From 73c65e7e245fdb873c1d2a4bc5edd01bd87faad0 Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 14:40:50 +0200 Subject: [PATCH 1/3] Prove sharper shifted Gaussian conditioning and inverse sharpness --- .../shifted-gaussian-conditioning/PROOF.md | 365 +++ .../shifted-gaussian-conditioning/README.md | 117 + .../shifted-gaussian-conditioning/RESULT.json | 2561 +++++++++++++++++ .../SOURCES.json | 15 + .../shifted-gaussian-conditioning/TASK.json | 64 + .../shifted-gaussian-conditioning/VERIFY.json | 27 + .../intervals.py | 164 ++ .../shifted-gaussian-conditioning/verify.py | 276 ++ 8 files changed, 3589 insertions(+) create mode 100644 research/shifted-gaussian-conditioning/PROOF.md create mode 100644 research/shifted-gaussian-conditioning/README.md create mode 100644 research/shifted-gaussian-conditioning/RESULT.json create mode 100644 research/shifted-gaussian-conditioning/SOURCES.json create mode 100644 research/shifted-gaussian-conditioning/TASK.json create mode 100644 research/shifted-gaussian-conditioning/VERIFY.json create mode 100644 research/shifted-gaussian-conditioning/intervals.py create mode 100644 research/shifted-gaussian-conditioning/verify.py diff --git a/research/shifted-gaussian-conditioning/PROOF.md b/research/shifted-gaussian-conditioning/PROOF.md new file mode 100644 index 0000000000..e34cd117b0 --- /dev/null +++ b/research/shifted-gaussian-conditioning/PROOF.md @@ -0,0 +1,365 @@ +# Shifted Gaussian conditioning + +Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance. +Apache-2.0. + +I use the normalized kernel + +\[ +k(u)=e^{-\gamma u^2+2\gamma su},\qquad +\Phi(w)=\sum_{u\in\mathbb Z}k(u)e^{-iuw},\qquad +\gamma>0,\quad |s|<1/2. +\] + +Write \(\delta=1-2|s|\) and \(m=\Phi(\pi)\). The unit diagonal is \(k(0)=1\). +For the unnormalized kernel \(e^{-\gamma(u-s)^2}\), multiply symbol and +accretivity bounds by \(e^{-\gamma s^2}\), and inverse bounds by +\(e^{\gamma s^2}\). I use Euclidean vector and induced operator norms throughout. + +I credit Jeremy Kothe/IdeaOuter and the PR1758 contributors for the product +and phase argument in `vendor/GAUGEFREE-INTERIORS.md`, Lemmas 3.1–3.3. +In Section 1 I make their minimum implication explicit. Vlado Vojdanovski +already used the unshifted Poisson estimate in `FORWARD-SMALL-A-AUDIT.md`. +I add the shifted uniform floor, its sharp exponential order, the +finite-section inverse limit, and the source-specific bound in Section 5. + +## 1. Exact minimum from the retained product argument + +The Gaussian decay gives absolute convergence, including after multiplication +by any fixed power of \(|u|\). Put \(q=e^{-\gamma}\). By conjugation we may +take \(s\ge0\), and by periodicity and conjugation take \(0\le w\le\pi\). +Jacobi's product gives + +\[ +\Phi(w)=C\prod_{n\ge1}(1+a_ne^{-iw})(1+b_ne^{iw}),\quad +C=\prod_{n\ge1}(1-q^{2n}), +\quad a_n=q^{2n-1-2s},\quad b_n=q^{2n-1+2s}. +\] + +We have \(1>a_n\ge b_n\ge a_{n+1}>0\), and summable deficits in the tails. +For \(f(r)=\arg(1+re^{iw})\), differentiation gives +\(f'(r)=\sin w/|1+re^{iw}|^2\ge0\). Finite sums telescope; convergence +then gives + +\[ +0\le-\arg\Phi(w)=\sum_{n\ge1}[f(a_n)-f(b_n)]\le f(a_1)<\pi/2. +\] + +Consequently, using positive real factors, + +\[ +\begin{aligned} +\Re\Phi(w) +&\ge |\Phi(w)|\cos f(a_1)\\ +&=C(1+a_1\cos w) + \prod_{n\ge2}|1+a_ne^{-iw}|\prod_{n\ge1}|1+b_ne^{iw}|\\ +&\ge C(1-a_1)\prod_{n\ge2}(1-a_n)\prod_{n\ge1}(1-b_n) +=\Phi(\pi)=m>0. +\end{aligned} +\] + +At \(w=\pi\) we attain equality. Away from \(\pi\) modulo \(2\pi\), +we have \(1+a_1\cos w>1-a_1\). Since \(|\Phi|\ge\Re\Phi\), we obtain + +\[ +\min_w\Re\Phi(w)=\min_w|\Phi(w)|=m. +\] + +I claim this minimum as an inherited corollary, not as a new phase method. + +## 2. A shift-uniform floor with exponent \(1/4\) + +I prove, for the entire stated domain, + +\[ +\boxed{\displaystyle +m\ge L(\gamma,s):=\delta\sqrt{\frac{\pi}{\gamma}}\, +e^{-\pi^2/(4\gamma)}.} +\] + +First suppose \(0<\gamma\le4\). Poisson summation for the shifted Gaussian, +followed by pairing opposite odd frequencies, gives + +\[ +m=2e^{\gamma s^2}\sqrt{\frac{\pi}{\gamma}}e^{-\pi^2/(4\gamma)} +\left[\cos(\pi s)+ +\sum_{r\ge1}e^{-\pi^2r(r+1)/\gamma}\cos((2r+1)\pi s)\right]. +\] + +For \(t=\pi/2-\pi|s|\in(0,\pi/2]\), use +\(|\sin(nt)|\le n\sin t\) for positive integers \(n\), obtained by summing +the \(n\) unit terms in the geometric formula for \(\sin(nt)/\sin t\). +For odd \(n=2r+1\) this gives + +\[ +|\cos((2r+1)\pi s)|\le(2r+1)\cos(\pi s). +\] + +With \(q_P=e^{-2\pi^2/\gamma}\), bound the relative remainder by + +\[ +D\le\sum_{r\ge1}(2r+1)q_P^r +=\frac{q_P(3-q_P)}{(1-q_P)^2}. +\] + +Here \(r(r+1)\ge2r\). Since \(\pi>3\) and \(e>2\), +\(q_P1/2\) in the paired expression proves \(m\ge L\). + +For \(\gamma\ge4\), put \(q=e^{-\gamma}<1/16\) and remove the first +near-unit deficit from the positive product: + +\[ +m=(1-q^\delta) +\prod_{n\ge1}(1-q^{2n}) +\prod_{n\ge2}(1-q^{2n-2+\delta}) +\prod_{n\ge1}(1-q^{2n-\delta}). +\] + +The sum of the remaining deficits is at most +\((q+2q^2)/(1-q^2)\le6/85\). Apply +\(\prod(1-r_j)\ge1-\sum r_j\) to finite products and pass to the +convergent limit. Concavity in \(\delta\) also gives +\(1-q^\delta\ge\delta(1-q)\ge15\delta/16\). Thus + +\[ +m\ge\frac{79}{85}\frac{15}{16}\delta +=\frac{237}{272}\delta>\frac23\delta. +\] + +For \(x=\pi^2/(2\gamma)>0\), + +\[ +\left(\sqrt{\frac{\pi}{\gamma}}e^{-\pi^2/(4\gamma)}\right)^2 +=\frac2\pi xe^{-x}\le\frac2{\pi e}<\frac13<\frac49. +\] + +This proves the remaining range. + +For \(0<\gamma\le1\), compare with the retained simplified floor +\(L_{\rm old}=(1-e^{-\gamma\delta})e^{-\pi^2/(3\gamma)}\). +Since \(1-e^{-\gamma\delta}\le\gamma\delta\), + +\[ +\frac{L}{L_{\rm old}}\ge +\sqrt\pi\,\gamma^{-3/2}e^{\pi^2/(12\gamma)}>1. +\] + +I restrict this dominance assertion to the stated range. For example, at +\(s=0\) and large \(\gamma\), the old floor approaches one and \(L\) approaches zero. +Reviewers can retain the maximum of the two valid floors. + +The relative Poisson remainder tends to zero uniformly in \(|s|<1/2\); +so does \(e^{\gamma s^2}-1\). Therefore, as \(\gamma\downarrow0\), + +\[ +\frac{m}{2\cos(\pi s)\sqrt{\pi/\gamma}\,e^{-\pi^2/(4\gamma)}}\longrightarrow1 +\quad\hbox{uniformly in }|s|<1/2. +\] + +For a coarse bound on \(0<\gamma\le1\), use +\(e^{\gamma s^2}\le e^{1/4}<4/3\), +\(\cos(\pi s)=\sin(\pi\delta/2)\le(\pi/2)\delta<2\delta\), and \(D<1/2\). +Then \(L\le m\le8L\). Thus we capture both the small-scale exponential +coefficient \(1/4\) and the linear dependence on the shift gap. + +## 3. Sharp uniform inverse constant for finite sections + +Let \(N\) be a positive integer and \(T_N[i,j]=k(j-i)\), \(0\le i,jm\) off \(\pi\). Compactness of the unit sphere gives a +strictly positive smallest eigenvalue for the Hermitian part minus \(mI\). +I use the Hermitian part for accretivity, not the eigenvalues of nonnormal \(T_N\). + +To prove sharpness uniformly over \(N\), take \(x_{N,j}=(-1)^j/\sqrt N\). +For an integer shift \(u\), zero extension loses +\(\min(|u|,N)\) entries of this vector. The triangle inequality yields + +\[ +\|T_Nx_N-mx_N\|_2 +\le\sum_{u\in\mathbb Z}k(u)\sqrt{\frac{\min(|u|,N)}N} +\le\frac{A}{\sqrt N},\quad +A=\sum_{u\in\mathbb Z}\sqrt{|u|}\,k(u). +\] + +For \(n\ge1\), \(\sqrt n\le n\) and +\(n^2-2|s|n\ge n(n-1)\ge n-1\). Thus + +\[ +A\le2\sum_{n\ge1}ne^{-\gamma(n-1)} +=\frac2{(1-e^{-\gamma})^2} +\le A_0:=\frac{2(1+\gamma)^2}{\gamma^2}. +\] + +The last step uses \(e^\gamma\ge1+\gamma\). Since +\(\sigma_{\min}(T_N)\le\|T_Nx_N\|_2\), we have + +\[ +\boxed{\displaystyle +\frac1{m+A_0/\sqrt N}\le\|T_N^{-1}\|_2<\frac1m,\qquad +\lim_{N\to\infty}\|T_N^{-1}\|_2 +=\sup_{N\ge1}\|T_N^{-1}\|_2=\frac1m.