Conditional κ = 0.001679745508802299: gauge-free (fractional-shift) interiors, structured boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier

[jeremykothe2]

Summary

Conditional κ = 0.001679745508802299051546533685199973672313473419790800679752457380932784242972585600630181672224983. Capped phase-cell fusion outer (PR1673, Vlado Vojdanovski) on this repository's T74 supplier (pivot-566 and pivot-554 helper-donor kernels, certified complex endpoint 0.00167975495612554452133567...), free strict slacks at 1e-64 or below, b = certified endpoint - 1e-64, with three NEW constructions (not in PR1673 or PR1699): a radix-distributed sparse-union schedule (outer/UNION-SCHEDULE.md) and structured boundary solves (outer/STRUCTURED-BOUNDARY.md): the Gaussian blocks of a phase cell are D_r T D_c with T Toeplitz, so segment, coupling and pivot blocks of the boundary algebra have constant displacement rank and are applied by generator representations and FFT in O(r W log W) per right-hand-side column, tau_eff = 2 + 1e-64, and gauge-free interiors (outer/GAUGEFREE-INTERIORS.md): the linear local gauge is an exact fractional centre shift of the Toeplitz kernel, the shifted Gaussian symbol is accretive (triple product), the shallow circuit lemma extends to accretive Toeplitz tiles, the precision guard is chirp-only (beta + 2 xi - eps <= 1) and the union tag work is paid per visit (1 - beta > 0), giving beta = 0.9966. The 58 strict inequalities, 11 rejecting controls (including the old union row and tau_eff at or below 2, which fail) and the endpoint rejection are regenerated with an independent balance derivation; four executable checks (union schedule, independent schedule simulation, exact displacement-rank/structured-solve check, gauge-free identities/symbol/circuit replay) are run; the supplier's full native admission is rerun. Conditional, with all inherited all-size hypotheses; all three new constructions need independent review; the precision argument of the structured solves is an error argument, not a verified theorem.

Verification

Fresh checkout of 7c2634c4 (follow-up to 8d975b6c: Theorem 4.1 of outer/GAUGEFREE-INTERIORS.md restated with the absolute-scale guard log(2+Lambda+1/mu), plus the unit-diagonal ratio corollary; credit to Mister-Meeseeks for the H=tI counterexample; kappa and every ledger number unchanged). Manifest: d541e74ede4930a15e1599b0e5e812a6c67906e4ddf1d9632a0b10e8831ce9f4.
PASS package manifest: 377 files

Supplier: the package's own unchanged verify.py (retained PR1297 provider and parent admission, owner reconstruction, parent lift, strict T74 admission, geometry controls): PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION, native reuse {'base': 'e8e6e78ff7422275c38324499e1f33fb2ecd30de9503aef2b8f1ee80d8c4df8e', 'extra': '7b0e9fbda32749aabc7e156d9fe09bceb0624a178628fe69e5dfea783462f905'}; its complex endpoint 419938739031386130333917748480... brackets the ledger's b = 4199387390313861303339177484801756442495778596256450738650818550407/2500000000000000000000000000000000000000000000000000000000000000000000. Outer (PR1673 capped phase-cell fusion): regenerated exact ledger (58 strict inequalities, 11 rejecting controls incl. the old g-factor union row, tau_eff at or below 2, the old linear gauge guard and the old 2*beta tag test, larger endpoint rejected by the native moment), independent balance derivation, the exact union-schedule simulation, the exact structured-boundary displacement-rank check, the gauge-free identity/symbol/circuit checks and 19 exact evidence replays all pass. Conditional kappa 0.001679745508802299051546533685199973672313473419790800679752457380932784242972585600630181672224983 (union schedule outer/UNION-SCHEDULE.md; structured boundary solves outer/STRUCTURED-BOUNDARY.md, tau_eff=2+1e-64; gauge-free interiors outer/GAUGEFREE-INTERIORS.md).

Full package-owned verification passed in 88.81 seconds. The fresh receipt binds the exact package manifest and score.

See research/capped-fusion-t74-k554/README.md for reproduction commands and proof qualifications.

Credits

PR1673 (Vlado Vojdanovski, with AI assistance): capped phase-cell fusion outer construction, its proof and exact evidence, vendored verbatim under vendor/pr1673. Gohberg and Semencul, Kailath, Kung and Morf, Bitmead and Anderson, Pan (structured Toeplitz-like matrix algorithms). Francois Le Gall, Alman, Duan, Vassilevska Williams, Xu, Xu, Zhou (rectangular exponents of the fallback packages), Chafik Boukhalfa (PR1699). The radix-distributed sparse-union schedule and the structured boundary solves are new in this package series; the independent schedule simulation outer/sched_review.py is by LeverRank. Supplier: PR1625 (mdumitrean) pivot-566 helper-donor kernel; PR1661 (midhatn) 400-digit endpoint-grid refinement of that stack; pivot-554 kernel (kernel566.py search by LeverRank/jeremykothe2 tooling, PR1672); PR1297, PR1319, PR1424, PR1485, PR1574 (Rohan Arun) T74 exact fill and cold kernels; PR1274, PR1479, PR1510, PR1513, PR1536, PR1578 (Jeremy Kothe) endpoint and outer-witness lineage; PR1534, PR1549 (midhatn); PR1519 (philippeVV) fine-Mercator soundness repair; PR1286, PR1193 (Jeremy Diviney), PR1148 (Thomas DiFiore) and the retained supplier lineage. Research direction, prioritisation and release strategy: jeremykothe2. Implementation assistance: oh-my-pi. The gauge-free interiors (fractional-shift kernels, accretive shallow-circuit lemma, chirp-only guard, per-visit tag accounting) are new in this package (IdeaOuter).

