diff --git a/research/rational-stein-generators/PROOF.md b/research/rational-stein-generators/PROOF.md
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+# Rational bounded generators at the dirty-shear interface
+
+Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance.
+Apache-2.0.
+
+I refine the coefficient generator in Vlado Vojdanovski's
+`SHALLOW-TOEPLITZ.md`, Sections 5–6. The author already gives a
+polynomial-time route through rational Gram–Schmidt and approximate
+square-root normalization. I replace that normalization with exact dyadic
+scaling and bind the resulting rational rows to the retained macro order.
+I do not claim that irrational normalization invalidates the original proof.
+
+I write \(E\in\mathbb Q(i)^{m\times m}\), \(m\ge1\), with rank \(r\).
+All vector norms and unqualified operator norms are Euclidean. I distinguish
+a coefficient **row** \(a\) from its adjoint column \(a^*\).
+
+## 1. Exact rational generators with bounded magnitude
+
+Process the columns of \(E\) in order by exact Gram–Schmidt. Skip a residual
+only when it is exactly zero. Denote the nonzero mutually orthogonal residuals
+by \(v_j\), and write \(n_j=v_j^*v_j>0\). Choose the least dyadic \(c_j=2^{e_j}\)
+such that \(c_j^2\ge n_j\). Equivalently,
+\[
+n_j\le c_j^2<4n_j.
+\]
+We can find this integer exponent by rational squared comparisons.
+Define
+\[
+g_j=v_j/c_j,\qquad a_j=(c_j/n_j)v_j^*E.
+\]
+Then all entries lie in \(\mathbb Q(i)\), and
+\[
+g_ja_j=\frac{v_jv_j^*}{v_j^*v_j}E.
+\]
+The orthogonal projectors sum to the identity on the column space of \(E\).
+Thus
+\[
+\boxed{E=\sum_{j=1}^r g_ja_j,\qquad
+\|g_j\|_2\le1,\qquad \|a_j\|_2<2\|E\|_2.}
+\]
+The last inequality follows from
+\(\|v_j^*E\|_2\le\|v_j\|_2\|E\|_2\) and \(c_j<2\|v_j\|_2\).
+A tiny nonzero residual therefore need not cause a large emitted coefficient.
+The zero matrix gives an empty generator list.
+
+For \(E=T-ZTZ^T\), where \(Z\) is the nilpotent lower shift,
+\[
+T=\sum_{k=0}^{m-1}Z^kE(Z^T)^k
+ =\sum_j L(g_j)U(a_j).
+\]
+We obtain the first identity by finite telescoping; the second follows by
+expanding each outer product. \(L\) uses a first column and \(U\) a first
+row. We apply no additional conjugation to \(a_j\).
+
+The source prose sometimes uses an adjoint column \(h=E^*g\), in which case
+the upper Toeplitz row is \(h^*\). The numerical source stores
+`vals[j] * Vh[j, :]`, which already is a row. I follow that executable
+convention. For \(T=\left[\begin{smallmatrix}a&i\tau\\0&a\end{smallmatrix}\right]\),
+\(a=\tau=1/16\), the Stein displacement has first row \((a,i\tau)\).
+Conjugating that emitted row changes the desired operator.
+This convention check does not establish a failure for the real Gaussian caller.
+
+## 2. Admission to the retained generator macro
+
+Let
+\[
+\ell(a)=\sum_k(|\Re a_k|+|\Im a_k|),\qquad
+\Gamma=\min\{2^e:2^e\ge\max(1,2\ell(a))\},\qquad
+P=I+U(a)/\Gamma.
+\]
+Then \(\|U(a)/\Gamma\|_1,\|U(a)/\Gamma\|_\infty\le1/2\).
+Hence \(\|P\|_2\le3/2\), \(\|P^{-1}\|_2\le2\), and its diagonal coefficient
+has modulus at least \(1/2\). Its inverse is a finite upper triangular
+Toeplitz series, with rational complex coefficients.
+For dyadic \(m\), these are the norm hypotheses of the retained triangular
+child. The generator and macro matrix identities hold for arbitrary \(m\);
+I do not extend the child's dyadic scheduling contract to other sizes.
+
+On disjoint banks \((x,y)\), execute the source's four macros:
+\[
+x\gets Px,\quad y\gets y+\Gamma L(g)x,\quad
+x\gets P^{-1}x,\quad y\gets y-\Gamma L(g)x.
+\]
+Substitution gives \(x_{\rm out}=x_{\rm in}\) and
+\[
+y_{\rm out}=y_{\rm in}+\Gamma L(g)(P-I)x_{\rm in}
+=y_{\rm in}+L(g)U(a)x_{\rm in}.
+\]
+No zero-initialized bank is required. Summing the completed generator shears
+therefore applies \(y\gets y+Tx\) while restoring arbitrary dirty \(x\).
+The dyadic division that constructs \(P\) must allocate a fresh coefficient
+vector; modifying the original row would change the final compensation.
+
+## 3. Macro prefixes and child boundaries
+
+Put \(B=\|E\|_2\), \(A_k=\sum_{j<k}L(g_j)U(a_j)\), and
+\(\alpha=2rmB\). Since
+\(\|L(g)\|_2\le\sqrt m\|g\|_2\) and
+\(\|U(a)\|_2\le\sqrt m\|a\|_2\), we have \(\|A_k\|_2\le\alpha\).
+Let \(K=\max_j\Gamma_j\sqrt m\), with \(K=0\) for an empty list.
+
+Before term \(k\), the four successive global prefixes are
+\[
+\begin{pmatrix}P&0\\A_k&I\end{pmatrix},\quad
+\begin{pmatrix}P&0\\A_k+\Gamma L(g)P&I\end{pmatrix},\quad
+\begin{pmatrix}I&0\\A_k+\Gamma L(g)P&I\end{pmatrix},\quad
+\begin{pmatrix}I&0\\A_{k+1}&I\end{pmatrix}.
+\]
+Block-triangular inversion and the preceding norm bounds give, for their
+norms and inverse norms,
+\[
+C_{\rm macro}=2+2\alpha+2K.