} +\] + +The limit also holds along dyadic \(N\). For \(\varepsilon>0\) and a +certified lower endpoint \(0<\underline m\le m\), any +\(N\ge(A_0/(\varepsilon\underline m))^2\) gives +\(\|T_N^{-1}\|_2\ge1/((1+\varepsilon)m)\). The verifier records such a +dyadic size descriptor for \(\varepsilon=1/8\); I do not execute those +large matrices or claim monotonicity in \(N\). This statement concerns fixed +\(\gamma,s\) and unrestricted sizes, not attainment within a caller's capped block. + +## 4. Rounding, truncation and certificate interpretation + +If an approximate kernel \(\widetilde k\) satisfies +\(\sum_u|\widetilde k(u)-k(u)|\le\varepsilonW}k(\pm u)\le +\frac{\exp[-\gamma((W+1)^2\mp2s(W+1))]} + {1-\exp[-\gamma(2W+3\mp2s)]}. +\] + +We derive this by bounding consecutive positive-term ratios by the first ratio. +At \(\pi\), retain \(R\) Jacobi factors of each kind and write their product +as \(P_R\). The omitted deficit sum obeys + +\[ +E_R= +\frac{e^{-\gamma(2R+2)}+ +e^{-\gamma(2R+1-2s)}+e^{-\gamma(2R+1+2s)}}{1-e^{-2\gamma}}. +\] + +For \(E_R<1\), \(P_R(1-E_R)\le m\le P_R\). +The verifier intersects this product enclosure with the direct alternating +series plus its two absolute tails. It also checks overlap with the Poisson +leading-term enclosure. Intersection corroborates consistency of formulas; +the explicit remainder bounds justify containment. + +For each real rounded finite section, the verifier forms +\((\widetilde T_N+\widetilde T_N^T)/2-\underline m I/2\), +performs exact rational LDL elimination, checks positive pivots and +reconstructs the entire matrix. These are positivity certificates for the +specified finite matrices. For a complex rounded kernel we would need +conjugate transpose instead. The universal claims rely on Sections 1–3. +The `kernel_error_majorant` interval encloses the computed upper-bound +expression; its lower endpoint is not a lower bound on the actual error. +The finite-section inverse lower/upper fields concern the ideal matrix. +The separately named rounded inverse bound uses \(1/(m-\varepsilon)\). + +I use precision-80 directed Decimal arithmetic for basic operations. +Python documents correctly rounded nearest `exp` and `sqrt`; I widen each +endpoint by one adjacent representable value. I enclose \(\pi\) through +Machin's identity and rational alternating series, and cosine through its +Taylor series with an explicit alternating remainder. I trap underflow, +subnormal values and invalid arithmetic. The checker rejects inconclusive +intervals; it never interprets a failed lower enclosure as a counterexample. +This contract trusts CPython Decimal and rational arithmetic, rather than a +formal proof kernel. See the [Python 3.12 Decimal documentation](https://docs.python.org/3.12/library/decimal.html). + +## 5. Consequence for the actual broad Gaussian caller + +In PR4100 `GAUSSIAN.md`, take \(p\ge2\) tending to infinity and use natural +logarithms. Fix \(0<\xi<1\), and take \(1\le N\le B\le p^\xi\), together with + +\[ +\frac1{4p^2}\le\theta\le\frac1{p^2},\quad +\rho=1+\theta,\quad\lambda=\rho^{-1},\quad +\alpha=\left\lceil\sqrt{\frac4\pi\log(16p)}\right\rceil,\quad +a=\pi\alpha^2,\quad b_0=\pi/\alpha^2. +\] + +For a centered constant-phase interior use \(|\Delta|\le1/2\), +\(s=-\Delta(1-\theta)\), \(\gamma=b_0\lambda\), and \(|x|,|y|\le B/2\). +Then \(\delta\ge\theta\), so + +\[ +m\ge\theta\alpha\sqrt\rho\,e^{-a\rho/4}. +\] + +The source's exact broad identity, including its raw \(\alpha^{-1}\) +normalization, gives + +\[ +K=\alpha^{-1}e^{-b_0\Delta^2}D_yT_N(k)D_x, +\] +\[ +D_x(x)=e^{-b_0(1-\lambda)x^2-2b_0(\Delta+\lambda s)x},\qquad +D_y(y)=e^{b_0\lambda(1-\lambda)y^2+2b_0\lambda(\Delta+s)y}. +\] + +The scalar \(e^{-\gamma s^2}\) from the unnormalized kernel cancels +the opposite constant in the separable gauge identity. These gauges are +positive real scalings. We must account for their inverse norms. +Since \(\Delta+\lambda s=2\lambda\theta\Delta\) and +\(\Delta+s=\theta\Delta\), + +\[ +G:=e^{b_0\Delta^2}\|D_x^{-1}\|_2\|D_y^{-1}\|_2 +\le\exp\left[\frac{b_0}4+ +b_0\lambda\theta\left(\frac{B^2}4+\frac{3B}2\right)\right]. +\] + +We get \(G=1+o(1)\), uniformly over the admitted \(\theta,\Delta,N,B\) at +fixed \(\xi\). Consequently + +\[ +\boxed{\displaystyle +\|K^{-1}\|_2\le Q_p:= +\frac{G}{\theta\sqrt\rho}e^{a\rho/4} +\le p^{3+o(1)}.} +\] + +To justify the last exponent, use the actual ceiling formula: + +\[ +a\le4\log(16p)+4\sqrt{\pi\log(16p)}+\pi. +\] + +Hence \(e^{a\rho/4}=p^{1+o(1)}\), while \(\theta^{-1}\le4p^2\). +The source's coarse \(a\le5\log(16p)\) alone would not give this exponent. +The ceiling term also prevents an \(O(p^3)\) conclusion from this argument. + +For a raw operator perturbation of norm \(\varepsilon_p\) with +\(\varepsilon_pQ_p\le1/2\), the inverse bound becomes +\(Q_p/(1-\varepsilon_pQ_p)\le2Q_p\), preserving \(p^{3+o(1)}\). +A claimed truncation/rounding implementation must prove that premise. + +I improve the source's loose \(p^6\) allowance for these capped interiors. +I do not assert optimality of \(p^{3+o(1)}\) for capped, gauged blocks. +The caller assigns holes, phase cuts and period cuts to the boundary, and +applies the broad Schur complement without inverting it. I retain those +restrictions. The shallow-circuit construction, coefficient-generation costs, +complete tensor error ledger and multiplication theorem need separate review. + +## References and priority + +For classical identities, see [NIST DLMF, theta products](https://dlmf.nist.gov/20.5) +and [theta transformations](https://dlmf.nist.gov/20.7). +Pasupathy and Damodar studied centered Gaussian Toeplitz matrices in +[*The Gaussian Toeplitz matrix*, Linear Algebra and its Applications 171 +(1992), 133–147](https://doi.org/10.1016/0024-3795(92)90255-9). +I make no worldwide priority claim for Gaussian Toeplitz conditioning, +Poisson summation or approximate Fourier vectors. The bounded repository +overlap search and exact predecessor credits appear in README.md. diff --git a/research/shifted-gaussian-conditioning/README.md b/research/shifted-gaussian-conditioning/README.md new file mode 100644 index 0000000000..3d2b306de5 --- /dev/null +++ b/research/shifted-gaussian-conditioning/README.md @@ -0,0 +1,117 @@ +# Sharper shifted Gaussian conditioning + +I prove an explicit conditioning bound for the normalized shifted Gaussian +kernel +\(k(u)=e^{-\gamma u^2+2\gamma su}\), with \(\gamma>0\) and \(|s|<1/2\). +For its Fourier symbol \(\Phi\), write \(m=\Phi(\pi)\). I obtain + +\[ +m\ge(1-2|s|)\sqrt{\pi/\gamma}\,e^{-\pi^2/(4\gamma)},\qquad +\lim_{N\to\infty}\|T_N(k)^{-1}\|_2 +=\sup_{N\ge1}\|T_N(k)^{-1}\|_2=1/m. +\] + +I include an explicit convergence bound using alternating vectors. +For the actual broad Gaussian caller in PR4100, I account for its +\(\alpha^{-1}\) normalization, positive diagonal gauges and ceiling in +\(\alpha\), and derive \(\|K^{-1}\|_2\le p^{3+o(1)}\). +The author used a looser \(p^6\) allowance for those capped interiors. + +I retain the original product/phase proof and its exact minimum-at-\(\pi\) +corollary. My shifted floor improves the source's simplified \(1/3\)-exponent +floor for \(0<\gamma\le1\). I prove the floor for all positive \(\gamma\), +but do not claim dominance over the old floor at large \(\gamma\). +For small \(\gamma\), I also prove \(L\le m\le8L\), uniformly in the shift, +where \(L\) denotes the displayed floor. + +## Scope and source + +I start from canonical main `12fa8c5c844be21bfefb38a19503fbd867a11244`, +priority revision 7, and pin the retained PR4100 source +`337d04939571a637aeab54a0f58f54fde811663a`. +[SOURCES.json](SOURCES.json) binds the actual source bytes. +[PROOF.md](PROOF.md) contains the universal arguments and caller substitution. +[Discussion4191](https://github.com/CrocSwap/integer-mult-bounds/discussions/4191) +records the claim, live overlap and publication handoff. + +I restrict the caller consequence to centered, constant-phase, unit-gap +capped interiors. The source assigns holes and phase/period cuts to the +boundary. I leave the broad Schur complement as an applied factor. +For a raw perturbation of norm \(\varepsilon_p\), the caller must establish +\(\varepsilon_pQ_p\le1/2\) before using the rounded inverse bound \(2Q_p\). + +The sharp constant \(1/m\) concerns exact normalized sections at fixed +\(\gamma,s\) as \(N\) grows without a cap. Each finite section has a strictly +smaller inverse norm. I make no optimality claim for capped gauged blocks, +no \(O(p^3)\) claim, and no selected-frontier change. + +## Verification + +I use CPython 3.12 and its standard library. The declared two commands have +180-second limits each. From the repository root: + +```sh +python3.12 -B scripts/research_package.py \ + --package research/shifted-gaussian-conditioning +``` + +The verifier checks these finite consequences: + +- I enclose 25 minima using both the direct alternating series and the + positive Jacobi product, with explicit omitted-tail bounds. I compare + the new floor, the old floor where improvement applies, and a separate + Poisson leading-term enclosure. Scales range from \(1/16\) to \(8\); + one shift lies \(2^{-40}\) below \(1/2\). +- I charge the complete \(\ell^1\) truncation and coefficient-rounding error + for half-width 64 and a 192-bit fractional grid. For 125 matrices of + sizes 1, 2, 4, 8 and 16, I reconstruct exact rational LDL factorizations + of the rounded Hermitian part minus half the certified margin. +- I check 150 frequency-circle points and alternating-vector inverse + bounds. I record analytic size thresholds for reaching \(8/9\) of the + limiting inverse constant, and mark those large matrices as unexecuted. +- I reject altered source bytes, half-shift/zero-scale parameters and an + indefinite matrix. I check exact cancellation at the half shift, loss + of accretivity outside the domain, the missing-normalization failure, + an unpaid truncation tail and JSON scalar-type corruption. + +[RESULT.json](RESULT.json) carries the complete typed receipt. +I use directed Decimal arithmetic with explicit transcendental widening, +rational Machin/Taylor bounds and exact rational LDL reconstruction. +I document the arithmetic trust boundary in PROOF.md. These finite +certificates support the formulas and implementation; reviewers must +assess the written universal