, self.p, + self.bits, self.stage_radius, self.gap) if kind == "narrow" else None) + if self.solver: + need(self.solver.error_bound <= self.stage_error, "funded complete boundary solve") + for cell in self.cells: + cell["program"] = RawGaussian(raw, reference, kind, cell["source_start"], len(cell["indices"]), + self.dense_radius, self.dense_target, self.bits) + cell["program"].compile() + need(cell["program"].data_bits == self.bits, "one core and raw data grid") + self.stages, self.word_ast = self.load_word() + self.amplitude_bound = max( + 16 * (k + 1) * self.stage_radius, + self.solver.temporary_bound if self.solver else Q(1), + *(cell["program"].bound["amplitude_bound"] for cell in self.cells)) + self.integer_bits = max(1, ceil_log2(self.amplitude_bound)) + 2 + self.temporary_bits = max([2 * self.bits + self.integer_bits + 2, + *(cell["program"].bound["temporary_bits"] for cell in self.cells)]) + self.total_error = 5 * k ** 4 * self.stage_error + need(self.total_error <= Q(1, 1 << (target + 1)), "complete five-factor error allowance") + + def load_carve(self): + tree = ast.parse(self.raw[CARVE]) + tree.body = [copy.deepcopy(node) for node in tree.body + if isinstance(node, ast.FunctionDef) and node.name == "carve"] + need(len(tree.body) == 1, "one retained carve") + namespace = {} + exec(compile(tree, "", "exec", optimize=0), namespace) + return namespace["carve"], hashlib.sha256(ast.dump(tree).encode()).hexdigest() + + def partition(self): + ref, pieces, cells, cursor = self.reference, [], [], 0 + while cursor < self.n: + source = ref.source_at.get(self.coordinates[cursor]) + if source is None: + pieces.append(dict(kind="B", start=cursor, stop=cursor + 1, phase=None)) + cursor += 1 + continue + phase, stop = ref.phases[source], cursor + 1 + while stop < self.n: + j = ref.source_at.get(self.coordinates[stop]) + if j != source + stop - cursor or ref.phases[j] != phase: + break + stop += 1 + for kind, a, b in self.carve(stop - cursor, self.width, self.cap): + need(0 <= a < b <= stop - cursor, "positive retained carve piece") + pieces.append(dict(kind=kind, start=cursor + a, stop=cursor + b, phase=phase)) + if kind == "I": + indices = list(range(cursor + a, cursor + b)) + coordinates, _, _ = ref.interior(source + a, b - a) + need(coordinates == tuple(self.coordinates[i] for i in indices), "carved source binding") + cells.append(dict(indices=indices, source_start=source + a)) + cursor = stop + need(pieces[0]["start"] == 0 and pieces[-1]["stop"] == self.n + and all(a["stop"] == b["start"] for a, b in zip(pieces, pieces[1:])), "complete carve coverage") + need(all(b["indices"][0] - a["indices"][-1] > self.width + for a, b in zip(cells, cells[1:])), "separated reference interiors") + return pieces, cells + + def prepare_schur(self): + self.upper_coupling = block(self.matrix, self.interior, self.boundary) + self.lower_coupling = block(self.matrix, self.boundary, self.interior) + self.schur = block(self.matrix, self.boundary, self.boundary) + owner = {i: j for j, cell in enumerate(self.cells) for i in cell["indices"]} + need(all(self.matrix[i][j] == 0 for i in self.interior for j in self.interior + if owner[i] != owner[j]), "actual reference interior direct sum") + for cell in self.cells: + indices = cell["indices"] + matrix = block(self.matrix, indices, indices) + inverse = self.adapter.solve(matrix, self.adapter.identity(len(indices))) + limit = 1 / self.gap if self.gap else 2 * self.raw_inverse_bound + need(row_norm(matrix) <= (2 if self.gap else 3) + and row_norm(inverse) <= limit, "reference interior factor bounds") + cell["matrix"], cell["inverse"] = matrix, inverse + left, right = block(self.matrix, self.boundary, indices), block(self.matrix, indices, self.boundary) + correction = product(left, product(inverse, right)) + self.schur = [[x - y for x, y in zip(a, b)] for a, b in zip(self.schur, correction)] + + def load_word(self): + tree = ast.parse(self.raw[WORD]) + assignments = [copy.deepcopy(node) for node in ast.walk(tree) + if isinstance(node, ast.Assign) and len(node.targets) == 1 + and isinstance(node.targets[0], ast.Name) and node.targets[0].id == "seq"] + need(len(assignments) == 1, "one retained six-stage factor word") + module = ast.Module(body=assignments, type_ignores=[]) + a, ai, si = object(), object(), object() + def embedded(value, indices, n): + need(n == self.n, "source word dimension") + if value is si: + need(indices == self.boundary, "source boundary coordinates") + return "boundary_inverse" + need(indices == self.interior and value in (a, ai), "source interior coordinates") + return "interior_forward" if value is a else "interior_inverse" + namespace = dict(A=a, Ai=ai, Si=si, I=self.interior, J=self.boundary, n=self.n, + lower="lower_negative", upper="upper_negative", embedded=embedded) + exec(compile(module, "", "exec", optimize=0), namespace) + inverse_word = namespace["seq"] + need(inverse_word == ["interior_inverse", "lower_negative", "boundary_inverse", + "interior_forward", "upper_negative", "interior_inverse"], + "retained inverse chronology") + if self.kind == "narrow": + stages = inverse_word + else: + inverse_names = {"interior_inverse": "interior_forward", "interior_forward": "interior_inverse", + "boundary_inverse": "boundary_forward", "lower_negative": "lower_positive", + "upper_negative": "upper_positive"} + stages = [inverse_names[name] for name in reversed(inverse_word)] + return stages, hashlib.sha256(ast.dump(module).encode()).hexdigest() + + def execute(self, rhs, capture=False): + state = Mat(rhs) + columns = len(state[0]) if state else 0 + need(len(state) == self.n and columns > 0 and all(len(row) == columns for row in state), + "complete core RHS shape") + need(max_complex(state) <= self.y0 and all( + (part * (1 << self.incoming)).denominator == 1 + for row in state for z in row for part in (z.re, z.im)), "declared core input grid and radius") + need(matrix_hash(real(self.reference.window(self.kind, self.coordinates))) == self.reference_hash, + "same immutable reference core at execution") + need(self.stages == self.load_word()[0], "authenticated core chronology") + records, traces = [], [] + stats = dict(dense_gate_applications=0, dense_rounded_outputs=0, dense_max_height=0, + boundary_scalar_products=0, sparse_real_products=0, sparse_complex_rounds=0, + sparse_fractional_bits=0, sparse_fixed_numerator_bits=0, + stored_grid_numerator_bits=0) + + def observe_state(): + need(max_complex(state) <= self.stage_radius, "actual stage input/output envelope") + for row in state: + for z in row: + for part in (z.re, z.im): + scaled = part * (1 << self.bits) + need(scaled.denominator == 1, "one completed core data grid") + stats["stored_grid_numerator_bits"] = max( + stats["stored_grid_numerator_bits"], abs(scaled.numerator).bit_length()) + need(stats["stored_grid_numerator_bits"] + 1 <= self.bits + self.integer_bits, + "signed core data words") + + def observe_temporary(value): + denominator = value.denominator + need(denominator & (denominator - 1) == 0 + and denominator.bit_length() - 1 <= 2 * self.bits, "sparse dyadic temporary grid") + stats["sparse_fractional_bits"] = max(stats["sparse_fractional_bits"], + denominator.bit_length() - 1) + encoded = value * (1 << (2 * self.bits)) + need(encoded.denominator == 1, "fixed sparse temporary numerator") + stats["sparse_fixed_numerator_bits"] = max( + stats["sparse_fixed_numerator_bits"], abs(encoded.numerator).bit_length()) + need(stats["sparse_fixed_numerator_bits"] + 