+\]
+For example, the second prefix has inverse lower-left block
+\(-A_kP^{-1}-\Gamma L(g)\), so we need no product of \(r\) large shear bounds.
+Completed generator contributions add in \(A_k\).
+
+By Cauchy–Schwarz,
+\(\ell(a)\le\sqrt{2m}\|a\|_2<2\sqrt{2m}B\).
+The least-power-of-two choice gives
+\(\Gamma<2\max(1,2\ell(a))\le2+8\sqrt{2m}B\).
+Consequently
+\[
+\boxed{C_{\rm macro}
+\le2+4\sqrt m+4rmB+16\sqrt2\,mB
+\le2+4m+4rmB+32mB.}
+\]
+The verifier uses the rational final expression with a row/column norm
+upper bound on \(B\), and checks a stronger Frobenius upper estimate for
+each executed prefix and inverse prefix.
+
+If a separately verified child implementation has prefix and inverse-prefix
+bound \(H\), one unfinished child combines with an already bounded macro
+prefix through \(HC_{\rm macro}\). This does not certify the child's bound.
+In particular, the replay here replaces `triangle_plan`, `toeplitz_shear`
+and `inverse` by exact matrix callbacks. I do not report FFT/butterfly
+execution or their internal prefixes.
+
+## 4. Explicit polynomial preprocessing height
+
+For a rational real component, define height as the maximum binary length
+of its reduced numerator and positive denominator. Suppose all input
+components have height at most \(b\ge1\). A conservative bound for the
+generator preprocessing and finite \(P^{-1}\) coefficients is
+\[
+\boxed{\mathcal B=4096(m+1)^6(b+1).}
+\]
+This bounds coefficient encodings; it is not a proposed online word width.
+
+Here are explicit estimates, including nearly dependent columns. Take
+the product \(D\) of the \(2m^2\) component denominators and put \(A=DE\).
+Then \(D\le2^{2m^2b}\) and the Gaussian-integer components of \(A\)
+have magnitude below \(2^H\), where
+\[
+H=(2m^2+1)b,\quad J=2H+2m+2,\quad
+T_0=m(J+m)+1,\quad V=T_0+J+H+2m+4.
+\]
+For any independent subset \(A_S\), entries of \(A_S^*A_S\) have
+complex modulus below \(2^J\). The determinant expansion, with
+\(k!\le2^{m^2}\), bounds its determinant and cofactors by \(2^{T_0}\).
+The determinant \(d\) is a positive integer.
+
+Each full or partial Gram–Schmidt residual has the form
+\[
+v=\frac{z}{Dd},\qquad
+z=dA_j-A_S\operatorname{adj}(A_S^*A_S)(A_S^*A_j).
+\]
+We can bound the components of \(z\) and the common denominator \(Dd\)
+by \(2^V\). This also controls intermediate residual vectors: subtracting
+the first accepted orthogonal residuals projects onto the span of an
+independent subset of original columns.
+
+Write \(Z_0=\sum_i|z_i|^2\), \(W=2V+2m+2\), \(C_0=V+m+2\).
+The squared norm \(Z_0/(D^2d^2)\) has numerator/denominator heights at most
+\(W\). For a nonzero residual,
+\(2^{-V}\le\|v\|_2\le\sqrt{2m}\,2^V\), so the exponent of \(c\)
+has absolute value at most \(C_0\). The components of \(g=v/c\)
+have height at most \(V+C_0+1\).
+
+For the coefficient row, cancel before estimating:
+\[
+a=\frac{c}{\|v\|_2^2}v^*E=\frac{cd\,z^*A}{Z_0}.
+\]
+Thus its components share a denominator and have height at most
+\(R=3V+3m+6\). This avoids an assumed lower bound on the nonzero singular
+values of \(E\).
+
+The norm estimate gives
+\(\ell(a)<4m^{3/2}2^b\), so the exponent of the adaptive \(\Gamma\)
+is at most \(b+2m+5\). Put \(H_P=R+b+2m+7\).
+The \(P\) coefficients have a common integer denominator and
+Gaussian-integer numerators of component height at most \(H_P\).
+If \(P(z)=a(z)/d_0\), its inverse coefficient of degree \(n<m\) has form
+\[
+u_n=\frac{d_0b_n}{a_0^{n+1}},\qquad
+|b_n|\le2^n(\sqrt2\,2^{H_P})^n.
+\]
+The finite composition expansion gives the displayed numerator estimate.
+Rationalizing the Gaussian denominator bounds the component heights by
+\(I_0=2mH_P+3m+4\).
+
+For operation bounds, retain the common-denominator invariants.
+Residual component denominators divide \(Dd\); dot-product partial sums divide the
+product of two vector denominators; inverse-series partial sums at degree
+\(n\) have Gaussian denominators dividing \(a_0^n\), with rationalized
+denominator \(|a_0|^{2n}\). Reduced `Fraction` storage preserves those
+divisibility statements. Raw cross-products in a scalar update, followed
+by gcd reduction, and dyadic squared comparisons fit \(16S\) bits, where
+\[
+S=H+J+T_0+V+W+C_0+R+H_P+I_0+m+b+1.
+\]
+For \(t=m+1\), \(b_1=b+1\), the respective estimates are
+\[
+H\le2t^2b_1,\ J\le5t^2b_1,\ T_0\le6t^3b_1,\ V\le12t^3b_1,\
+W\le25t^3b_1,\ C_0\le13t^3b_1,\ R\le38t^3b_1,\
+H_P\le40t^3b_1,\ I_0\le81t^4b_1.
+\]
+Summing gives \(S\le150t^4b_1\), hence
+\(16S\le2400t^4b_1<\mathcal B\).
+There are polynomially many scalar operations. With schoolbook arithmetic
+and gcd reduction, \(O(m^4\mathcal B^3)\) bit operations and
+\(O(m^2\mathcal B)\) coefficient-storage bits suffice.
+
+The executable records observed component, named-norm and final-scale
+heights during generator extraction. It does not instrument every internal
+`Fraction` temporary; the universal intermediate bound relies on the proof.