proof separately. + +## Overlap and attribution + +I searched 5,236 local records and inspected the retained shifted product +proof, unshifted Poisson estimate, fine-prime audit and accretive-scale +correction. Live checks covered 100 recent PRs and 60/691 hub comments, +then 60 PRs and 40/692 comments before the claim. I found no equivalent +shifted floor or sharp finite-section inverse statement in that coverage. +Before publication I refreshed unchanged canonical main and inspected a +further 100 PR records and 45 hub comments. ToikY's new PR4192 concerns +sequential compiler arena scans, a separate obligation. +I did not exhaust older comments, arbitrary diffs, unpublished work or +the mathematical literature. + +I credit Jeremy Kothe/IdeaOuter and the PR1758 contributors for the shifted +product/phase argument; Vlado Vojdanovski/PR1673 for the unshifted Poisson +estimate and capped framework; and Chafik Boukhalfa/PR1786 for the +fine-prime extension. Cinna Davis supplied the later caller I audit. +I checked the original PR1758/1786 credits instead of repeating the later +abbreviated attribution. I preserve the frozen notices and input bytes. +I also retain the classical Jacobi, Poisson, Toeplitz and Gaussian +literature attribution in PROOF.md. + +Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance, +including source-analysis agents. I count those agents within the author +process. I claim no independent human or formal-kernel acceptance. +Apache-2.0. + +## Open next questions + +In [TASK.json](TASK.json) I specify two remaining questions with first steps +and blockers: a sharper bound that uses the actual cap \(N\le B(p)\), and +integration into the complete Gaussian circuit and error ledger. +Reviewers must retain the coefficient-generation, circuit-prefix and +physical execution obligations when using this conditioning estimate. diff --git a/research/shifted-gaussian-conditioning/RESULT.json b/research/shifted-gaussian-conditioning/RESULT.json new file mode 100644 index 0000000000..f82a506f78 --- /dev/null +++ b/research/shifted-gaussian-conditioning/RESULT.json @@ -0,0 +1,2561 @@ +{ + "schema_version": 1, + "scope": "finite certified evidence; universal proof in PROOF.md", + "sources": [ + { + "path": "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md", + "sha256": "d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116" + }, + { + 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b/research/shifted-gaussian-conditioning/SOURCES.json new file mode 100644 index 0000000000..4dd6b9f52b --- /dev/null +++ b/research/shifted-gaussian-conditioning/SOURCES.json @@ -0,0 +1,15 @@ +{ + "schema_version": 1, + "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "files": [ + {"path":"certificates/selected-result.json","usage":"provenance","git_blob":"b1fea0c94436b236cc2b7d41a7be032811b94c81","sha256":"bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"}, + {"path":"docs/research/agent-task-catalog.json","usage":"provenance","git_blob":"64bfc5af6709ed9811171e0a2108f05e1219cb65","sha256":"352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"}, + {"path":"research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md","usage":"input","git_blob":"20585e44fa4e49b13b5d8495bbf6ed254459a362","sha256":"d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116"}, + 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{"path":"research/capped-fusion-t74-k554/README.md","usage":"input","git_blob":"9b3ebbf298044413214f0275ba6a2fee055ccbde","sha256":"c0003664b62f671e089d8f470f79ba533691d46c552d32e7b0d4124ed100b930"}, + {"path":"research/unconditional-batched-crt/vendor/FINE-PRIME.md","usage":"input","git_blob":"bf9899ee21be4058e4aac0b64bfa67d01ee8ddd3","sha256":"47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce"} + ] +} diff --git a/research/shifted-gaussian-conditioning/TASK.json b/research/shifted-gaussian-conditioning/TASK.json new file mode 100644 index 0000000000..6991793cfe --- /dev/null +++ b/research/shifted-gaussian-conditioning/TASK.json @@ -0,0 +1,64 @@ +{ + "schema_version": 1, + "task_id": "uniform-transfer-audit", + "priority_revision": 7, + "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244", + "original_card": { + "id": "uniform-transfer-audit", + "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?", + "starting_points": ["docs/research/research-family-status.md", "certificates/selected-result.json", "research/unconditional-batched-crt/README.md"], + "methods": [ + "Prove one missing implication with explicit quantified hypotheses.", + "Find an exact counterexample at the claimed interface boundary." + ], + "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.", + "first_step": { + "command": ["python3", "-B", "scripts/research_baseline.py"], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.", + "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately." + }, + "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.", + "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.", + "allowed_variations": ["One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."], + "scope_change_triggers": [ + "Replacing a quantified obligation with another finite smoke test.", + "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses." + ], + "overlap_terms": ["uniform transfer", "all-size", "precision", "4100", "unconditional-batched-crt"], + "existing_checks": [ + { + "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.", + "evidence": "research/unconditional-batched-crt/README.md", + "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.", + "status": "author-checked", + "source_commit": "337d04939571a637aeab54a0f58f54fde811663a", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + } + ] + }, + "scope_status": "refined", + "actual_scope": "Prove a shift-uniform Gaussian accretivity floor with exponential coefficient 1/4, the sharp finite-section inverse limit, and a p^(3+o(1)) upper bound at PR4100's capped broad caller; certify finite consequences with directed enclosures and exact rational positivity.", + "scope_change_reason": "The retained source already supplies the product/phase argument and its minimum-at-pi corollary. The separate shifted floor and inverse sharpness improve its conditioning estimate without changing the arithmetic mechanism or paid supplier profile.", + "bottleneck_link": "The broad Gaussian caller needs a correctly normalized inverse bound before truncation, rounding and circuit-prefix estimates can be integrated into the full transform.", + "budget_minutes": 120, + "stop_rule": "Stop at an invalid normalization, unjustified infinite-product limit, nonnormal section treated as Hermitian, or fixed-cap optimality inferred from unrestricted-size sharpness. Keep complete circuit and multiplication acceptance outside this scope.", + "outcome": "completed", + "evidence": "PROOF.md gives quantified analytic statements and the exact caller substitution. RESULT.json records 25 interval cases, 125 exact rational finite-section positivity certificates, 150 circle samples and domain/normalization/tail controls. Hosted and exported receipts are separate from scientific acceptance.", + "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 caller obtain a smaller inverse bound using the restriction N<=B(p), rather than the unrestricted-size sharp constant?", + "why_it_matters": "The p^(3+o(1)) upper bound is not an optimality theorem for finite capped, gauged blocks.", + "first_step": "Compare an explicit finite-section singular-value lower bound against the retained symbol floor at the actual scale, shift gap and block cap.", + "blocked_by": "A size-dependent argument that remains uniform near half shifts and preserves the real diagonal gauges." + }, + { + "question": "Can the complete Gaussian circuit and error ledger consume the improved bound with source-bound coefficient and perturbation costs?", + "why_it_matters": "Conditioning alone does not establish the shallow circuit, physical coefficient generation or complete multiplication theorem.", + "first_step": "Substitute the explicit Q_p from PROOF.md into one actual interior's perturbation and prefix bounds, retaining the incoming-height and rounded-boundary obligations.", + "blocked_by": "Separate all-size circuit/precision review and physical caller integration." + } + ] +} diff --git a/research/shifted-gaussian-conditioning/VERIFY.json b/research/shifted-gaussian-conditioning/VERIFY.json new file mode 100644 index 0000000000..97a5700123 --- /dev/null +++ b/research/shifted-gaussian-conditioning/VERIFY.json @@ -0,0 +1,27 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md", + "research/unconditional-batched-crt/GAUSSIAN.md", + "research/unconditional-batched-crt/PROOF.md", + "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md", + "research/capped-phase-fusion/README.md", + "research/capped-fusion-t74-k554/README.md", + "research/unconditional-batched-crt/vendor/FINE-PRIME.md" + ], + "commands": [ + { + "argv": ["{python}", "-I", "-B", "{package}/verify.py"], + "timeout_seconds": 180, + "checks": "Authenticate sources, enclose shifted Gaussian minima by two convergent representations, check Poisson bounds, charge coefficient/tail perturbations, reconstruct exact rational LDL certificates and reject domain/normalization controls." + }, + { + "argv": ["{python}", "-I", "-O", "-B", "{package}/verify.py"], + "timeout_seconds": 180, + "checks": "Repeat the complete certificate with optimized Python and explicit acceptance checks; compare the full typed receipt." + } + ] +} diff --git a/research/shifted-gaussian-conditioning/intervals.py b/research/shifted-gaussian-conditioning/intervals.py new file mode 100644 index 0000000000..c2d1cfb655 --- /dev/null +++ b/research/shifted-gaussian-conditioning/intervals.py @@ -0,0 +1,164 @@ +"""Directed decimal enclosures with rational input and explicit transcendental widening. