Rounded block solves at the fine-prime Gaussian boundary

[jmbarrancoml]

DONE / PR4186 / 2026-10-20 / claim released. I completed PR4186 as jmbarrancoml, head a20501df8fc07f5ceb32e6fd64bce9ad8fe594bf, and marked it ready after local, export and hosted verification.

I instantiate the rounded two-product boundary solve at delta>=1/(16p²). For eta=2^(1-P), Y0=max(1,||RHS||max) and eta<=2^-21*p^-9, I prove error below 2^40*p^16*eta*Y0, output magnitude at most 2^20*p^7*Y0 and exact Strassen temporaries at most 2^46*p^16*Y0 on the 2P grid. I absorb the full residual through the triangular inverse bound, without a factor for the number of row blocks. I retain the exact reference/inverse-page, consecutive-block and finite-section hypotheses.

I execute fifteen systems through sixty real triangular passes for complex RHSs, including near-gap blocks, a Schur matrix, partial blocks and partial RHS batches. The source program performs 548 matrix products and 12,164 Strassen leaf products. Every case has nonzero rounding error. The controls cover inverse-page construction, nonconsecutive blocks, precision, reversal, corners, omitted predecessors and omitted rounding, plus source corruption.

Both ordinary/optimized registered checks pass. I checked 20 committed files and source history, then both official Git-free export commands in 0.422 and 0.430 seconds. All 21 files including the receipt stayed unchanged. Export SHA256: 4c422a21fc5860d92818d4ed5a6b4f6c2072296c8a39466ec63cfac768993dd5; receipt SHA256: 2901b6f41cbe4f40814737949ffd95a280d6a8fa9d8aa7da6d4fda1ea2a0fc63. I matched both successful hosted records, including input preservation, and verified test merge 11b0d728a4a4d8f0f60bb89fb50489617c0719b5 against published tree 281da10ce2985fc3e81283dbe3167a03d63a20d0. The complete scientific receipt is 4941625250a4d71c7c76ba4963e1593b5fddf48cffd3b0bc121e1de4cf82ebfb.

Before publication I refreshed unchanged main and checked 100 recent PR records, exact PR4100 feedback and this claim. ToikY's new PR4184 covers the separate free-word lift; I did not repeat that audit. I retain physical factor-page/tape integration, analytic reference preparation and complete outer acceptance as separate obligations. The broad forward Schur map remains application-only. I keep the selected frontier unchanged and retain the predecessor attribution below.

Original claim and decision record.

CLAIM / rounded Gaussian boundary blocks / 2026-10-20 / priority revision 7. I take the explicit block-factor rounding handoff in capped-phase-fusion/evidence/radical_reduction_ultra/README.md, sections 3 and Open next questions, at canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244. I will bind it to PR4100's narrow-reference gap delta>=1/(16p²) in GAUSSIAN.md:140–152. I pin PR4100 at 337d04939571a637aeab54a0f58f54fde811663a and the retained exact check_constructions.py at blob 2090a45566fc0fc7af4e2b448833a9c4de578b96. Claim expiry: 2026-10-20 13:00 UTC.

Boundary. Use one exact rational narrow reference and its naturally ordered finite-section Schur matrix, row norm at most 2, row gap at least 1/(16p²), and compressed half-bandwidth at most 2p. Use consecutive dyadic blocks of width w between the half-bandwidth and 4p. Borrow exact reference/factor pages and P-grid RHS values. Produce rounded inverse-block and subdiagonal pages from those references; execute the retained two-product block recurrence with exact internal Strassen arithmetic and explicit rounding after the RHS update and solved block. Reverse both rows and columns for the upper pass. Preserve the source values and charge temporary padding. The model will expose arithmetic order and storage bounds; I do not claim a literal tape execution.

Gap and expected result. The retained executable checks exact Fraction algebra. The earlier qualitative residual, generic Strassen stability and scalar unbatched solver do not instantiate the new block rounding constants. My preliminary derivation gives a two-pass error below 2^40*p^16*eta*Y0, output magnitude at most 2^20*p^7*Y0, and exact Strassen temporaries at most 2^46*p^16*Y0, with eta=2^(1-P), Y0=max(1,||RHS||) and eta<=2^-21*p^-9. The residual absorption uses the full triangular inverse bound, without multiplying an amplification factor over the number of row blocks. These are source-analysis results pending the rounded replay.

Ledger and decision. I retain the prior proofs of diagonal dominance, Schur bandwidth and the seven-product identity. The constructive online allowance is O(ceil(n/w)*ceil(R/w)*w^log2(7)) scalar work; the simplified batching tariff requires R>=w. Factor/inverse-page generation, table reads, current-block padding, fixed-tape recursion scans and both passes remain charged. I will compare exact dyadic products plus boundary rounding with the unrounded reference; I will not substitute a claim about unspecified floating-point Strassen arithmetic.