+Its exact matrix callbacks verify macro identities; I do not claim that
+their generic Gaussian elimination is the proposed online implementation
+or that its temporary allocation follows the coefficient-vector storage bound.
+
+## 5. Actual caller and proof boundary
+
+The retained full Toeplitz recursion gives shear Stein rank \(O(\log N)\).
+This replacement emits exactly that rank, with generator norms at most
+twice the source's orthonormal bounds. Its adaptive \(\Gamma\) admits
+the same near-identity triangular child. Therefore it preserves the
+source's \(O(\log^3 N)\) shear depth and \(O(\log^4 N)\) full depth,
+conditional on the retained child constructions.
+
+For the full recursion, leading-section inverse bounds control contiguous
+Schur quotients. The parity recursion inside \(P\) can use noncontiguous
+principal sets; it needs its own \(\|P-I\|_{1,\infty}\le1/2\) argument.
+I do not substitute leading-section conditioning for that near-identity
+hypothesis.
+
+At PR4100 `GAUSSIAN.md:102–117`, rational reference Schur matrices provide
+the inputs, while bounded coefficients and prefix norms determine later
+data precision. This package makes the generator stage exact over
+\(\mathbb Q(i)\) and removes a fixed numerical rank threshold from that
+stage. FFT roots, approximate child gates, rounding across the complete
+circuit, physical coefficient delivery and the outer theorem remain
+separate obligations.
+
+## 6. Source-bound finite evidence
+
+I authenticate the original Python sources before parsing or executing them.
+I execute `check_exact_algebra.py`'s Schur walk with one collection call
+after its \(K,L,S\) calculation; I preserve its rank/displacement assertions
+under optimized verifier execution.
+
+From `lowdisp_shear` I extract the seven statements beginning with
+`p = h / gamma`, preserving their child-call order and
+`source[::-1]` convention. I replace SVD and scale selection by the
+rational producer, and replace numerical children by exact matrix callbacks.
+I check both generator identities, exact rank, bounded norms/heights,
+the complete macro endpoint, inverse-prefix identities and a norm envelope.
+I retain the macro AST hash in the receipt.
+
+A \(2^{-200}\) pivot lies below the source replay's fixed SVD threshold,
+but is nonzero in the new producer. Omitting it changes the exact endpoint.
+This is a precision-sensitive control, not a claim that the predecessor
+violated its stated floating tolerance. Other controls reverse the complex
+row convention, omit compensation or source restoration, and reject zero
+normalization and floating input.
+
+I credit Vlado Vojdanovski/PR1673 for the Schur producer, bounded-generator
+compiler and original preprocessing proof, and Cinna Davis for the later
+Gaussian caller. The earlier core review already discusses offline versus
+online precision. I retain classical Gram–Schmidt and structured-matrix
+attribution, and claim no worldwide novelty for dyadic normalization.
diff --git a/research/rational-stein-generators/README.md b/research/rational-stein-generators/README.md
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+# Exact rational Stein generators
+
+I implement bounded rational generators for the coefficient stage of the
+retained Gaussian dirty-shear compiler. For a rational complex Stein
+displacement \(E\), I emit exactly \(\operatorname{rank}E\) pairs
+\[
+E=\sum_j g_ja_j,\qquad
+\|g_j\|_2\le1,\quad\|a_j\|_2<2\|E\|_2,
+\]
+where \(a_j\) is an explicit row and every coefficient lies in \(\mathbb Q(i)\).
+I use exact orthogonal residuals and dyadic norm scaling. I compute neither
+a numerical rank cutoff nor a square root.
+
+The source already gives a polynomial-time route through rational
+Gram–Schmidt and approximate square-root normalization. I refine that
+construction with an exact producer, explicit bit bounds and source-bound
+macro evidence. I do not report irrational normalization as an error.
+
+## Result and caller
+
+I prove a conservative preprocessing height bound
+\(4096(m+1)^6(b+1)\) for \(m\times m\) input with component height \(b\),
+and retain the source's generator count. I choose a dyadic
+\(\Gamma\ge2\sum_k(|\Re a_k|+|\Im a_k|)\), making
+\(P=I+U(a)/\Gamma\) admissible to the original triangular child on its
+dyadic tiles.
+
+I replay the source's order
+\[
+Px;\quad y+\Gamma L(g)x;\quad P^{-1}x;\quad y-\Gamma L(g)x
+\]
+with exact matrix callbacks. I prove a polynomial macro-prefix and
+inverse-prefix bound using additive completed shears, and show how one
+unfinished child's separate bound combines with it.
+[PROOF.md](PROOF.md) gives the normalization, conjugation convention,
+bit estimates and full scope.
+
+At PR4100 `GAUSSIAN.md:102–117`, this producer can replace the bounded-generator
+normalization stage for rational reference Schur matrices. The source's
+numerical implementation already uses a correct coefficient-row convention;
+I make its relation to the adjoint-column notation explicit. FFT roots and
+complete child circuits remain separate.
+
+I start from main `12fa8c5c844be21bfefb38a19503fbd867a11244`, priority 7,
+and pin PR4100's retained source
+`337d04939571a637aeab54a0f58f54fde811663a`.
+[SOURCES.json](SOURCES.json) records the actual bytes.
+[Discussion4196](https://github.com/CrocSwap/integer-mult-bounds/discussions/4196)
+records the claim, live overlap and publication handoff.
+
+## Verification
+
+From the repository root, with CPython 3.12 and its standard library:
+
+```sh
+python3.12 -B scripts/research_package.py \
+  --package research/rational-stein-generators
+```
+
+I run two ordinary/optimized commands, each with a 180-second limit.
+I authenticate the retained exact Schur producer and preserve its assertions,
+adding one collection hook. I then extract the original generator macro's
+seven-statement tail, preserve its order and reversed source indices, and
+replace SVD/scale selection with the new producer.
+
+The finite evidence covers 56 matrices, 67 generators and 268 macro stages.
+I check exact displacement/product reconstruction, ranks, norm bounds,
+observed component/norm/scale heights, full endpoints, inverse-prefix
+identities and a conservative prefix envelope. The fixtures include
+actual recursive Schur shears, a tiny second independent residual, both
+real and complex matrices, zero rank and a large scale.