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. Exp/sqrt rely on Python Decimal's correctly-rounded contract. +""" +from decimal import (Decimal as D, Context, ROUND_FLOOR, ROUND_CEILING, + ROUND_HALF_EVEN, Underflow, Subnormal, Clamped, FloatOperation) +from fractions import Fraction as Q +from math import factorial + +PRECISION = 80 +DOWN = Context(prec=PRECISION, rounding=ROUND_FLOOR) +UP = Context(prec=PRECISION, rounding=ROUND_CEILING) +NEAR = Context(prec=PRECISION, rounding=ROUND_HALF_EVEN) +for context in (DOWN, UP, NEAR): + for signal in (Underflow, Subnormal, Clamped, FloatOperation): + context.traps[signal] = True + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +class I: + __slots__ = ("lo", "hi") + + def __init__(self, value, upper=None): + if isinstance(value, D): + lo, hi = value, value if upper is None else upper + else: + need(type(value) in (int, Q), "exact rational interval input") + q = Q(value) + lo = DOWN.divide(D(q.numerator), D(q.denominator)) + hi = UP.divide(D(q.numerator), D(q.denominator)) + need(upper is None, "single rational input") + need(isinstance(lo, D) and isinstance(hi, D) + and lo.is_finite() and hi.is_finite() and lo <= hi, "finite ordered interval") + self.lo, self.hi = lo, hi + + @staticmethod + def wrap(value): + return value if isinstance(value, I) else I(value) + + def __add__(self, other): + other = I.wrap(other) + return I(DOWN.add(self.lo, other.lo), UP.add(self.hi, other.hi)) + + __radd__ = __add__ + + def __neg__(self): + return I(self.hi.copy_negate(), self.lo.copy_negate()) + + def __sub__(self, other): + return self + -I.wrap(other) + + def __rsub__(self, other): + return I.wrap(other) + -self + + def __mul__(self, other): + other = I.wrap(other) + pairs = [(x, y) for x in (self.lo, self.hi) for y in (other.lo, other.hi)] + return I(min(DOWN.multiply(x, y) for x, y in pairs), + max(UP.multiply(x, y) for x, y in pairs)) + + __rmul__ = __mul__ + + def __truediv__(self, other): + other = I.wrap(other) + need(other.lo > 0, "positive interval divisor") + pairs = [(x, y) for x in (self.lo, self.hi) for y in (other.lo, other.hi)] + return I(min(DOWN.divide(x, y) for x, y in pairs), + max(UP.divide(x, y) for x, y in pairs)) + + def __rtruediv__(self, other): + return I.wrap(other) / self + + def power(self, n): + need(type(n) is int and n >= 0, "nonnegative integer power") + value, factor = I(1), self + while n: + if n & 1: + value = value * factor + factor = factor * factor + n //= 2 + return value + + def square(self): + upper = max(UP.multiply(x, x) for x in (self.lo, self.hi)) + lower = D(0) if self.lo <= 0 <= self.hi else min( + DOWN.multiply(x, x) for x in (self.lo, self.hi)) + return I(lower, upper) + + def exp(self): + if self.lo == self.hi == 0: + return I(1) + # Decimal exp/sqrt round to nearest. One adjacent value on each side + # encloses the exact endpoint before monotonicity propagates the interval. + return I(DOWN.next_minus(NEAR.exp(self.lo)), UP.next_plus(NEAR.exp(self.hi))) + + def sqrt(self): + need(self.lo >= 0, "nonnegative square root") + lower = D(0) if self.lo == 0 else DOWN.next_minus(NEAR.sqrt(self.lo)) + return I(lower, UP.next_plus(NEAR.sqrt(self.hi))) + + def widen(self, error): + error = I.wrap(error) + need(error.lo >= 0, "nonnegative enclosure radius") + return I(DOWN.subtract(self.lo, error.hi), UP.add(self.hi, error.hi)) + + def intersect(self, other): + return I(max(self.lo, other.lo), min(self.hi, other.hi)) + + def record(self): + return [str(self.lo), str(self.hi)] + + +def atan_rational(x, terms=80): + need(0 < x < 1, "alternating arctangent domain") + total = sum(((-1) ** k * x ** (2 * k + 1) / (2 * k + 1) for k in range(terms)), Q(0)) + adjacent = total + (-1) ** terms * x ** (2 * terms + 1) / (2 * terms + 1) + return min(total, adjacent), max(total, adjacent) + + +def pi_interval(): + a, b = atan_rational(Q(1, 5)), atan_rational(Q(1, 239)) + lower, upper = 16 * a[0] - 4 * b[1], 16 * a[1] - 4 * b[0] + need(Q(3) < lower < upper < Q(4), "Machin enclosure") + return I(I(lower).lo, I(upper).hi) + + +def cos_interval(x, terms=100): + x = I.wrap(x) + total, term = I(1), I(1) + square = x.square() + for n in range(1, terms): + term = -term * square / ((2 * n - 1) * (2 * n)) + total = total + term + radius = max(x.lo.copy_abs(), x.hi.copy_abs()) + # The alternating tail decreases from this index; bound its first term. + need(Q(radius) ** 2 < (2 * terms + 1) * (2 * terms + 2), + "decreasing cosine tail") + error = I(radius).power(2 * terms) / factorial(2 * terms) + return total.widen(error) + + +def arithmetic_controls(): + for x in (Q(1, 8), Q(1, 2), Q(1)): + total = sum(((-x) ** k / factorial(k) for k in range(200)), Q(0)) + adjacent = total + (-x) ** 200 / factorial(200) + interval = I(-x).exp() + need(Q(interval.lo) <= min(total, adjacent) <= max(total, adjacent) <= Q(interval.hi), + "exp against independent rational alternating bounds") + for x in (Q(1, 4), Q(2), Q(17, 3)): + root = I(x).sqrt() + need(Q(root.lo) ** 2 <= x <= Q(root.hi) ** 2, "sqrt endpoint squares") + for x in (Q(-7, 3), Q(0), Q(5, 7)): + for y in (Q(2, 9), Q(11, 4)): + for candidate, exact in ((I(x) + I(y), x + y), + (I(x) * I(y), x * y), + (I(x) / I(y), x / y)): + need(Q(candidate.lo) <= exact <= Q(candidate.hi), "directed rational arithmetic") + return {"exp_cases": 3, "sqrt_cases": 3, "rational_operations": 18, + "decimal_digits": PRECISION} diff --git a/research/shifted-gaussian-conditioning/verify.py b/research/shifted-gaussian-conditioning/verify.py new file mode 100644 index 0000000000..48e5659984 --- /dev/null +++ b/research/shifted-gaussian-conditioning/verify.py @@ -0,0 +1,276 @@ +"""Certify finite consequences of the shifted Gaussian conditioning theorem. + +Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. +Apache-2.0. The universal argument is in PROOF.md. +""" +import argparse +from hashlib import sha256 +import importlib.util +import json +from pathlib import Path +from fractions import Fraction as Q + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +spec = importlib.util.spec_from_file_location("shifted_intervals", HERE / "intervals.py") +arithmetic = importlib.util.module_from_spec(spec) +spec.loader.exec_module(arithmetic) +I, D, DOWN, UP, need = (getattr(arithmetic, name) for name in ("I", "D", "DOWN", "UP", "need")) +WIDTH, BITS = 64, 192 + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(",", ":"), ensure_ascii=True).encode() + + +def sign(u): + return 1 if u % 2 == 0 else -1 + + +def check_source(raw, row): + need(sha256(raw).hexdigest() == row["sha256"], "source SHA256: " + row["path"]) + + +def authenticate(): + sources = json.loads((HERE / "SOURCES.json").read_text()) + rows = [row for row in sources["files"] if row["usage"] == "input"] + for row in rows: + raw = (ROOT / row["path"]).read_bytes() + check_source(raw, row) + rejected(lambda: check_source(raw + b"\x00", row), "changed source rejects its pin") + return [{"path": row["path"], "sha256": row["sha256"]} for row in rows] + + +def validate(gamma, shift): + need(type(gamma) is Q and type(shift) is Q, "rational parameters") + need(gamma > 0 and abs(shift) < Q(1, 2), "strict shifted Gaussian domain") + + +def coefficient(gamma, shift, u): + return I(gamma * (-u * u + 2 * shift * u)).exp() + + +def tails(gamma, shift, width): + need(width >= 0 and abs(shift) < Q(1, 2), "Gaussian tail domain") + return sum((coefficient(gamma, sign * shift, width + 1) / + (1 - I(-gamma * (2 * width + 3 - 2 * sign * shift)).exp()) + for sign in (-1, 1)), I(0)) + + +def minimum(gamma, shift, coefficients): + direct = sum((sign(u) * coefficients[u] for u in range(-WIDTH, WIDTH + 1)), I(0)) + direct = direct.widen(tails(gamma, shift, WIDTH)) + # Jacobi's product, with the omitted positive deficits bounded geometrically. + count = max(4, (40 * gamma.denominator + gamma.numerator - 1) // gamma.numerator) + step = I(-2 * gamma).exp() + c, a, b = step, I(-gamma * (1 - 2 * shift)).exp(), I(-gamma * (1 + 2 * shift)).exp() + product = I(1) + for _ in range(count): + product = product * (1 - c) * (1 - a) * (1 - b) + c, a, b = c * step, a * step, b * step + deficit = (c + a + b) / (1 - step) + need(0 <= deficit.lo <= deficit.hi < 1, "positive product tail") + product = I(DOWN.multiply(product.lo, DOWN.subtract(D(1), deficit.hi)), product.hi) + result = direct.intersect(product) + need(result.lo > 0, "positive minimum enclosure") + return result, {"direct": direct.record(), "product": product.record(), + "product_terms": count, "tail": tails(gamma, shift, WIDTH).record()} + + +def circle_checks(coefficients, minimum_bound, tail): + points = [(Q(1), Q(0)), (Q(4, 5), Q(3, 5)), (Q(0), Q(1)), + (Q(-4, 5), Q(3, 5)), (Q(-1), Q(0)), (Q(0), Q(-1))] + for x, y in points: + real, imag = I(1), I(0) + total = coefficients[0] + for u in range(1, WIDTH + 1): + real, imag = real * x - imag * y, real * y + imag * x + total = total + (coefficients[u] + coefficients[-u]) * real + total = total.widen(tail) + if x == -1: + total.intersect(minimum_bound) + else: + need(total.lo > minimum_bound.hi, "sampled real part exceeds minimum") + return len(points) + + +def rounded_kernel(coefficients, tail): + rounded, error = {}, tail + scale = 1 << BITS + for u, value in coefficients.items(): + lower = Q(value.lo) + rounded[u] = Q((lower * scale).