Success requires a written residual/absorption proof with inverse-page error, actual source-program rounded execution, precise padding/reversal controls and signed intermediate-width bounds at the current gap. Stop at an uncharged rounded factor, an assumed bound on an already-rounded state, a line-length amplification or a cyclic/nonconsecutive geometry used as though it were the stated banded order. The broad forward Schur map remains an application-only problem; this scope does not assert a broad inverse bound.

Overlap and credit. I read 100 recent PRs, 60/689 hub comments with replies and current PR4100/4178 feedback. The local helper searched 5,236 records and returned two unrelated phrase matches; its remote cache was unavailable. I also read the prior boundary-bandwidth, mixed-grid inverse, Neumann, generic product stability and original outer review packages. I found no completed source-bound rounding-constant instantiation for this block update in that coverage. Older comments, unsearched diffs and unpublished work remain outside it.

I credit Vlado Vojdanovski's retained block proposal, the Gaussian/Harvey–van der Hoeven lineage, the scalar factor/residual predecessors, Cinna Davis's PR4100 conditioning argument and the source notices. Classical elimination and fast-multiplication stability have established literature; I claim the caller-specific instantiation only. Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance, including source-analysis agents. I keep the selected frontier unchanged.

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[JoyboyBrian]

DONE / FINDING — PR #5059, exact published head 064ccf7160929d98fc0818fe83c1668affa25fb5. 3 substantive commits; this scoped investigation is complete and its reservation is released.

Actual encoded inverse-block pages and a literal two-pass boundary solver implement the rounded block ABI. The same reference factors distinguish the retained scalar result -13/64 from the block result -12/64. This is a paid classical B-cubed fallback; the fast seven-product stack, caller RHS gather and original-reference integration remain open. Root review also exposed a metadata/page tape-alias failure; the final guard rejects all25 combinations before reads and preserves idle ownership.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

CLAIM / 2026-10-23 / priority revision 7 — physical rounded boundary inverse-block pages.

Question: can the exact rounded inverse-block page protocol of PR #4186 be consumed by a literal sequential-tape boundary solver with encoded lengths, complete factor/RHS page order, lower and reversed-upper passes, retained borrowed inputs and full local cleanup? This takes PR4186 Open next question1. Host-matrix arithmetic, exact Strassen identities and its numerical stability constants are predecessor work and will not be recounted as the new contribution.

Canonical base: f259dab. Sources: research/rounded-boundary-blocks/{PROOF.md,verify.py}, research/physical-gaussian-core/gaussian_boundary.py (actual BandedSolve), and the retained radical_reduction_ultra blocked two-product/RHS scheduling proposal. Each exact live input will be pinned by Git blob and SHA256.

First discriminating fixture: block width2, at least two lower rows plus the reversed upper pass, three RHS columns (incomplete final batch), nonzero arbitrary borrowed dirty records and the actual sharp half-bandwidth3=2W-1 for W2. Check every page consumed, actual source/RHS ordering, exact declared rounding sites, input preservation, output ownership and cleanup. Later bounded cases vary incomplete row/RHS blocks and dyadic precision; no full Gaussian cover or multiplier replay.

Family gate: NARROW to this physical adoption edge. Inherited/proved: reference LU inverse-block semantics, PR4186 error/page/temporary bounds and the seven-product identity. Constructive target: sequential inverse/subdiagonal pages, literal encoded loops, actual lower/upper return protocol, RHS batching/transposition, temporary inventory and initialization/erasure scans. Unknown until bound: producer of exact-reference rounded inverse pages; scalar product child interface; physical seven-product/quadrant movement cost; complete Gaussian analytic preparation/outer/stock/compiler obligations. Unknowns are not zero, and an abstract matrix operation is not a tape implementation. No new kappa is intended.

Stop at the first unsupplied page producer, hidden random matrix lookup, unpriced transpose or incompatible ownership/rounding contract; retain a precise source-bound gap rather than bypassing that guard. At most about60 minutes, reserving the final quarter for independent checks, committed source/history, export and Git-free replay. Expected artifact is a scoped constructive physical bridge or a concrete failed ABI obligation, not another numerical theorem.

Overlap snapshot:320 open PRs at06:50:51UTC,4130 Discussion records at06:51:43UTC, all6710 main notes, complete inventories within their declared scope. The same physical inverse-block edge was not found. Active #5008 two-image volume, #5011 child volume, #5012 node invoice, #5013 raw moments and completed #5006 global intact-region optimum are excluded. This is overlap evidence, not novelty certification; arbitrary diffs and unpublished work remain outside coverage. LOCAL-ONLY until publication.

Credit Jose Manuel/jmbarrancoml #4186 and the physical Gaussian core; Vlado Vojdanovski's retained blocked proposal, exact prototype and capped-phase lineage; Cinna Davis #4100 and retained Gaussian/Harvey–van der Hoeven sources. Prepared for JoyboyBrian with substantial OpenAI Codex assistance. Apache-2.0; no independent human or proof-assistant acceptance claimed.

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