+
+Controls reject omission of a \(2^{-200}\) nonzero rank term, an extra
+complex conjugation, missing compensation, missing source restoration,
+zero normalization, floating inputs and altered source bytes.
+I compare the complete typed [RESULT.json](RESULT.json).
+The observed heights are not an exhaustive profile of arithmetic temporaries.
+
+I use exact matrix callbacks for `triangle_plan`, `toeplitz_shear` and
+`inverse`. This is a source-bound macro-schedule replay. I do not report
+FFT/butterfly execution, physical tape behavior or complete Gaussian
+acceptance. The separate written height and norm proofs carry the
+universal statements.
+
+## Prior work and attribution
+
+I inspected 100 recent PR records and 35 hub comments, the original
+shallow-Toeplitz preprocessing proof, earlier core/outer reviews and
+local generator searches. I found no matching rational bounded-generator
+implementation in that coverage. I did not exhaust arbitrary diffs,
+older comments, unpublished work or literature priority.
+The repository overlap helper searched 5,236 local records and returned
+106 broad terminology matches; it had no usable remote cache. I used
+separate live snapshots, including 60 PRs and 10/696 hub comments before
+publication. The new PR4195 concerns CRT shared rows and PR4197 concerns
+compensation forests.
+
+I retain PR4178's incoming-height/rounded-caller result, PR4186's boundary
+rounding result, PR4193's conditioning result and PR4192's compiler arena
+audit as separate contributions. The new source adapter does not replace
+their respective obligations.
+
+I credit Vlado Vojdanovski/PR1673 for the bounded-generator compiler,
+exact Schur producer and polynomial-time preprocessing argument; Cinna
+Davis for the later caller; and classical Gram–Schmidt and structured-matrix
+methods. I preserve the frozen source bytes and notices.
+Jose Manuel (`jmbarrancoml`), with substantial OpenAI Codex assistance.
+Source-analysis agents belong to the author process. I claim no independent
+human or formal-kernel acceptance, worldwide novelty or selected-frontier change.
+Apache-2.0.
+
+## Open next questions
+
+In [TASK.json](TASK.json) I specify two next questions with first steps
+and blockers: certified rounded child-circuit execution using these
+coefficients, and source-bound delivery into the full Gaussian caller's
+coefficient pages and error budget. Reviewers must retain those obligations
+when using this generator.
diff --git a/research/rational-stein-generators/RESULT.json b/research/rational-stein-generators/RESULT.json
new file mode 100644
index 0000000000..272748119e
--- /dev/null
+++ b/research/rational-stein-generators/RESULT.json
@@ -0,0 +1,2914 @@
+{
+  "schema_version": 1,
+  "scope": "rational coefficient generation and exact source macro replay",
+  "sources": [
+    {
+      "path": "research/unconditional-batched-crt/GAUSSIAN.md",
+      "sha256": "2f63c91c58f2d1cf31294812c75970ea390c61866a8da7ed700d4316a033a242"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md",
+      "sha256": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md",
+      "sha256": "d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116"
+    },
+    {
+      "path": "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_circuits.py",
+      "sha256": "06d1861bfe8af543908e26557ffda7e451ff176bceb9bacfaf0f04f234b336d5"
+    },
+    {
+      "path": "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py",
+      "sha256": "5ff7f25c55b370a1d0cda6c84ab9cdf2c7b0de3266d3f2e7a43280a6491074ab"
+    },
+    {
+      "path": "research/capped-phase-fusion/README.md",
+      "sha256": "d3fdbf5642ab684755bdecab12af247244a25b54765d6866bf5af5c342140aba"
+    },
+    {
+      "path": "research/capped-phase-fusion/evidence/independent_fusion_review_ultra/core_review.md",
+      "sha256": "13f2df4d5b240f9b89ca0ae44bba198c935532098ff4c1161631fcbd86b3e8b1"
+    }
+  ],
+  "macro_ast_sha256": "2a2e0f16a779597fa555857307451831bace11e66861fdc3caf26d2373a21210",
+  "source_schur_trace": [
+    {
+      "n": 2,
+      "depth": 0,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
+      "L_rank": 0
+    },
+    {
+      "n": 2,
+      "depth": 0,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
+      "L_rank": 0
+    },
+    {
+      "n": 4,
+      "depth": 0,
+      "rank": 2,
+      "A_rank": 2,
+      "S_rank": 2,
+      "K_rank": 2,
+      "L_rank": 2
+    },
+    {
+      "n": 2,
+      "depth": 1,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
+      "L_rank": 0
+    },
+    {
+      "n": 2,
+      "depth": 1,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
+      "L_rank": 0
+    },
+    {
+      "n": 4,
+      "depth": 0,
+      "rank": 2,
+      "A_rank": 2,
+      "S_rank": 2,
+      "K_rank": 2,
+      "L_rank": 2
+    },
+    {
+      "n": 2,
+      "depth": 1,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
+      "L_rank": 0
+    },
+    {
+      "n": 2,
+      "depth": 1,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
+      "L_rank": 0
+    },
+    {
+      "n": 8,
+      "depth": 0,
+      "rank": 2,
+      "A_rank": 2,
+      "S_rank": 2,
+      "K_rank": 2,
+      "L_rank": 2
+    },
+    {
+      "n": 4,
+      "depth": 1,
+      "rank": 2,
+      "A_rank": 2,
+      "S_rank": 2,
+      "K_rank": 2,
+      "L_rank": 2
+    },
+    {
+      "n": 2,
+      "depth": 2,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
+      "L_rank": 0
+    },
+    {
+      "n": 2,
+      "depth": 2,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
+      "L_rank": 0
+    },
+    {
+      "n": 4,
+      "depth": 1,
+      "rank": 2,
+      "A_rank": 2,
+      "S_rank": 2,
+      "K_rank": 2,
+      "L_rank": 2
+    },
+    {
+      "n": 2,
+      "depth": 2,
+      "rank": 2,
+      "A_rank": 0,
+      "S_rank": 0,
+      "K_rank": 0,
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+      "stein_sha256": "3c3ae9f38dc3a64532dce0bb9d7db955c735f5bf3bba945462850fa08d04c158",
+      "generators": [
+        {
+          "column": 0,