__floor__(), scale) + error = error + (value - I(rounded[u])) + need(error.lo >= 0 and rounded[0] == 1, "rounded normalized coefficients") + return rounded, error + + +def ldl(matrix): + size = len(matrix) + lower = [[Q(0) for _ in range(size)] for _ in range(size)] + pivots = [] + for i in range(size): + pivot = matrix[i][i] - sum((lower[i][k] ** 2 * pivots[k] for k in range(i)), Q(0)) + need(pivot > 0, "strict positive rational LDL pivot") + pivots.append(pivot) + lower[i][i] = 1 + for j in range(i + 1, size): + lower[j][i] = (matrix[j][i] - + sum((lower[j][k] * lower[i][k] * pivots[k] + for k in range(i)), Q(0))) / pivot + # Verify the full factorization, rather than treating pivot signs alone as a receipt. + for i in range(size): + for j in range(size): + need(sum((lower[i][k] * pivots[k] * lower[j][k] + for k in range(min(i, j) + 1)), Q(0)) == matrix[i][j], + "exact rational LDL reconstruction") + return pivots + + +def fraction_hash(values): + digest = sha256() + for value in values: + 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 sections(coefficients, rounded, m): + result = [] + for size in (1, 2, 4, 8, 16): + margin = Q(m.lo) / 2 + gram = [[(rounded[j - i] + rounded[i - j]) / 2 - + (margin if i == j else 0) for j in range(size)] for i in range(size)] + pivots = ldl(gram) + norm_squared = I(0) + for i in range(size): + row = sum(((-1) ** j * coefficients[j - i] for j in range(size)), I(0)) + norm_squared = norm_squared + row.square() / size + need(norm_squared.lo > 0, "positive alternating-vector image") + inverse_lower = (1 / norm_squared.sqrt()).lo + inverse_upper = (1 / m).hi + need(inverse_lower <= inverse_upper, "finite inverse bounds consistent") + result.append({"size": size, "ldl_pivot_sha256": fraction_hash(pivots), + "smallest_pivot": I(min(pivots)).record(), + "ideal_inverse_norm_lower": str(inverse_lower), + "ideal_uniform_inverse_norm_upper": str(inverse_upper)}) + return result + + +def size_descriptor(gamma, m): + # This certifies a lemma's size threshold; no matrix of this size is executed. + a0, relative = 2 * (1 + gamma) ** 2 / gamma ** 2, Q(1, 8) + threshold = (a0 / (relative * Q(m.lo))) ** 2 + power = max(0, threshold.numerator.bit_length() - threshold.denominator.bit_length()) + if (1 << power) < threshold: + power += 1 + need(a0 ** 2 <= relative ** 2 * Q(m.lo) ** 2 * (1 << power), + "dyadic near-sharp size threshold") + if power: + need((1 << (power - 1)) < threshold, "smallest certified dyadic threshold") + return {"log2_size": power, "relative_error": "1/8", + "inverse_fraction_of_limit_at_least": "8/9", "executed": False} + + +def rejected(operation, label): + try: + operation() + except ValueError: + return label + raise ValueError("control unexpectedly passed: " + label) + + +def controls(): + rejected_cases = [ + rejected(lambda: validate(Q(1), Q(1, 2)), "positive half-shift"), + rejected(lambda: validate(Q(1), Q(-1, 2)), "negative half-shift"), + rejected(lambda: validate(Q(0), Q(0)), "zero scale"), + rejected(lambda: ldl([[Q(1), Q(4)], [Q(4), Q(1)]]), "indefinite off-diagonal"), + ] + # Exact cancellation at the endpoint pairs u with 1-u. + for u in range(-WIDTH, WIDTH + 2): + v = 1 - u + need(-u * u + u == -v * v + v and sign(u) == -sign(v), + "half-shift pair cancellation") + need((coefficient(Q(8), Q(3, 4), 1) + + coefficient(Q(8), Q(3, 4), -1)).lo > 2, + "outside-domain two-by-two Hermitian part is indefinite") + coeff = {u: coefficient(Q(8), Q(1, 4), u) for u in range(-WIDTH, WIDTH + 1)} + m, _ = minimum(Q(8), Q(1, 4), coeff) + raw_diagonal = I(-Q(8) * Q(1, 4) ** 2).exp() + need(m.lo > raw_diagonal.hi, "missing normalization overstates raw one-by-one margin") + need((coefficient(Q(1, 16), Q(0), 1) + + coefficient(Q(1, 16), Q(0), -1)).lo > 1, + "zero-width truncation cannot meet half-minimum budget") + need(canonical({"size": 1}) != canonical({"size": True}), + "receipt comparison preserves scalar types") + return {"rejected": rejected_cases, "half_shift_pairs": 2 * WIDTH + 2, + "outside_domain_indefinite": True, "normalization_control": True, + "unpaid_tail_control": True, "typed_receipt_control": True} + + +def run(): + sources = authenticate() + arithmetic_record = arithmetic.arithmetic_controls() + pi = arithmetic.pi_interval() + cases = [(g, s) for g in (Q(1, 16), Q(1, 8), Q(1, 2), Q(1), Q(4), Q(8)) + for s in (Q(0), Q(1, 4), Q(-1, 4), Q(1, 2) - Q(1, 2 ** 20))] + cases.append((Q(1, 16), Q(1, 2) - Q(1, 2 ** 40))) + records = [] + for gamma, shift in cases: + validate(gamma, shift) + coefficients = {u: coefficient(gamma, shift, u) for u in range(-WIDTH, WIDTH + 1)} + m, methods = minimum(gamma, shift, coefficients) + delta = 1 - 2 * abs(shift) + floor = delta * (pi / gamma).sqrt() * (-pi.square() / (4 * gamma)).exp() + old_floor = (1 - I(-gamma * delta).exp()) * (-pi.square() / (3 * gamma)).exp() + need(m.lo >= floor.hi, "new lower bound below certified minimum") + if gamma <= 1: + need(floor.lo > old_floor.hi, "small-scale improvement") + lead = (2 * I(gamma * shift ** 2).exp() * (pi / gamma).sqrt() * + (-pi.square() / (4 * gamma)).exp() * arithmetic.cos_interval(pi * shift)) + qp = (-2 * pi.square() / gamma).exp() + relative_tail = qp * (3 - qp) / (1 - qp).square() + need(relative_tail.hi < Q(1, 2), "positive Poisson leading bound for fixture") + poisson = I((lead * (1 - relative_tail)).lo, (lead * (1 + relative_tail)).hi) + poisson.intersect(m) + tail = tails(gamma, shift, WIDTH) + circle_count = circle_checks(coefficients, m, tail) + rounded, error = rounded_kernel(coefficients, tail) + need(error.hi < DOWN.divide(m.lo, D(2)), "complete kernel perturbation budget") + records.append({"gamma": str(gamma), "shift": str(shift), "gap": str(delta), + "minimum": m.record(), "enclosures": methods, + "new_floor": floor.record(), "old_floor": old_floor.record(), + "poisson_enclosure": poisson.record(), + "circle_samples": circle_count, "kernel_error_majorant": error.record(), + "rounded_inverse_norm_upper": str((1 / (m - error)).hi), + "finite_sections": sections(coefficients, rounded, m), + "near_sharp_size_descriptor": size_descriptor(gamma, m)}) + return {"schema_version": 1, "scope": "finite certified evidence; universal proof in PROOF.md", + "sources": sources, "arithmetic": arithmetic_record, "pi": pi.record(), + "kernel_half_width": WIDTH, "coefficient_fractional_bits": BITS, + "cases": records, "controls": controls()} + + +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, ensure_ascii=True) + "\n") + else: + need(canonical(result) == canonical(json.loads(path.read_text())), "complete typed receipt") + print(json.dumps({"status": "PASS_SHIFTED_GAUSSIAN_CONDITIONING", + "case_count": len(result["cases"]), + "finite_sections": sum(len(row["finite_sections"]) for row in result["cases"]), + "circle_samples": sum(row["circle_samples"] for row in result["cases"]), + "result_sha256": sha256(canonical(result)).hexdigest()}, sort_keys=True)) + + +if __name__ == "__main__": + main() From 1c874dd66b3e68b43937a1bb81ddc5205d48954e Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 14:55:48 +0200 Subject: [PATCH 2/3] Use finite Gaussian factors to remove the shift-gap loss --- .../shifted-gaussian-conditioning/PROOF.md | 118 +- .../shifted-gaussian-conditioning/README.md | 32 +- .../shifted-gaussian-conditioning/RESULT.json | 1709 +++++++++++++++-- .../SOURCES.json | 4 +- .../shifted-gaussian-conditioning/TASK.json | 16 +- .../shifted-gaussian-conditioning/VERIFY.json | 6 +- .../shifted-gaussian-conditioning/verify.py | 84 +- 7 files changed, 1818 insertions(+), 151 deletions(-) diff --git a/research/shifted-gaussian-conditioning/PROOF.md b/research/shifted-gaussian-conditioning/PROOF.md index e34cd117b0..b0110758f3 100644 --- a/research/shifted-gaussian-conditioning/PROOF.md +++ b/research/shifted-gaussian-conditioning/PROOF.md @@ -21,7 +21,11 @@ and phase