[Mister-Meeseeks]

Scoped batch triage at 8d975b6caba8d261f1924e527225c193219a9967. This PR remains open.

Whole-head supplier/outer admission remains pending. Review fractional-shift accretivity, the shallow accretive Toeplitz circuit extension, chirp-only guard and per-visit union tagging, plus structured boundary and T74 admission. These new analytic interfaces are not established by finite symbol/circuit samples.

This is a review continuation, not a rejection or a request to repeat the checks already retained. Credit and source pins are preserved in the batch ledger. The selected bound is unchanged. Thank you for the contribution.

[Mister-Meeseeks]

Continuing the shared dependency review at unchanged head 8d975b6caba8d261f1924e527225c193219a9967 (this note is also retained by #2265): there is a scale-normalization issue in outer/GAUGEFREE-INTERIORS.md, Theorem 4.1.

As a uniform statement in Λ and μ, the claimed

log(max prefix / suffix / inverse-prefix norm) = O(log N · log(2 + Λ/μ) + log N)

cannot hold for every accretive Toeplitz matrix. Take N=2 and H=tI, t>0, with Λ=μ=t. The ratio stays 1, while the completed circuit has norm t and its inverse has norm 1/t. Thus the left side must be at least |log t|, unbounded as t varies. The corresponding scale-free precision assertion needs the same qualification.

A repair appears available for the intended application: the normalized shifted kernels G′ have diagonal 1, hence Λ≥1 and μ≤1. Then Λ+1/μ ≤ 2Λ/μ, so the absolute-scale dependence can be controlled by the stated ratio. Alternatively state the general theorem with the absolute-scale guard used in SHALLOW-TOEPLITZ §9, log(2 + Λ + 1/μ), and derive the ratio-only corollary for unit-diagonal kernels. Please carry that qualification through truncated/rounded kernels and descendant/shear bounds.

This is a correction to the general theorem statement, not a counterexample to the normalized Gaussian application or an acceptance/rejection of the full multiplication claim. The remaining shallow-circuit, locality, precision and fixed-tape composition review is still in progress.

[jeremykothe2]

Thank you, Mister-Meeseeks. The H = tI counterexample is correct: with Λ = μ = t the ratio is 1, but the completed circuit has norm t and its inverse 1/t, so a statement in Λ/μ alone cannot hold. This was an error in our general statement of Theorem 4.1.

We pushed 7c2634c to this PR, which:

• Restates Theorem 4.1 with the absolute-scale guard log(2 + Λ + 1/μ), as in SHALLOW-TOEPLITZ §9, and adds Remark 4.1a with your counterexample.
• Adds Corollary 4.1b, the ratio-only form for unit-diagonal kernels. Diagonal entries d with |d−1| ≤ 1/2 give Λ ≥ 1/2 and μ ≤ 3/2, hence Λ + 1/μ ≤ (7/2)·Λ/μ. For d = 1 exactly this is Λ ≥ 1 ≥ μ and Λ + 1/μ ≤ 2Λ/μ.
• Carries the guard through the descendant/shear bounds in (b)–(d), where every quantity is polynomial in M = 2 + Λ + 1/μ. It also carries it through the Stein-generator bound γ_P, Corollary 4.2 (truncation/rounding), the Section 5 precision assertion and the obligations list. Corollary 4.3 was already stated with absolute bounds (‖P_j⁻¹‖ ≤ 1/μ, ‖P_j‖ ≤ Λ + Λ²/μ).
• Checks that truncation and rounding keep the diagonal exactly 1. The u = 0 entry of the normalized kernel G′_s is exactly representable and is always inside the band, so the circuit kernels stay exactly unit-diagonal. If a conversion ever drifted the diagonal, any drift up to 1/2 is covered by the 1/2-hypothesis. check_gaugefree.py stage F asserts this for γ from 0.02 to 189, s up to 0.4999, band W from 1 up to the tail cutoff, and fixed or floating point rounding at 64 to 1024 bits.

The application is unchanged: the normalized shifted Gaussian kernels are unit-diagonal, so the ratio-only form still holds there. The exponent β + 2ξ − ε, every ledger number and κ are untouched. verify_release.py was rerun fresh and gives PASS_RELEASE with the same κ.

We understand the remaining shallow-circuit, locality, precision and fixed-tape composition review is still in progress, and will address whatever it turns up.