+          "scale": "1/1606938044258990275541962092341162602522202993782792835301376",
+          "scale_exponent": -200,
+          "gamma": "1",
+          "g": [
+            [
+              "1",
+              "0"
+            ],
+            [
+              "0",
+              "0"
+            ]
+          ],
+          "row": [
+            [
+              "1/1606938044258990275541962092341162602522202993782792835301376",
+              "0"
+            ],
+            [
+              "0",
+              "0"
+            ]
+          ]
+        }
+      ],
+      "heights": {
+        "input_height": 201,
+        "max_observed_height": 801
+      },
+      "declared_height_bound": 603168768,
+      "macro_stages": 4,
+      "prefix_norm_bound": "2008672555323737844427452615426453253152753742228491044126729/200867255532373784442745261542645325315275374222849104412672",
+      "observed_frobenius_squared": "3873374817130362884483878758004517811494558689243835269245992248687059951005812875868652712245411791559842323544791842817/645562469521727147413979793000752968582426448207305878207664839135161905504210298657411338320034457858975792993186873344",
+      "endpoint_sha256": "1a117d4da9905831e2f799f291545dacd07aee54bf06bb8920f7a1af189b6667"
+    },
+    {
+      "label": "near-dependent-columns",
+      "size": 2,
+      "rank": 2,
+      "input_sha256": "392f464f5272d053bef732869c75ae418d5026a54b7efbf0affaab2823e6dbc4",
+      "stein_sha256": "441ae999837655670dd7c71862a1a0ad4e56fa1ee62cb473fc731d763dfd5172",
+      "generators": [
+        {
+          "column": 0,
+          "scale": "1",
+          "scale_exponent": 0,
+          "gamma": "4",
+          "g": [
+            [
+              "1",
+              "0"
+            ],
+            [
+              "0",
+              "0"
+            ]
+          ],
+          "row": [
+            [
+              "1",
+              "0"
+            ],
+            [
+              "1",
+              "0"
+            ]
+          ]
+        },
+        {
+          "column": 1,
+          "scale": "1/1606938044258990275541962092341162602522202993782792835301376",
+          "scale_exponent": -200,
+          "gamma": "1",
+          "g": [
+            [
+              "0",
+              "0"
+            ],
+            [
+              "1",
+              "0"
+            ]
+          ],
+          "row": [
+            [
+              "0",
+              "0"
+            ],
+            [
+              "1/1606938044258990275541962092341162602522202993782792835301376",
+              "0"
+            ]
+          ]
+        }
+      ],
+      "heights": {
+        "input_height": 201,
+        "max_observed_height": 401
+      },
+      "declared_height_bound": 603168768,
+      "macro_stages": 8,
+      "prefix_norm_bound": "170",
+      "observed_frobenius_squared": "899/16",
+      "endpoint_sha256": "e1bd19607cbf2f34c06f04a050e25e440edb796371450dcec53902e3f93f4cd9"
+    },
+    {
+      "label": "complex-row",
+      "size": 2,
+      "rank": 1,
+      "input_sha256": "461768c10e5e555db99fbcf9b2d4b801f65a8ee0969967fda395e077f19a9d43",
+      "stein_sha256": "4c022553513d75f65183d3679d460d6d37ff3263426b3e16dc0aa1a952c5cb2f",
+      "generators": [
+        {
+          "column": 0,
+          "scale": "1/16",
+          "scale_exponent": -4,
+          "gamma": "1",
+          "g": [
+            [
+              "1",
+              "0"
+            ],
+            [
+              "0",
+              "0"
+            ]
+          ],
+          "row": [
+            [
+              "1/16",
+              "0"
+            ],
+            [
+              "0",
+              "1/16"
+            ]
+          ]
+        }
+      ],
+      "heights": {
+        "input_height": 5,
+        "max_observed_height": 17
+      },
+      "declared_height_bound": 17915904,
+      "macro_stages": 4,
+      "prefix_norm_bound": "19",
+      "observed_frobenius_squared": "835/128",
+      "endpoint_sha256": "c73f3e5937a764c4b1daa40fcd9e447cc573d680b1f4893fc7b8739a1223b7e3"
+    },
+    {
+      "label": "large-scale",
+      "size": 2,
+      "rank": 1,
+      "input_sha256": "7be036b50e786ac4cfc340cdc57339afb0f5a9215abdf1e8b57ad021443ddb1a",
+      "stein_sha256": "69da144118ac753898d397bac895394c3b67ef17a5c51e98d630c2f00a885ce9",
+      "generators": [
+        {
+          "column": 0,
+          "scale": "1208925819614629174706176",
+          "scale_exponent": 80,
+          "gamma": "4835703278458516698824704",
+          "g": [
+            [
+              "1",
+              "0"
+            ],
+            [
+              "0",
+              "0"
+            ]
+          ],
+          "row": [
+            [
+              "1208925819614629174706176",
+              "0"
+            ],
+            [
+              "1208925819614629174706176",
+              "0"
+            ]
+          ]
+        }
+      ],
+      "heights": {
+        "input_height": 81,
+        "max_observed_height": 321
+      },
+      "declared_height_bound": 244850688,
+      "macro_stages": 4,
+      "prefix_norm_bound": "174085318024506601157689354",
+      "observed_frobenius_squared": "1192585336062016781254206823496486944039240955068499/16",
+      "endpoint_sha256": "267977618b72e63d5ec8610e25f8e6d48f9df3ce037314b27aff52009bdc68e6"
+    }
+  ],
+  "controls": [
+    "tiny-nonzero-rank",
+    "omitted-compensation",
+    "omitted-source-restoration",
+    "complex-row-conjugation",
+    "zero residual normalization",
+    "floating input",
+    "typed-receipt"
+  ],
+  "fft_or_parity_gate_execution": false
+}
diff --git a/research/rational-stein-generators/SOURCES.json b/research/rational-stein-generators/SOURCES.json
new file mode 100644
index 0000000000..09133fedf6
--- /dev/null
+++ b/research/rational-stein-generators/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/GAUSSIAN.md","usage":"input","git_blob":"adfb28c93e6b06178c901caaac39ede41a9e541b","sha256":"2f63c91c58f2d1cf31294812c75970ea390c61866a8da7ed700d4316a033a242"},