argument in `vendor/GAUGEFREE-INTERIORS.md`, Lemmas 3.1–3.3. In Section 1 I make their minimum implication explicit. Vlado Vojdanovski already used the unshifted Poisson estimate in `FORWARD-SMALL-A-AUDIT.md`. I add the shifted uniform floor, its sharp exponential order, the -finite-section inverse limit, and the source-specific bound in Section 5. +finite-section inverse limit, and the source-specific bounds in Section 5. +For the stronger finite-cap estimate I use Vlado's retained exact q-LDU +identities from `HOLES-AND-Q-LDU.md`, Section 2, and the finite-origin +Newton argument in `BILATERAL-LDU.md`. I claim the conditioning corollary +and caller substitution, not the factorization identities. ## 1. Exact minimum from the retained product argument @@ -219,6 +223,93 @@ dyadic size descriptor for \(\varepsilon=1/8\); I do not execute those large matrices or claim monotonicity in \(N\). This statement concerns fixed \(\gamma,s\) and unrestricted sizes, not attainment within a caller's capped block. +## 3a. A finite-size bound through the half-shift endpoints + +We can use the finite cap to avoid the loss as \(\delta\downarrow0\). +For this section allow \(|s|\le1/2\), including both endpoints, and let +\(N\ge1\) be an integer. With \(q=e^{-\gamma}\), \(Q=q^2\), put +\[ +P_j=\prod_{r=1}^j(1-Q^r),\quad P_0=1,\quad +{i\brack j}_Q=\frac{P_i}{P_jP_{i-j}}\quad(0\le j\le i). +\] +I prove the finite bound +\[ +\boxed{\displaystyle +|(T_N(s)^{-1})_{ij}| +\le \frac{q^{|i-j|+2s(i-j)}}{P_{N-1}^3} +\le\frac1{P_{N-1}^3},\qquad +\|T_N(s)^{-1}\|_2\le\frac{N}{P_{N-1}^3} +\le N e^{\pi^2/(4\gamma)}.} +\] +Unlike the uniform-over-all-sizes accretivity bound, this estimate remains +finite at the half shifts for a fixed \(N\). + +Start from the source's exact finite factorization for +\(G_{ij}=q^{(i-j)^2}\): +\[ +G=L\,\operatorname{diag}(P_j)L^T,\qquad +L_{ij}=q^{(i-j)^2}{i\brack j}_Q,\qquad +(L^{-1})_{ij}=(-q)^{i-j}{i\brack j}_Q \quad(i\ge j). +\] +For completeness, factor \(G_{ij}=q^{i^2}(Q^{-i})^jq^{j^2}\). +Vandermonde determinants give the leading minor +\(\Delta_n=\prod_{j=0}^{n-1}P_j\); the powers of \(q\) cancel. +The minor with rows \(0,\ldots,j-1,i\) and columns \(0,\ldots,j\), +divided by \(\Delta_{j+1}\), gives the stated \(L_{ij}\). +The positive pivots are \(P_j\), so symmetry supplies the upper factor. +To check the inverse, use +\({i\brack k}_Q{k\brack j}_Q={i\brack j}_Q{i-j\brack k-j}_Q\). +For \(n=i-j\), the product reduces to a nonzero scalar times +\[ +\sum_{r=0}^n(-1)^rQ^{r(r-1)/2}{n\brack r}_Q +=\prod_{r=0}^{n-1}(1-Q^r). +\] +For \(n>0\) the product vanishes; for \(n=0\) it equals one. This is the +classical [finite q-binomial theorem, DLMF17.2.35](https://dlmf.nist.gov/17.2.E35). + +Write \(D_s[i,i]=q^{2si}\). Then +\(T_N(s)=D_sGD_s^{-1}\), which proves invertibility for these finite +matrices without a positive symbol minimum. At \(s=1/2\), the absolute +entries of the shifted lower inverse are +\[ +|(D_{1/2}L^{-1}D_{1/2}^{-1})_{ij}| +=Q^{i-j}{i\brack j}_Q +\le Q^{i-j}/P_{i-j}. +\] +Here \(P_i/P_j\le1\). Its column sums are at most +\[ +\sum_{r=0}^{N-1}\frac{Q^r}{P_r}=\frac1{P_{N-1}}, +\] +because \(Q^r/P_r=1/P_r-1/P_{r-1}\) for \(r\ge1\). +The shifted upper inverse has absolute entries +\({j\brack i}_Q\le1/P_{N-1}\), and each inverse diagonal pivot is also +at most \(1/P_{N-1}\). Multiplying the three finite factors gives +\[ +|(T_N(1/2)^{-1})_{ij}|\le P_{N-1}^{-3}. +\] +The negative endpoint is the transpose, so it satisfies the same entry bound. + +For \(i\ge j\), use +\((T_N(-1/2)^{-1})_{ij}=q^{-(i-j)}(G^{-1})_{ij}\); +for \(i0\). Expanding the logarithm and summing +nonnegative terms gives +\[ +-\log P_\infty +=\sum_{h\ge1}\frac1{h(e^{2\gamma h}-1)} +\le\frac1{2\gamma}\sum_{h\ge1}\frac1{h^2} +=\frac{\pi^2}{12\gamma}. +\] +This proves \(P_{N-1}^{-3}\le e^{\pi^2/(4\gamma)}\). +For strict shifts, reviewers may use the smaller of this finite-size +bound and \(1/m\). I do not claim an optimal finite-size constant or an +inverse algorithm with controlled intermediate precision from this norm bound. + ## 4. Rounding, truncation and certificate interpretation If an approximate kernel \(\widetilde k\) satisfies @@ -267,6 +358,11 @@ The `kernel_error_majorant` interval encloses the computed upper-bound expression; its lower endpoint is not a lower bound on the actual error. The finite-section inverse lower/upper fields concern the ideal matrix. The separately named rounded inverse bound uses \(1/(m-\varepsilon)\). +The additional finite-cap fixtures use exact rational \(q\), including +both half shifts. I reconstruct the finite q-LDU and triangular-inverse +identities, check both products with the direct shifted matrix, and +check the entry decay and row/column bounds. These checks concern completed +inverses; they do not certify a physical implementation of the factorization. I use precision-80 directed Decimal arithmetic for basic operations. Python documents correctly rounded nearest `exp` and `sqrt`; I widen each @@ -341,13 +437,27 @@ Hence \(e^{a\rho/4}=p^{1+o(1)}\), while \(\theta^{-1}\le4p^2\). The source's coarse \(a\le5\log(16p)\) alone would not give this exponent. The ceiling term also prevents an \(O(p^3)\) conclusion from this argument. +For the actual finite cap, Section 3a gives the stronger bound +\[ +\boxed{\displaystyle +\|K^{-1}\|_2\le Q_p^{\rm cap}:= +\alpha G N e^{a\rho/4} +\le p^{1+\xi+o(1)}\qquad (N\le B\le p^\xi).} +\] +Here \(\alpha=p^{o(1)}\), \(G=1+o(1)\) and +\(e^{a\rho/4}=p^{1+o(1)}\), uniformly over the admitted blocks. +This avoids the \(\theta^{-1}\) loss in the unrestricted-size symbol estimate. +At fixed \(\xi<1\), we obtain an inverse allowance below \(p^2\) eventually. +We can use \(\min(Q_p,Q_p^{\rm cap})\) in a perturbation argument. + For a raw operator perturbation of norm \(\varepsilon_p\) with -\(\varepsilon_pQ_p\le1/2\), the inverse bound becomes -\(Q_p/(1-\varepsilon_pQ_p)\le2Q_p\), preserving \(p^{3+o(1)}\). +\(\varepsilon_pQ_p^{\rm cap}\le1/2\), the inverse bound becomes +\(Q_p^{\rm cap}/(1-\varepsilon_pQ_p^{\rm cap})\le2Q_p^{\rm cap}\), +preserving \(p^{1+\xi+o(1)}\). A claimed truncation/rounding implementation must prove that premise. I improve the source's loose \(p^6\) allowance for these capped interiors. -I do not assert optimality of \(p^{3+o(1)}\) for capped, gauged blocks. +I do not assert optimality of \(p^{1+\xi+o(1)}\) for capped, gauged blocks. The caller assigns holes, phase cuts and period cuts to the boundary, and applies the broad Schur complement without inverting it. I retain those restrictions. The shallow-circuit construction, coefficient-generation costs, diff --git a/research/shifted-gaussian-conditioning/README.md b/research/shifted-gaussian-conditioning/README.md index 3d2b306de5..063d4fa860 100644 --- a/research/shifted-gaussian-conditioning/README.md +++ b/research/shifted-gaussian-conditioning/README.md @@ -12,9 +12,18 @@ m\ge(1-2|s|)\sqrt{\pi/\gamma}\,e^{-\pi^2/(4\gamma)},\qquad \] I include an explicit convergence bound using alternating vectors. +Using the predecessor's exact finite q-LDU identities, I also prove +\[ +\|T_N(s)^{-1}\|_2\le N/P_{N-1}^3\le N e^{\pi^2/(4\gamma)} +\quad (|s|\le1/2),\qquad +P_j=\prod_{r=1}^j(1-e^{-2\gamma r}). +\] +This second estimate includes both half-shift endpoints. I use the actual +block cap instead of losing a factor inverse to the shift gap. For the actual broad Gaussian caller in PR4100, I account for its \(\alpha^{-1}\) normalization, positive diagonal gauges and ceiling in -\(\alpha\), and derive \(\|K^{-1}\|_2\le p^{3+o(1)}\). +\(\alpha\), and derive \(\|K^{-1}\|_2\le p^{1+\xi+o(1)}\) for +\(N\le B\le p^\xi\), fixed \(0<\xi<1\). This is below \(p^2\) eventually. The author used a looser \(p^6\) allowance for those capped interiors. I retain the original product/phase proof and its exact minimum-at-\(\pi\) @@ -38,12 +47,13 @@ I restrict the caller consequence to centered, constant-phase, unit-gap capped interiors. The source assigns holes and phase/period cuts to the boundary. I leave the broad Schur complement as an applied factor. For a raw perturbation of norm \(\varepsilon_p\), the caller must establish -\(\varepsilon_pQ_p\le1/2\) before using the rounded inverse bound \(2Q_p\). +\(\varepsilon_pQ_p^{\rm cap}\le1/2\) before using the rounded inverse bound +\(2Q_p^{\rm cap}\). The sharp constant \(1/m\) concerns exact normalized sections at fixed \(\gamma,s\) as \(N\) grows without a cap. Each finite section has a strictly -smaller inverse norm. I make no optimality claim for capped gauged blocks, -no \(O(p^3)\) claim, and no selected-frontier change. +smaller inverse norm. I make no optimality claim for capped gauged blocks +or selected-frontier change. ## Verification @@ -69,6 +79,10 @@ The verifier checks these finite consequences: - I check 150 frequency-circle points and alternating-vector inverse bounds. I record analytic size thresholds for reaching \(8/9\) of the limiting inverse constant, and mark those large matrices as unexecuted. +- I