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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[JoyboyBrian]

DONE — contribution submitted: 5483. Maintainer review remains pending.

Exact head f51ca6ecb16117eea340448d2b9488eddc41dd9a; 2 substantive commits. Local validation and independent agent review are evidence for this scoped contribution, not human or formal-kernel acceptance.

The physical inverse-block boundary consumer required batch-major RHS input, while the actual Gaussian Core produces separate scalar component streams. This package executes the unchanged Core.gather/Core.first_cell pair, physically packs real and imaginary RHSs, invokes the unchanged BoundaryPages consumer and recovers the original component/row order. Padding, scans, rewinds, protected output consumption, parked input and retained backing are charged.

The source-relative proof covers all positive n/B/w for this actual R=2 layout. The P16/w32 five-row fixture executes both interior first passes and distinguishes skipping the consumer. B1 and B4 cases cover separate batches and incomplete rows/columns. Exact LU(round_P(S)) is the supplied reference; old scalar BandedSolve has different rounding. Second-cell reconstruction, normalized Core installation, production coefficient preparation and the fast seven-product kernel remain open. No new κ or selected-frontier change.

Validation:

• Registered normal and optimized scientific replays pass three actual compositions and nine physical rejection/discrimination controls.
• Four real optimized CLI source/receipt corruption controls pass in a Git-free relocation.
• Final committed-byte/source-history preflight, 31-file export with explicit base input ref, offline archive check and final Git-free optimized replay pass.
• Separate Codex peer review found no blockers after entry-head and partial-acquisition cleanup fixes; reviewed proof/code hashes are unchanged at the final head. No independent human or proof-kernel verification is claimed.

Canonical claim: #4182 (comment)

Prepared by JoyboyBrian with substantial OpenAI Codex assistance. Credit Jose Manuel/jmbarrancoml's Core/runtime/fixture and block handoff, the physical-boundary-pages predecessor, Vlado Vojdanovski, Cinna Davis, Chafik Boukhalfa and the retained Gaussian/FFT/tape lineage. Apache-2.0; frozen sources and notices unchanged.

Claim base 32890bc5de034ab8fc93ac88ead7279a6a416d04. Selected frontier unchanged.

[JoyboyBrian]

DONE — contribution submitted: 5499. Maintainer review remains pending.

Exact head ab307e8027fed0d7a5ec4d4f2fdc17c80af6b086; 2 substantive commits. Local validation and independent agent review are evidence for this scoped contribution, not human or formal-kernel acceptance.

The actual Gaussian first-pass caller supplies real and imaginary RHS columns. The padded inverse-block consumer nevertheless computes all B columns. This package adds a physical two-column kernel that checks inactive inputs, preserves the original active-term order and rounding, and emits the complete padded output. B≤2 delegates to the original kernel.

The source-relative proof gives the zero-column invariant and complete local arithmetic/movement tariff. For B>2, the complete scalar count falls from 2(2K−1)B³ to 4(2K−1)B². The actual-Core B=4 fixture lowers complete consumer reads/writes/unit moves from 7,017,713 to 4,200,707 and retained owner backing from 9,953 to 9,731 cells. Exact padded output, protected tapes and same-owner reuse agree. The cross-block B4 case lowers total work from 21,413,925 to 12,862,809 and backing from 13,697 to 13,511 cells. B1/B2 preserve output with small dispatch overhead; no saving is claimed there.

Validation: normal four-case execution passed in 16.176s; the final committed Git-free export passed optimized full typed-receipt replay in 16.601s. Committed bytes/source history (33 files), explicit-input export integrity, package metadata and four actual CLI source-binding rejection controls pass. Independent optimized B4 replay and six physical controls also pass. The larger fixture tests a nonzero cross-block dependency; B1/B2 test fallback. Retained/adapted PR5483 helper hashes are authenticated separately, with predecessor attribution and self-contained runtime bytes.

This is a scoped ordinary arithmetic improvement, not a recursive paid-profile or κ improvement. LU(round_P(S)) is the supplied factor reference; no equivalence with the scalar BandedSolve rounding contract or complete normalized Core integration is claimed. Owned scratch retains the original blank-entry premise; arbitrary disjoint dirty spectators are preserved.

Claim: #4182 (comment)

JoyboyBrian, with substantial OpenAI Codex assistance; Apache-2.0 and predecessor attribution retained.

Claim base 32890bc5de034ab8fc93ac88ead7279a6a416d04. Selected frontier unchanged.

[JoyboyBrian]

DONE — contribution submitted: 5506. Maintainer review remains pending.

Exact head fd3d814ce4ae8e1767db50e5cd52ef4826a7a8a8; 2 substantive commits. Local validation and independent agent review are evidence for this scoped contribution, not human or formal-kernel acceptance.

The existing physical Gaussian block pipeline stopped after recovering the two scalar boundary solutions. This package carries those streams through unchanged Core.second_cell, normalize_cell, and the final complex disk guard, preserving source order, cropping, protected returns and cleanup.

The source-relative proof derives explicit magnitude and perturbation recurrences for the actual seven Interior stages (five diagonals, two completed-coordinate rounded convolutions). It gives sufficient intermediate width and final disk conditions without an unspecified inverse constant or an incorrect equality with scalar BandedSolve. The reference remains the supplied rounded block-factor function; actual global-Gaussian coefficient admission and first-pass/block error remain separate.

The nonzero five-row supplied-factor calibration compares two complete schedules. Retaining selections costs 29,995,126 tape operations and 22,910 retained owner cells; regathering costs 30,220,120 and 22,478. Retain therefore saves 224,994 operations at 432 additional retained cells. Both return the exact same cropped complex stream, execute four actual interior calls, and preserve borrowed source/page/factor bytes and heads. Common physical factor preparation costs 946,116 operations separately. The deliberately changed 1/64 boundary diagonal is calibration, not a global-Gaussian matrix claim.

Validation: normal complete gate PASS (8.131s), final committed Git-free optimized typed receipt PASS (13.693s), metadata/source pins, 33-file committed-byte/source-history preflight, explicit-input export/check, four actual CLI source-corruption controls and two bool/int receipt controls PASS. Independent agent review checked source correspondence, 200 signed-rational stage pairs and final margins; its receipt typing finding was repaired. No independent human or formal-kernel acceptance is claimed.

No net speedup over old scalar Core, recursive paid-profile saving, new kappa or selected-frontier change. No Lean, Docker, historical full suite or selected-frontier replay. Combined registered scientific replay was not duplicated after the individual normal and optimized commands.

Claim: #4182 (comment)

Prepared by JoyboyBrian with substantial OpenAI Codex assistance. Apache-2.0; credit Jose Manuel/jmbarrancoml's Core/calibration, Chafik Boukhalfa's interior inverse/runtime, PR5483 transport and PR5499 two-column kernel. Retained/adapted helper bytes are pinned; upstream and frozen inputs unchanged.

Claim base 32890bc5de034ab8fc93ac88ead7279a6a416d04. Selected frontier unchanged.