+    {"path":"research/unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md","usage":"input","git_blob":"bc4349b2daea548ffa09e1f38a31f1df86b0b7be","sha256":"b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310"},
+    {"path":"research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md","usage":"input","git_blob":"20585e44fa4e49b13b5d8495bbf6ed254459a362","sha256":"d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116"},
+    {"path":"research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_circuits.py","usage":"input","git_blob":"f80c141c8116b163c2bb5a028c28212077c8d5a9","sha256":"06d1861bfe8af543908e26557ffda7e451ff176bceb9bacfaf0f04f234b336d5"},
+    {"path":"research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py","usage":"input","git_blob":"4f0180c88f6ea25b85814883ef7cf2ad41e67110","sha256":"5ff7f25c55b370a1d0cda6c84ab9cdf2c7b0de3266d3f2e7a43280a6491074ab"},
+    {"path":"research/capped-phase-fusion/README.md","usage":"input","git_blob":"a8c26bf5320acc8f7be48aa5120b52f94b80a1d2","sha256":"d3fdbf5642ab684755bdecab12af247244a25b54765d6866bf5af5c342140aba"},
+    {"path":"research/capped-phase-fusion/evidence/independent_fusion_review_ultra/core_review.md","usage":"input","git_blob":"6a5ecf4229d6d654457e55d7c157a09230705f1b","sha256":"13f2df4d5b240f9b89ca0ae44bba198c935532098ff4c1161631fcbd86b3e8b1"}
+  ]
+}
diff --git a/research/rational-stein-generators/TASK.json b/research/rational-stein-generators/TASK.json
new file mode 100644
index 0000000000..453d097eb7
--- /dev/null
+++ b/research/rational-stein-generators/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": "Construct exact rational bounded Stein generators, prove polynomial preprocessing height and macro-prefix bounds, and bind them to the retained Schur producer and dirty-shear macro order.",
+  "scope_change_reason": "The source already handles irrational normalization offline. The distinct contribution implements a rational replacement with exact rank, tiny-pivot and explicit complex-row behavior at that interface.",
+  "bottleneck_link": "The Gaussian interior compiler needs bounded generated coefficient rows before child rounding and physical coefficient delivery can be integrated into its complete error ledger.",
+  "budget_minutes": 120,
+  "stop_rule": "Stop at rank-dependent norm loss, an unjustified bit bound, wrong conjugation, source-protocol mismatch or an existing equivalent implementation. Do not equate macro endpoints with numerical child-circuit execution.",
+  "outcome": "completed",
+  "evidence": "PROOF.md gives exact rational reconstruction, norm and bit bounds. RESULT.json records 56 cases, 67 generators, 268 source macro stages and exact rank/endpoint/prefix/control evidence. 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 FFT and triangular children consume the new coefficients with certified rounded gate execution?",
+      "why_it_matters": "The exact macro replay does not execute the child gates or establish their internal numerical prefixes.",
+      "first_step": "Replace one macro callback with the actual retained child compiler and directed coefficient/rounding bounds, retaining the exact macro endpoint as a reference.",
+      "blocked_by": "A separate child-level numerical refinement and its complete prefix/rounding contract."
+    },
+    {
+      "question": "Can the rational producer feed the actual Gaussian coefficient-page and physical caller interface?",
+      "why_it_matters": "Polynomial offline generation alone does not prove the complete data precision, page format or physical setup bill.",
+      "first_step": "Bind one generated row and Gamma to the existing caller's table allocation and declared rounding protocol, accounting for coefficient delivery and normalization.",
+      "blocked_by": "Source-bound integration with the complete Gaussian error and setup ledger."
+    }
+  ]
+}
diff --git a/research/rational-stein-generators/VERIFY.json b/research/rational-stein-generators/VERIFY.json
new file mode 100644
index 0000000000..3d7b41b4b5
--- /dev/null
+++ b/research/rational-stein-generators/VERIFY.json
@@ -0,0 +1,27 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/unconditional-batched-crt/GAUSSIAN.md",
+    "research/unconditional-batched-crt/vendor/SHALLOW-TOEPLITZ.md",
+    "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md",
+    "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_circuits.py",
+    "research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py",
+    "research/capped-phase-fusion/README.md",
+    "research/capped-phase-fusion/evidence/independent_fusion_review_ultra/core_review.md"
+  ],
+  "commands": [
+    {
+      "argv": ["{python}", "-I", "-B", "{package}/verify.py"],
+      "timeout_seconds": 180,
+      "checks": "Authenticate the Schur producer and generator macro, extract exact rational generators, verify ranks/norms/heights, replay source macro order with inverse prefixes and reject tiny-pivot/conjugation/restoration controls."
+    },
+    {
+      "argv": ["{python}", "-I", "-O", "-B", "{package}/verify.py"],
+      "timeout_seconds": 180,
+      "checks": "Repeat the complete typed certificate under optimized Python, keeping authenticated subject assertions enabled and numerical children outside the declared scope."
+    }
+  ]
+}
diff --git a/research/rational-stein-generators/rational.py b/research/rational-stein-generators/rational.py
new file mode 100644
index 0000000000..71b240a460
--- /dev/null
+++ b/research/rational-stein-generators/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]
diff --git a/research/rational-stein-generators/verify.py b/research/rational-stein-generators/verify.py
new file mode 100644
index 0000000000..87933aa322
--- /dev/null
+++ b/research/rational-stein-generators/verify.py
@@ -0,0 +1,274 @@
+"""Source-bound exact rational generator and dirty-shear macro certificates.
+
+Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance.
+Apache-2.0. Numerical FFT/parity children are outside this macro replay.