check 60 exact rational finite-cap cases, including both half shifts, + against direct Gaussian matrices. I reconstruct the inherited q-LDU + identities, multiply each shifted inverse on both sides, and check the + new entry-decay and row/column bounds. I reject reversed shift conjugation. - I reject altered source bytes, half-shift/zero-scale parameters and an indefinite matrix. I check exact cancellation at the half shift, loss of accretivity outside the domain, the missing-normalization failure, @@ -91,12 +105,16 @@ shifted floor or sharp finite-section inverse statement in that coverage. Before publication I refreshed unchanged canonical main and inspected a further 100 PR records and 45 hub comments. ToikY's new PR4192 concerns sequential compiler arena scans, a separate obligation. +During final review I followed the finite-cap question to the retained +`HOLES-AND-Q-LDU.md` and `BILATERAL-LDU.md`. The author already supplies +the exact factors and a different narrow-cell estimate. I derive the +closed-shift finite-cap bound as a corollary and extend this same package. I did not exhaust older comments, arbitrary diffs, unpublished work or the mathematical literature. I credit Jeremy Kothe/IdeaOuter and the PR1758 contributors for the shifted product/phase argument; Vlado Vojdanovski/PR1673 for the unshifted Poisson -estimate and capped framework; and Chafik Boukhalfa/PR1786 for the +estimate, exact q-LDU identities and capped framework; and Chafik Boukhalfa/PR1786 for the fine-prime extension. Cinna Davis supplied the later caller I audit. I checked the original PR1758/1786 credits instead of repeating the later abbreviated attribution. I preserve the frozen notices and input bytes. @@ -111,7 +129,7 @@ Apache-2.0. ## Open next questions In [TASK.json](TASK.json) I specify two remaining questions with first steps -and blockers: a sharper bound that uses the actual cap \(N\le B(p)\), and -integration into the complete Gaussian circuit and error ledger. +and blockers: sharpness of the finite-cap estimate, and integration into +the complete Gaussian circuit and error ledger. Reviewers must retain the coefficient-generation, circuit-prefix and physical execution obligations when using this conditioning estimate. diff --git a/research/shifted-gaussian-conditioning/RESULT.json b/research/shifted-gaussian-conditioning/RESULT.json index f82a506f78..0e7abec695 100644 --- a/research/shifted-gaussian-conditioning/RESULT.json +++ b/research/shifted-gaussian-conditioning/RESULT.json @@ -29,6 +29,14 @@ { "path": "research/unconditional-batched-crt/vendor/FINE-PRIME.md", "sha256": "47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce" + }, + { + "path": "research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md", + "sha256": "c6a032ed495325806ddf491c5c85bc822a7fdd613aff906a7be553960d0f85a3" + }, + { + "path": "research/capped-phase-fusion/BILATERAL-LDU.md", + "sha256": "0d65e66d227464dda1a19f781ae0a0be671cbfd347c41e37ec954f3d629d8a77" } ], "arithmetic": { @@ -94,7 +102,8 @@ 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a/research/shifted-gaussian-conditioning/SOURCES.json +++ b/research/shifted-gaussian-conditioning/SOURCES.json @@ -10,6 +10,8 @@ {"path":"research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md","usage":"input","git_blob":"0f7f80b3f74d469c2f72f3b9f338d32568b4eae3","sha256":"e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0"}, {"path":"research/capped-phase-fusion/README.md","usage":"input","git_blob":"a8c26bf5320acc8f7be48aa5120b52f94b80a1d2","sha256":"d3fdbf5642ab684755bdecab12af247244a25b54765d6866bf5af5c342140aba"}, {"path":"research/capped-fusion-t74-k554/README.md","usage":"input","git_blob":"9b3ebbf298044413214f0275ba6a2fee055ccbde","sha256":"c0003664b62f671e089d8f470f79ba533691d46c552d32e7b0d4124ed100b930"}, - {"path":"research/unconditional-batched-crt/vendor/FINE-PRIME.md","usage":"input","git_blob":"bf9899ee21be4058e4aac0b64bfa67d01ee8ddd3","sha256":"47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce"} + {"path":"research/unconditional-batched-crt/vendor/FINE-PRIME.md","usage":"input","git_blob":"bf9899ee21be4058e4aac0b64bfa67d01ee8ddd3","sha256":"47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce"}, + {"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/BILATERAL-LDU.md","usage":"input","git_blob":"5cad6ee19f60d626beb9f6d7be78cbc22fdd08c6","sha256":"0d65e66d227464dda1a19f781ae0a0be671cbfd347c41e37ec954f3d629d8a77"} ] } diff --git a/research/shifted-gaussian-conditioning/TASK.json b/research/shifted-gaussian-conditioning/TASK.json index 6991793cfe..7014715ac5 100644 --- a/research/shifted-gaussian-conditioning/TASK.json +++ b/research/shifted-gaussian-conditioning/TASK.json @@ -38,26 +38,26 @@ ] }, "scope_status": "refined", - "actual_scope": "Prove a shift-uniform Gaussian accretivity floor with exponential coefficient 1/4, the sharp finite-section inverse limit, and a p^(3+o(1)) upper bound at PR4100's capped broad caller; certify finite consequences with directed enclosures and exact rational positivity.", - "scope_change_reason": "The retained source already supplies the product/phase argument and its minimum-at-pi corollary. The separate shifted floor and inverse sharpness improve its conditioning estimate without changing the arithmetic mechanism or paid supplier profile.", + "actual_scope": "Prove a shift-uniform Gaussian accretivity floor with exponential coefficient 1/4, the sharp unrestricted-size inverse limit, and a finite-cap q-LDU bound through half shifts giving p^(1+xi+o(1)) at PR4100's broad caller; certify finite consequences with directed enclosures and exact rational identities.", + "scope_change_reason": "The retained source already supplies the product/phase argument and exact q-LDU factors. The new conditioning corollaries improve the actual caller estimate without changing the arithmetic mechanism or paid profile. Final review sharpened the first p^(3+o(1)) estimate by using the finite cap in the same package.", "bottleneck_link": "The broad Gaussian caller needs a correctly normalized inverse bound before truncation, rounding and circuit-prefix estimates can be integrated into the full transform.", "budget_minutes": 120, "stop_rule": "Stop at an invalid normalization, unjustified infinite-product limit, nonnormal section treated as Hermitian, or fixed-cap optimality inferred from unrestricted-size sharpness. Keep complete circuit and multiplication acceptance outside this scope.", "outcome": "completed", - "evidence": "PROOF.md gives quantified analytic statements and the exact caller substitution. RESULT.json records 25 interval cases, 125 exact rational finite-section positivity certificates, 150 circle samples and domain/normalization/tail controls. Hosted and exported receipts are separate from scientific acceptance.", + "evidence": "PROOF.md gives quantified analytic statements and the exact caller substitution. RESULT.json records 25 interval cases, 125 rational positivity certificates, 150 circle samples, 60 exact finite-cap inverse cases including half shifts, and domain/normalization/tail/conjugation controls. Hosted and exported receipts are separate from scientific acceptance.", "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 caller obtain a smaller inverse bound using the restriction N<=B(p), rather than the unrestricted-size sharp constant?", - "why_it_matters": "The p^(3+o(1)) upper bound is not an optimality theorem for finite capped, gauged blocks.", - "first_step": "Compare an explicit finite-section singular-value lower bound against the retained symbol floor at the actual scale, shift gap and block cap.", - "blocked_by": "A size-dependent argument that remains uniform near half shifts and preserves the real diagonal gauges." + "question": "Is the finite-cap inverse estimate sharp in N near the half shifts, and can the actual caller obtain a smaller bound?", + "why_it_matters": "The p^(1+xi+o(1)) upper bound is not an optimality theorem for capped, gauged blocks.", + "first_step": "Construct a finite-section singular vector or refine the source q-LDU entry sum at the actual scale, shift gap and block cap.", + "blocked_by": "A matching size-dependent estimate that remains uniform near half shifts and preserves the real diagonal gauges." }, { "question": "Can the complete Gaussian circuit and error ledger consume the improved bound with source-bound coefficient and perturbation costs?", "why_it_matters": "Conditioning alone does not establish the shallow circuit, physical coefficient generation or complete multiplication theorem.", - "first_step": "Substitute the explicit Q_p from PROOF.md into one actual interior's perturbation and prefix bounds, retaining the incoming-height and rounded-boundary obligations.", + "first_step": "Substitute the explicit Q_p^cap from PROOF.md into one actual interior's perturbation and prefix bounds, retaining the incoming-height and rounded-boundary obligations.", "blocked_by": "Separate all-size circuit/precision