1 <= self.temporary_bits, + "signed sparse temporary width") + + def sparse(page, targets, sources, sign, add): + output, scale = [], 1 << self.bits + for i, row in zip(targets, page): + new = [] + for col in range(columns): + re, im = (state[i][col].re, state[i][col].im) if add else (Q(0), Q(0)) + for j, coefficient in zip(sources, row): + if coefficient: + x, y = coefficient * state[j][col].re, coefficient * state[j][col].im + stats["sparse_real_products"] += 2 + observe_temporary(x) + observe_temporary(y) + re += sign * x + im += sign * y + observe_temporary(re) + observe_temporary(im) + new.append(C(Q(int(re * scale), scale), Q(int(im * scale), scale))) + stats["sparse_complex_rounds"] += 1 + output.append(new) + for i, row in zip(targets, output): + state[i] = row + + observe_state() + for number, stage in enumerate(self.stages): + before = Mat([list(row) for row in state]) if capture else None + stage_stats = {} + if stage.startswith("interior_"): + for cell in self.cells: + program = cell["program"] + if stage == "interior_inverse": + plan = program.inverse_plan + else: + plan = program.plan + for col in range(columns): + vector = Vec(state[i][col] for i in cell["indices"]) + need(sum((z.norm2() for z in vector), Q(0)) <= self.dense_radius ** 2, + "actual dense input radius") + output, measured = apply(plan, vector, self.bits) + for i, value in zip(cell["indices"], output): + state[i][col] = value + stats["dense_gate_applications"] += sum(map(len, plan)) + stats["dense_rounded_outputs"] += measured["rounded_outputs"] + stats["dense_max_height"] = max(stats["dense_max_height"], measured["max_observed_height"]) + need(measured["max_observed_height"] <= program.bound["temporary_bits"], + "dense arithmetic temporary width") + elif stage == "boundary_inverse": + boundary_rhs = Mat([state[i] for i in self.boundary]) + output, stage_stats = self.solver.run(boundary_rhs) + for i, row in zip(self.boundary, output): + state[i] = row + stats["boundary_scalar_products"] += ( + stage_stats["lower"]["scalar_products"] + stage_stats["upper"]["scalar_products"]) + need(max(stage_stats["lower"]["fractional_bits"], + stage_stats["upper"]["fractional_bits"]) <= 2 * self.bits, + "boundary temporary fractional widths") + elif stage == "boundary_forward": + sparse(self.schur_page, self.boundary, self.boundary, 1, False) + elif stage.startswith("lower_"): + sparse(self.lower_page, self.boundary, self.interior, + -1 if stage.endswith("negative") else 1, True) + elif stage.startswith("upper_"): + sparse(self.upper_page, self.interior, self.boundary, + -1 if stage.endswith("negative") else 1, True) + else: + raise ValueError("unknown core factor") + observe_state() + records.append(dict(index=number, factor_index=(0, 1, 2, 2, 3, 4)[number], + stage=stage, output_sha256=matrix_hash(state), + maximum_upper=str(upper(max_complex(state))), boundary=stage_stats)) + if capture: + traces.append((stage, before, Mat([list(row) for row in state]))) + return state, dict(stages=records, statistics=stats), traces + + def record(self): + return dict(kind=self.kind, first=self.first, last=self.last, rows=self.n, cap=self.cap, + physical_band=self.width, + common_band=self.width >= max(2 * self.reference.spec.source_halfwidth + 2, + self.reference.spec.target_halfwidth), + cap_exceeds_band=self.cap > self.width, + reference_description_sha256=self.reference.description_hash, + reference_matrix_sha256=self.reference_hash, + carve_ast_sha256=self.carve_ast, word_ast_sha256=self.word_ast, + boundary_adapter_ast_sha256=self.adapter.source_ast_sha256, + pieces=self.pieces, interior=self.interior, boundary=self.boundary, + interior_blocks=[dict(source_start=cell["source_start"], indices=cell["indices"], + reference_sha256=matrix_hash(cell["matrix"]), + reference_inverse_sha256=matrix_hash(cell["inverse"]), + program=raw_core_record(cell["program"])) for