+"""
+import argparse
+import ast
+import copy
+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("rational_stein", HERE / "rational.py")
+rat = importlib.util.module_from_spec(spec)
+spec.loader.exec_module(rat)
+C, Matrix, Vector, need = rat.C, rat.Matrix, rat.Vector, rat.need
+SOURCE = "research/capped-phase-fusion/evidence/inplace_algebra_ultra/"
+
+
+def canonical(value):
+    return json.dumps(value, sort_keys=True, separators=(",", ":")).encode()
+
+
+def digest(matrix):
+    return sha256(canonical(rat.exact_record(matrix))).hexdigest()
+
+
+def rejected(operation, label):
+    try:
+        operation()
+    except ValueError:
+        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"]
+    for row in rows:
+        raw = (ROOT / row["path"]).read_bytes()
+        need(sha256(raw).hexdigest() == row["sha256"], "source hash: " + row["path"])
+        rejected(lambda: need(sha256(raw + b"\0").hexdigest() == row["sha256"], "source corruption"),
+                 "source corruption")
+    return [{"path": r["path"], "sha256": r["sha256"]} for r in rows]
+
+
+def load_subject():
+    path = ROOT / (SOURCE + "check_exact_algebra.py")
+    namespace = {"__name__": "_retained_exact_subject", "__file__": str(path)}
+    exec(compile(path.read_bytes(), str(path), "exec", optimize=0), namespace)
+    return namespace
+
+
+def macro_template():
+    tree = ast.parse((ROOT / (SOURCE + "check_circuits.py")).read_bytes())
+    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 tail")
+    calls = [n.func.id for statement in tail for n in ast.walk(statement)
+             if isinstance(n, ast.Call) and isinstance(n.func, ast.Name)]
+    need(calls == ["triangle_plan", "toeplitz_shear", "lower", "inverse",
+                   "toeplitz_shear", "lower"], "retained child-call order")
+    identity = sha256(ast.dump(ast.Module(body=tail, type_ignores=[])).encode()).hexdigest()
+    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, "<authenticated generator macro>", "exec", optimize=0), identity
+
+
+class MacroEmitter:
+    def __init__(self, size, subject, code):
+        self.size, self.subject = size, subject
+        namespace = {"triangle_plan": self.triangle, "toeplitz_shear": self.shear,
+                     "inverse": self.inverse, "lower": rat.lower}
+        exec(code, namespace)
+        self.emit = namespace["emit"]
+        self.calls = []
+
+    def triangle(self, indices, coefficients):
+        need(len(indices) == self.size and len(set(indices)) == self.size, "ordered source indices")
+        matrix = Matrix(self.subject["eye"](2 * self.size))
+        lower = rat.lower(coefficients)
+        for i, row in enumerate(indices):
+            for j, column in enumerate(indices):
+                matrix[row][column] = lower[i][j]
+        self.calls.append("triangle")
+        return [matrix]
+
+    def shear(self, target, source, coefficients):
+        need(len(target) == len(source) == self.size
+             and len(set(target + source)) == 2 * self.size, "disjoint complete banks")
+        matrix = Matrix(self.subject["eye"](2 * self.size))
+        for i, row in enumerate(target):
+            for j, column in enumerate(source):
+                matrix[row][column] = C.wrap(coefficients[i][j])
+        self.calls.append("shear")
+        return [matrix]
+
+    def inverse(self, plan):
+        self.calls.append("inverse")
+        return [Matrix(self.subject["inv"](a)) for a in reversed(plan)]
+
+
+def collect_shears(subject):
+    """Execute the retained exact producer with one read-only collection hook."""
+    tree = ast.parse((ROOT / (SOURCE + "check_exact_algebra.py")).read_bytes())
+    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 = []
+    namespace = dict(subject)
+    namespace["trace"] = []
+    namespace["capture"] = lambda k, l, n, depth: collected.extend(
+        [(f"source-K-n{n}-d{depth}", Matrix(k)), (f"source-L-n{n}-d{depth}", Matrix(l))])
+    exec(compile(ast.fix_missing_locations(ast.Module(body=[walk], type_ignores=[])),
+                 "<authenticated Schur producer with collection hook>", "exec", optimize=0), namespace)
+    for size in (2, 4, 8):
+        for complex_case in (False, True):
+            column = [C(1)] + [C(Q((-1) ** i, 64 * size),
+                                  Q(i, 128 * size) if complex_case else 0) for i in range(1, size)]
+            row = [C(1)] + [C(Q(1, 64 * size),
+                               Q((-1) ** j, 128 * size) if complex_case else 0) for j in range(1, size)]
+            matrix = Matrix([[column[i - j] if i >= j else row[j - i]
+                              for j in range(size)] for i in range(size)])
+            namespace["walk"](matrix)
+    # Source-admissible complex corner fixture from the row/conjugate convention check.