review and physical caller integration." } ] diff --git a/research/shifted-gaussian-conditioning/VERIFY.json b/research/shifted-gaussian-conditioning/VERIFY.json index 97a5700123..8e72db3ffb 100644 --- a/research/shifted-gaussian-conditioning/VERIFY.json +++ b/research/shifted-gaussian-conditioning/VERIFY.json @@ -10,13 +10,15 @@ "research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md", "research/capped-phase-fusion/README.md", "research/capped-fusion-t74-k554/README.md", - "research/unconditional-batched-crt/vendor/FINE-PRIME.md" + "research/unconditional-batched-crt/vendor/FINE-PRIME.md", + "research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md", + "research/capped-phase-fusion/BILATERAL-LDU.md" ], "commands": [ { "argv": ["{python}", "-I", "-B", "{package}/verify.py"], "timeout_seconds": 180, - "checks": "Authenticate sources, enclose shifted Gaussian minima by two convergent representations, check Poisson bounds, charge coefficient/tail perturbations, reconstruct exact rational LDL certificates and reject domain/normalization controls." + "checks": "Authenticate sources, enclose shifted Gaussian minima and perturbations, reconstruct rational positivity certificates, and check finite q-LDU inverse bounds through both half-shift endpoints with exact kernel/inverse identities." }, { "argv": ["{python}", "-I", "-O", "-B", "{package}/verify.py"], diff --git a/research/shifted-gaussian-conditioning/verify.py b/research/shifted-gaussian-conditioning/verify.py index 48e5659984..112aa87431 100644 --- a/research/shifted-gaussian-conditioning/verify.py +++ b/research/shifted-gaussian-conditioning/verify.py @@ -138,7 +138,7 @@ def fraction_hash(values): return digest.hexdigest() -def sections(coefficients, rounded, m): +def sections(coefficients, rounded, m, finite_constant): result = [] for size in (1, 2, 4, 8, 16): margin = Q(m.lo) / 2 @@ -156,7 +156,8 @@ def sections(coefficients, rounded, m): result.append({"size": size, "ldl_pivot_sha256": fraction_hash(pivots), "smallest_pivot": I(min(pivots)).record(), "ideal_inverse_norm_lower": str(inverse_lower), - "ideal_uniform_inverse_norm_upper": str(inverse_upper)}) + "ideal_uniform_inverse_norm_upper": str(inverse_upper), + "ideal_gap_free_inverse_norm_upper": str((size * finite_constant).hi)}) return result @@ -183,6 +184,79 @@ def rejected(operation, label): raise ValueError("control unexpectedly passed: " + label) +def multiply(left, right): + return [[sum((left[i][k] * right[k][j] for k in range(len(right))), Q(0)) + for j in range(len(right[0]))] for i in range(len(left))] + + +def finite_shift_validate(shift): + need(abs(shift) <= Q(1, 2), "closed half-shift domain") + + +def exact_power(base, exponent): + exponent = Q(exponent) + need(exponent.denominator == 1, "integer power in rational Gaussian fixture") + return base ** exponent.numerator + + +def finite_cap_certificates(): + records = [] + wrong_shift_rejections = 0 + for root_q in (Q(1, 2), Q(3, 4), Q(15, 16)): + q = root_q ** 2 + for size in (1, 2, 4, 8): + product = [Q(1)] + for j in range(1, size): + product.append(product[-1] * (1 - q ** (2 * j))) + choose = [[product[i] / (product[j] * product[i - j]) if j <= i else Q(0) + for j in range(size)] for i in range(size)] + lower = [[q ** ((i - j) ** 2) * choose[i][j] if j <= i else Q(0) + for j in range(size)] for i in range(size)] + lower_inverse = [[(-q) ** (i - j) * choose[i][j] if j <= i else Q(0) + for j in range(size)] for i in range(size)] + identity = [[Q(i == j) for j in range(size)] for i in range(size)] + centered = [[q ** ((i - j) ** 2) for j in range(size)] for i in range(size)] + diagonal_times_transpose = [[product[i] * lower[j][i] for j in range(size)] + for i in range(size)] + need(multiply(lower, diagonal_times_transpose) == centered, "retained q-LDU identity") + need(multiply(lower_inverse, lower) == identity, "finite q-binomial triangular inverse") + centered_inverse = [[sum((lower_inverse[k][i] * lower_inverse[k][j] / product[k] + for k in range(max(i, j), size)), Q(0)) + for j in range(size)] for i in range(size)] + # The finite product is larger than P_infinity, so this is a stronger + # finite-fixture bound than the universal P_infinity expression. + constant = 1 / product[-1] ** 3 + for shift in (Q(-1, 2), Q(-1, 4), Q(0), Q(1, 4), Q(1, 2)): + finite_shift_validate(shift) + matrix = [[exact_power(root_q, 2 * ((j - i) ** 2 - 2 * shift * (j - i))) + for j in range(size)] for i in range(size)] + inverse = [[centered_inverse[i][j] * exact_power(root_q, 4 * shift * (i - j)) + for j in range(size)] for i in range(size)] + need(multiply(matrix, inverse) == identity and multiply(inverse, matrix) == identity, + "exact shifted Gaussian left and right inverse") + for i in range(size): + for j in range(size): + decay = exact_power(root_q, 2 * abs(i - j) + 4 * shift * (i - j)) + need(abs(inverse[i][j]) <= constant * decay <= constant, + "gap-free inverse entry bound") + row_norm = max(sum(abs(x) for x in row) for row in inverse) + column_norm = max(sum(abs(inverse[i][j]) for i in range(size)) for j in range(size)) + need(max(row_norm, column_norm) <= size * constant, "finite Euclidean norm majorant") + if shift and size > 1: + wrong = [[centered_inverse[i][j] * exact_power(root_q, -4 * shift * (i - j)) + for j in range(size)] for i in range(size)] + need(multiply(matrix, wrong) != identity, "reversed shift conjugation rejects") + wrong_shift_rejections += 1 + records.append({"q": str(q), "shift": str(shift), "size": size, + "inverse_sha256": fraction_hash([x for row in inverse for x in row]), + "row_norm": I(row_norm).record(), "column_norm": I(column_norm).record(), + "finite_product_constant": I(constant).record(), + "gap_free_inverse_norm_upper": I(size * constant).record()}) + rejected(lambda: finite_shift_validate(Q(3, 4)), "outside closed half-shift domain") + return {"cases": records, "reversed_shift_rejections": wrong_shift_rejections, + "outside_domain_rejected": True} + + def controls(): rejected_cases = [ rejected(lambda: validate(Q(1), Q(1, 2)), "positive half-shift"), @@ -241,18 +315,19 @@ def run(): circle_count = circle_checks(coefficients, m, tail) rounded, error = rounded_kernel(coefficients, tail) need(error.hi < DOWN.divide(m.lo, D(2)), "complete kernel perturbation budget") + finite_constant = (pi.square() / (4 * gamma)).exp() records.append({"gamma": str(gamma), "shift": str(shift), "gap": str(delta), "minimum": m.record(), "enclosures": methods, "new_floor": floor.record(), "old_floor": old_floor.record(), "poisson_enclosure": poisson.record(), "circle_samples": circle_count, "kernel_error_majorant": error.record(), "rounded_inverse_norm_upper": str((1 / (m - error)).hi), - "finite_sections": sections(coefficients, rounded, m), + "finite_sections": sections(coefficients, rounded, m, finite_constant), "near_sharp_size_descriptor": size_descriptor(gamma, m)}) return {"schema_version": 1, "scope": "finite certified evidence; universal proof in PROOF.md", "sources": sources, "arithmetic": arithmetic_record, "pi": pi.record(), "kernel_half_width": WIDTH, "coefficient_fractional_bits": BITS, - "cases": records, "controls": controls()} + "cases": records, "finite_cap": finite_cap_certificates(), "controls": controls()} def main(): @@ -269,6 +344,7 @@ def main(): "case_count": len(result["cases"]), "finite_sections": sum(len(row["finite_sections"]) for row in result["cases"]), "circle_samples": sum(row["circle_samples"] for row in result["cases"]), + "exact_finite_cap_cases": len(result["finite_cap"]["cases"]), "result_sha256": sha256(canonical(result)).hexdigest()}, sort_keys=True)) From 52e962e318423be1fe8ed9d705dba92067795b61 Mon Sep 17 00:00:00 2001 From: Jose Manuel <118126596+jmbarrancoml@users.noreply.github.com> Date: Tue, 20 Oct 2026 14:57:06 +0200 Subject: [PATCH 3/3] Clarify the half-shift domains of the two conditioning bounds --- research/shifted-gaussian-conditioning/README.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/research/shifted-gaussian-conditioning/README.md b/research/shifted-gaussian-conditioning/README.md index 063d4fa860..90beab7d54 100644 --- a/research/shifted-gaussian-conditioning/README.md +++ b/research/shifted-gaussian-conditioning/README.md @@ -83,8 +83,9 @@ The verifier checks these finite consequences: against direct Gaussian matrices. I reconstruct the inherited q-LDU identities, multiply each shifted inverse on both sides, and check the new entry-decay and row/column bounds. I reject reversed shift conjugation. -- I reject altered source bytes, half-shift/zero-scale parameters and an - indefinite matrix. I check exact cancellation at the half shift, loss +- I reject altered source bytes, zero scales and an indefinite matrix. + I reject half shifts for positive symbol-margin admission while admitting + them in the finite-cap theorem. I check exact cancellation at the half shift, loss of accretivity outside the domain, the missing-normalization failure, an unpaid truncation tail and JSON scalar-type corruption.