cell in self.cells], + schur_sha256=matrix_hash(self.schur), schur_band=self.schur_band, + upper_coupling_sha256=matrix_hash(self.upper_coupling), + lower_coupling_sha256=matrix_hash(self.lower_coupling), + upper_page_sha256=matrix_hash(self.upper_page), lower_page_sha256=matrix_hash(self.lower_page), + boundary_solver=self.solver.record() if self.solver else None, + stage_order=self.stages, + budget=dict(input_bound=str(self.y0), target_bits=self.target, + incoming_bits=self.incoming, data_bits=self.bits, + factor_bound=str(self.factor_bound), stage_radius=str(self.stage_radius), + stage_error=str(self.stage_error), dense_radius=str(self.dense_radius), + dense_target_bits=self.dense_target, + core_gap_lower=str(self.gap) if self.gap else None, + raw_inverse_bound=str(self.raw_inverse_bound), + total_error=str(self.total_error), amplitude_bound=str(self.amplitude_bound), + integer_bits=self.integer_bits, temporary_bits=self.temporary_bits)) diff --git a/research/rational-unitary-fft/verify.py b/research/rational-unitary-fft/verify.py index c0320350b1..6cef36660f 100644 --- a/research/rational-unitary-fft/verify.py +++ b/research/rational-unitary-fft/verify.py @@ -33,7 +33,8 @@ def rejected(operation, label): raise ValueError("control unexpectedly passed: " + label) -def authenticate(include_generator=False, include_full=False, include_gaussian=False, include_raw=False): +def authenticate(include_generator=False, include_full=False, include_gaussian=False, include_raw=False, + include_core=False): rows = [r for r in json.loads((HERE / "SOURCES.json").read_text())["files"] if r["usage"] == "input"] raw = {} for row in rows: @@ -65,11 +66,19 @@ def authenticate(include_generator=False, include_full=False, include_gaussian=F "research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md", "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md", } + core = { + "research/rational-unitary-fft/vendor/boundary_adapter.py", + "research/rational-unitary-fft/vendor/BOUNDARY-PROVENANCE.json", + "research/capped-phase-fusion/evidence/radical_reduction_ultra/check_constructions.py", + "research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py", + "research/capped-phase-fusion/evidence/phase_layout_audit_ultra/check_union.py", + } return raw, [{"path": r["path"], "sha256": r["sha256"]} for r in rows if (include_generator or r["path"] not in added) and (include_full or r["path"] not in full) and (include_gaussian or r["path"] not in gaussian) - and (include_raw or r["path"] not in raw_gaussian)] + and (include_raw or r["path"] not in raw_gaussian) + and (include_core or r["path"] not in core)] def operator(plan, n): diff --git a/research/rational-unitary-fft/verify_core.py b/research/rational-unitary-fft/verify_core.py new file mode 100644 index 0000000000..ee143c3ccd --- /dev/null +++ b/research/rational-unitary-fft/verify_core.py @@ -0,0 +1,229 @@ +"""Execute complete finite-core Gaussian words against their shared references. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +sys.path.insert(0, str(HERE)) +from exact import C, Mat, apply, ceil_log2, need, upper +from core import Core, allocated_reference, matrix_hash, max_complex, row_gap +from gaussian_reference import Entry, Reference +from verify import authenticate, canonical, rejected + + +def exact_action(matrix, state): + columns = len(state[0]) if state else 0 + return Mat([[C(sum((a * state[k][j].re for k, a in enumerate(row) if a), Q(0)), + sum((a * state[k][j].im for k, a in enumerate(row) if a), Q(0))) + for j in range(columns)] for row in matrix]) + + +def difference(left, right): + need(len(left) == len(right), "reference comparison rows") + return Mat([[a - b for a, b in zip(x, y)] for x, y in zip(left, right)]) + + +def report(value, allowance): + need(0 <= value <= allowance, "reported error allowance") + bits = max(160, ceil_log2(1 / allowance) + 16) + return str(min(allowance, upper(value, bits))) + + +def stage_certificate(core, stage, before, after): + columns = len(before[0]) + if stage == "boundary_inverse": + residual = difference(exact_action(core.schur, Mat([after[i] for i in core.boundary])), + Mat([before[i] for i in core.boundary])) + error = max_complex(residual) / core.gap + need(all(before[i] == after[i] for i in core.interior), "boundary stage preserves interiors") + else: + wanted = Mat([list(row) for row in before]) + if stage.startswith("interior_"): + for cell in core.cells: + matrix = cell["inverse"] if stage.endswith("inverse") else cell["matrix"] + output = exact_action(matrix, Mat([before[i] for i in cell["indices"]])) + for i, row in zip(cell["indices"], output): + wanted[i] = row + elif stage == "boundary_forward": + output = exact_action(core.schur, Mat([before[i] for i in core.boundary])) + for i, row in zip(core.boundary, output): + wanted[i] = row + else: + lower = stage.startswith("lower_") + matrix = core.lower_coupling if lower else core.upper_coupling + targets, sources = ((core.boundary, core.interior) if lower + else (core.interior, core.boundary)) + if sources: + output = exact_action(matrix, Mat([before[i] for i in sources])) + sign = -1 if stage.endswith("negative") else 1 + for i, row in zip(targets, output): + wanted[i] = [before[i][j] + sign * row[j] for j in range(columns)] + error = max_complex(difference(after, wanted)) + need(error <= core.stage_error, "actual factor error against the current reference input") + return dict(stage=stage, input_sha256=matrix_hash(before), output_sha256=matrix_hash(after), + error_upper=report(error, core.stage_error), nonzero_error=error > 0) + + +def endpoint_certificate(core, rhs, answer): + if core.kind == "narrow": + residual = max_complex(difference(exact_action(core.matrix, answer), rhs)) + error = residual / core.gap + method = "exact reference residual divided by certified row gap" + else: + error = max_complex(difference(answer, exact_action(core.matrix, rhs))) + residual = None + method = "exact reference forward product" + need(error <= core.total_error, "complete finite-core reference error") + return dict(method=method, error_upper=report(error, core.total_error), + residual_upper=str(upper(residual)) if residual is not None else None, + nonzero_error=error > 0) + + +def one_case(label, raw, reference, kind, first, last, cap, columns, target, incoming): + print("PREPARE_GAUSSIAN_CORE", label, last - first, flush=True) + core = Core(raw, reference, kind, first, last, cap, 1, target, incoming, + physical_band=max(2 * reference.spec.source_halfwidth + 2, + reference.spec.target_halfwidth)) + print("EXECUTE_GAUSSIAN_CORE", label, len(core.cells), len(core.boundary), flush=True) + rhs = Mat([[C(Q((3 * i + 2 * j) % 7 - 3, 8), Q((5 * i + j) % 5 - 2, 8)) + for j in range(columns)] for i in range(core.n)]) + if incoming > 3: + rhs[0][0] = C(Q(1, 1 << incoming)) + answer, execution, traces = core.execute(rhs, capture=True) + need(matrix_hash(rhs) == matrix_hash(traces[0][1]), "borrowed initial core data") + need(all(matrix_hash(a[2]) == matrix_hash(b[1]) for a, b in zip(traces, traces[1:])), + "current-value chronological stage inputs") + factors = [stage_certificate(core, *trace) for trace in traces] + endpoint = endpoint_certificate(core, rhs, answer) + record = core.record() + record.update(label=label, columns=columns, input_sha256=matrix_hash(rhs), + output_sha256=matrix_hash(answer), execution=execution, + factor_certificates=factors, endpoint=endpoint) + print("CHECK_GAUSSIAN_CORE", label, flush=True) + return record, core, rhs, answer, traces + + +def controls(raw, reference, cases): + results = [] + narrow = next(row for row in cases if row[1].kind == "narrow" and row[1].cells) + edge = next(row for row in cases if row[1].kind == "narrow" and not row[1].cells) + _, core, rhs, _, traces = narrow + _, small, small_rhs, _, small_traces = edge + results.append(rejected(lambda: Core(raw, reference, "narrow", 0, reference.target_size, + 16, 1, 20), "unhandled-periodic-corner")) + results.append(rejected(lambda: Core(raw, reference, "narrow", 0, 20, 3, 1, 20), + "nondyadic-core-cap")) + results.append(rejected(lambda: Core(raw, reference, "narrow", 0, 20, 16, 1, 20, + physical_band=1), "understated-core-physical-band")) + poor = Reference(raw, 384, 1, 2, 4, 18, 8) + results.append(rejected(lambda: Core(raw, poor, "narrow", core.first, core.last, 16, 1, 20), + "reference-not-funded-for-amplified-dense-inputs")) + saved = list(small.stages) + small.stages[1], small.stages[4] = small.stages[4], small.stages[1] + results.append(rejected(lambda: small.execute(small_rhs), "wrong-retained-factor-order")) + small.stages = saved + upper_trace = next(trace for trace in traces if trace[0] == "upper_negative") + results.append(rejected(lambda: stage_certificate(core, upper_trace[0], upper_trace[1], + upper_trace[1]), "omitted-original-sparse-coupling")) + boundary_trace = next(trace for trace in small_traces if trace[0] == "boundary_inverse") + boundary_rhs = Mat([boundary_trace[1][i] for i in small.boundary]) + prepared, _ = small.solver.run(boundary_rhs) + baseline = small.solver.baseline(boundary_rhs) + need(prepared == baseline, "prepared dyadic pages match the unchanged boundary adapter") + results.append("prepared-page-source-value-equivalence") + value = small.solver.inverse_pages["lower"][0][0][0] + small.solver.inverse_pages["lower"][0][0][0] += 1 + bad, _ = small.solver.run(boundary_rhs) + bad_state = Mat([list(row) for row in boundary_trace[1]]) + for i, row in zip(small.boundary, bad): + bad_state[i] = row + results.append(rejected(lambda: stage_certificate(small, "boundary_inverse", boundary_trace[1], bad_state), + "corrupted-prepared-inverse-page")) + small.solver.inverse_pages["lower"][0][0][0] = value + key = "narrow", small.coordinates[0], small.coordinates[0] + old_entry = reference.entries[key] + reference.entries[key] = Entry(Q(2), "unit-diagonal", 0, Q(0)) + results.append(rejected(lambda: small.execute(small_rhs), "changed-core-reference-after-setup")) + reference.entries[key] = old_entry + off_grid = Mat([list(row) for row in small_rhs]) + off_grid[0][0] = C(Q(1, 1 << (small.incoming + 1))) + results.append(rejected(lambda: small.execute(off_grid), "discarded-core-incoming-grid")) + oversized = Mat([list(row) for row in small_rhs]) + oversized[0][0] = C(2) + results.append(rejected(lambda: small.execute(oversized), "underdeclared-core-input-radius")) + need(all(row[1].solver is None for row in cases if row[1].kind == "broad"), + "broad boundary requires no inverse solver") + results.append("broad-schur-is-applied") + need(canonical({"columns": 1}) != canonical({"columns": True}), "typed core receipt") + return results + ["typed-core-receipt"] + + +def run(quick=False): + raw, sources = authenticate(include_generator=True, include_full=True, include_gaussian=True, + include_raw=True, include_core=True) + reference = Reference(raw, 384, 1, 2, 4, 18, 320) + length = 7 * reference.spec.target_halfwidth + 16 + 1 + specifications = [ + ("narrow-core", "narrow", 8, 8 + length, 1, 20, 3), + ("broad-core", "broad", 8, 8 + length, 1, 20, 3), + ("narrow-hole-fine", "narrow", 187, 198, 5, 24, 768), + ("broad-hole", "broad", 187, 198, 3, 20, 3), + ] + if quick: + specifications = specifications[2:] + cases = [one_case(label, raw, reference, kind, first, last, 16, columns, target, incoming) + for label, kind, first, last, columns, target, incoming in specifications] + if quick: + return dict(scope="diagnostic boundary-only core replay", cases=[row[0] for row in cases]) + need(all(len(row[1].cells) >= 2 for row in cases[:2]), "multiple actual interior blocks") + need(any(row[0]["endpoint"]["nonzero_error"] for row in cases), "nonzero complete rounded error") + auto, allocation = allocated_reference(raw, 512, 1, 2, 2, 1, 8) + need(auto.narrow_gap > 0 and auto.allocation is not None, "noncircular common-reference allocation") + return dict(schema_version=1, scope="complete ordinary finite-core Gaussian factor words", + sources=sources, reference=reference.record(), cases=[row[0] for row in cases], + allocation_only=dict(reference=auto.record(), budget=allocation), + controls=controls(raw, reference, cases), + finite_core_numerical_execution=True, physical_tape_execution=False, + complete_gaussian_execution=False, core_halo_comparison=False, + tensor_execution=False, stored_global_inverse_pair=False, + independent_transcendental_backend=False, universal_proof_kernel=False) + + +def verify_result(write=False): + result = run() + path = HERE / "CORE-RESULT.json" + if write: + path.write_text(json.dumps(result, indent=2) + "\n") + else: + need(canonical(result) == canonical(json.loads(path.read_text())), "complete typed core receipt") + print(json.dumps(dict(status="PASS_FINITE_GAUSSIAN_CORE", cases=len(result["cases"]), + interiors=sum(len(row["interior_blocks"]) for row in result["cases"]), + dense_gate_applications=sum(row["execution"]["statistics"]["dense_gate_applications"] + for row in result["cases"]), + boundary_scalar_products=sum(row["execution"]["statistics"]["boundary_scalar_products"] + for row in result["cases"]), + controls=len(result["controls"]), + result_sha256=sha256(canonical(result)).hexdigest()), sort_keys=True)) + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write-result", action="store_true") + parser.add_argument("--quick", action="store_true") + args = parser.parse_args() + if args.quick: + result = run(quick=True) + print(json.dumps(dict(status="PASS_CORE_DIAGNOSTIC", cases=len(result["cases"])))) + else: + verify_result(args.write_result) + + +if __name__ == "__main__": + main() diff --git a/research/rational-unitary-fft/verify_full.py b/research/rational-unitary-fft/verify_full.py index 0720dda749..ad92e3c48d 100644 --- a/research/rational-unitary-fft/verify_full.py +++ b/research/rational-unitary-fft/verify_full.py @@ -257,8 +257,10 @@ def main(): parser.add_argument("--write-result", action="store_true") parser.add_argument("--with-gaussian", action="store_true") parser.add_argument("--with-raw", action="store_true") + parser.add_argument("--with-core", action="store_true") args = parser.parse_args() need(not args.with_raw or args.with_gaussian, "raw verification follows normalized verification") + need(not args.with_core or args.with_raw, "core verification follows raw verification") result = run() path = HERE / "FULL-RESULT.json" if args.write_result: @@ -277,6 +279,9 @@ def main(): if args.with_raw: from verify_raw_gaussian import verify_result as verify_raw_result verify_raw_result(args.write_result) + if args.with_core: + from verify_core import verify_result as verify_core_result + verify_core_result(args.write_result) if __name__ == "__main__":