+    matrix = Matrix(subject["eye"](4))
+    matrix[0][2] = matrix[1][3] = C(Q(1, 16))
+    matrix[0][3] = C(0, Q(1, 16))
+    namespace["walk"](matrix)
+    return collected, namespace["trace"]
+
+
+def one_case(label, matrix, subject, code):
+    size = len(matrix)
+    z = subject["shift"](size)
+    displacement = Matrix(subject["add"](matrix, subject["mm"](
+        subject["mm"](z, matrix), rat.transpose(z)), -1))
+    generators, heights = rat.generators(displacement)
+    rank = subject["rank"](displacement)
+    need(len(generators) == rank, "exact generator rank")
+    reconstructed_e = Matrix(subject["zeros"](size))
+    reconstructed_t = Matrix(subject["zeros"](size))
+    bound_e = rat.norm_bound(displacement)
+    emitter = MacroEmitter(size, subject, code)
+    plan = []
+    summaries = []
+    for item in generators:
+        g, row, gamma = item["g"], item["row"], item["gamma"]
+        need(rat.norm2(g) <= 1 and rat.norm2(row) < 4 * bound_e ** 2,
+             "bounded rational generator norms")
+        reconstructed_e = Matrix(subject["add"](reconstructed_e, [[x * y for y in row] for x in g]))
+        reconstructed_t = Matrix(subject["add"](reconstructed_t,
+            subject["mm"](rat.lower(g), rat.transpose(rat.lower(row)))))
+        before = list(row)
+        plan.extend(emitter.emit(list(range(size, 2 * size)), list(range(size)), g, row, gamma))
+        need(before == row, "source vector division preserves original coefficient row")
+        summaries.append({"column": item["column"], "scale": str(item["scale"]),
+                          "scale_exponent": item["scale_exponent"], "gamma": str(gamma),
+                          "g": [x.record() for x in g], "row": [x.record() for x in row]})
+    need(reconstructed_e == displacement and reconstructed_t == matrix, "exact Stein and Toeplitz-product reconstruction")
+    need(emitter.calls == ["triangle", "shear", "inverse", "shear"] * rank, "actual macro call sequence")
+    expected = Matrix(subject["block"](subject["eye"](size), subject["zeros"](size),
+                                      matrix, subject["eye"](size)))
+    prefix = Matrix(subject["eye"](2 * size))
+    inverse_prefix = Matrix(subject["eye"](2 * size))
+    identity = Matrix(subject["eye"](2 * size))
+    prefix_bound = 2 + 4 * size + 4 * rank * size * bound_e + 32 * size * bound_e
+    peak = rat.frobenius2(prefix)
+    for gate in plan:
+        prefix = Matrix(subject["mm"](gate, prefix))
+        inverse_prefix = Matrix(subject["mm"](inverse_prefix, subject["inv"](gate)))
+        need(subject["mm"](prefix, inverse_prefix) == identity, "exact macro inverse-prefix identity")
+        peak = max(peak, rat.frobenius2(prefix), rat.frobenius2(inverse_prefix))
+        need(peak <= prefix_bound ** 2, "certified macro norm envelope")
+    need(prefix == expected, "actual source macro endpoint and dirty-source restoration")
+    polynomial_height = 4096 * (size + 1) ** 6 * (heights["input_height"] + 1)
+    need(heights["max_observed_height"] <= polynomial_height, "polynomial generator height envelope")
+    return {"label": label, "size": size, "rank": rank, "input_sha256": digest(matrix),
+            "stein_sha256": digest(displacement), "generators": summaries, "heights": heights,
+            "declared_height_bound": polynomial_height, "macro_stages": len(plan),
+            "prefix_norm_bound": str(prefix_bound), "observed_frobenius_squared": str(peak),
+            "endpoint_sha256": digest(prefix)}, plan, expected
+
+
+def controls(subject, code):
+    controls = []
+    tiny = Matrix([[C(Q(1, 2 ** 200)), C()], [C(), C(Q(1, 2 ** 200))]])
+    row, plan, wanted = one_case("tiny-control", tiny, subject, code)
+    need(row["rank"] == 1 and Q(1, 2 ** 200) < Q(1, 10 ** 11), "tiny exact pivot below source SVD cutoff")
+    need(Matrix(subject["eye"](4)) != wanted, "dropping tiny term changes exact endpoint")
+    controls.append("tiny-nonzero-rank")
+    complex_matrix = Matrix([[C(Q(1, 16)), C(0, Q(1, 16))], [C(), C(Q(1, 16))]])
+    record, plan, wanted = one_case("complex-control", complex_matrix, subject, code)
+    item = record["generators"][0]
+    g = Vector(C(Q(x[0]), Q(x[1])) for x in item["g"])
+    wrong_row = Vector(C(Q(x[0]), -Q(x[1])) for x in item["row"])
+    emitter = MacroEmitter(2, subject, code)
+    wrong = emitter.emit([2, 3], [0, 1], g, wrong_row, Q(item["gamma"]))
+    actual = Matrix(subject["eye"](4))
+    for step in wrong:
+        actual = Matrix(subject["mm"](step, actual))
+    need(actual != wanted, "conjugated coefficient row rejects")
+    for label, altered in (("omitted-compensation", plan[:-1]),
+                           ("omitted-source-restoration", plan[:2] + plan[3:])):
+        actual = Matrix(subject["eye"](4))
+        for step in altered:
+            actual = Matrix(subject["mm"](step, actual))
+        need(actual != wanted, label)
+        controls.append(label)
+    controls.append("complex-row-conjugation")
+    controls.append(rejected(lambda: rat.norm_scale(Q(0)), "zero residual normalization"))
+    controls.append(rejected(lambda: C(0.5), "floating input"))
+    need(canonical({"rank": 1}) != canonical({"rank": True}), "typed receipt comparison")
+    return controls + ["typed-receipt"]
+
+
+def run():
+    sources = authenticate()
+    subject = load_subject()
+    code, ast_hash = macro_template()
+    cases, source_trace = collect_shears(subject)
+    epsilon = Q(1, 2 ** 200)
+    cases += [("zero", Matrix([[C(), C()], [C(), C()]])),
+              ("scalar", Matrix([[C(Q(1, 32))]])),
+              ("tiny-diagonal", Matrix([[C(epsilon), C()], [C(), C(epsilon)]])),
+              ("near-dependent-columns", Matrix([[C(1), C(1)], [C(), C(1 + epsilon)]])),
+              ("complex-row", Matrix([[C(Q(1, 16)), C(0, Q(1, 16))], [C(), C(Q(1, 16))]])),
+              ("large-scale", Matrix([[C(2 ** 80), C(2 ** 80)], [C(), C(2 ** 80)]]))]
+    records = [one_case(label, matrix, subject, code)[0] for label, matrix in cases]
+    return {"schema_version": 1, "scope": "rational coefficient generation and exact source macro replay",
+            "sources": sources, "macro_ast_sha256": ast_hash, "source_schur_trace": source_trace,
+            "cases": records, "controls": controls(subject, code),
+            "fft_or_parity_gate_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_STEIN_GENERATORS", "cases": len(result["cases"]),
+                      "generators": sum(x["rank"] for x in result["cases"]),
+                      "macro_stages": sum(x["macro_stages"] for x in result["cases"]),
+                      "result_sha256": sha256(canonical(result)).hexdigest()}, sort_keys=True))
+
+
+if __name__ == "__main__":
+    main()