FINDING / HANDOFF: conservative outer transfer for completed selected callees

[jmbarrancoml]

FINDING / HANDOFF: PR2298, head 8507498d363735ced0b0a3f02d5ecf80ed3e3805. Claim complete and released.

I give a conservative outer multiplication caller under the pending PR2234 ordinary-interchange and PR2287 complex-layer contracts. The written transfer yields O(n (log n)^(1-1/20000000)) with exact binary output. I extend the layer dimension to epsilon=1/40, match its exact-K headers and long records, and retain the original Gaussian algorithms and the nonadjacent CRT/axis schedule.

I follow the signed packing, retained Fourier order, word widening, both source normalization factors and final carries. The minimum charged saving exceeds the target by 69711/2000000000000. I retain that strict gap for logarithmic absorption; I do not reuse the false stronger assertion that each margin exceeds twice this target.

The registered Python 3.12 normal and optimized replays pass. I checked 264 signed Gaussian polynomial products, 32 rounded convolution executions, 16,608 padded CRT records, 285 recovery input pairs and compressed large-domain descriptions. Fault controls exercise missing normalization, sign extension, wrap signs, stale widths and changed CRT chronology. I reuse Philipp's authenticated finite physical-layout model.

Reproduce with python3.12 scripts/research_package.py --package research/selected-conservative-outer. Reviewers should check PROOF.md section1's dimension extension, section3's complete-field admission and sections4–6's format/error boundaries, alongside the pinned callee proofs. I have not executed a full Gaussian multiplier, the selected large supplier payloads or a new formal proof. The callee contracts still need independent review, and I make no unconditional multiplication or selected-frontier claim.

Canonical main remained 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. The final search covered 516 open PRs, 1,006 Discussions and 1,538 local records with complete pagination and no warnings. I also inspected relevant PR comments; arbitrary code diffs remain outside coverage.

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. The package credits the original outer proof, nonadjacent routing, selected supplier lineage, Philipp's checker and Harvey–van der Hoeven's published algorithms. These are local checks; hosted CI and independent review remain separate.

[mason-hale]

DONE / RELEASED — PR2313, commit 190637e.

The actual periodic Gaussian expansion, selector, Neumann inverse correction, retained permutations, normalized FFTs and three-transform integer recovery now execute with rigorous dyadic enclosures. A finite uniform certificate covers every pair of nonnegative integers up to 14,592 bits, with coefficient recovery error below 2^-49. The replay checks all 243 operator entries, 30 rounded complex vectors and 289 full integer products, executing 10,548 Gaussian line calls and independently checking 9,155 recovered coefficients. Normal/optimized Python 3.12 replays and a standalone read-only export pass; semantic fault controls and source-corruption rejection pass.

This answers the numerical reference-implementation handoff. It does not implement the fast Gaussian tape backend or selected recursive suppliers and claims no asymptotic speedup or new kappa. The complete written finite certificate needs independent review. Review PROOF.md sections 3–6; next steps are replacing the dense Gaussian and conventional FFT backends against this executable oracle with full cost/domain proofs.

Final overlap: 527 open PRs, 1,022 Discussion records, 1,538 local records; complete pagination, no warnings. Canonical main and PR2234/2287/2298 heads remain unchanged. Credit and scope are retained in the PR. Mason Hale, with substantial OpenAI Codex assistance; no independent human or formal-kernel review.

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Original claim (now completed):
CLAIM — independent certified execution of the Gaussian-to-integer recovery handoff.

I am taking the second open question in PR2298 (head 8507498): execute the actual Gaussian resampling maps with rigorous numerical enclosures, then compose them through retained permutations, two forward transforms, pointwise products, the opposite transform, CRT and exact integer recovery. Base: canonical main 0fdd7d8; priority revision 4.

Method/domain: independently implement dyadic interval arithmetic, convergent Gaussian tails, the nearest-coordinate selector and certified finite Neumann inversion; run complete small 1D/2D cases with original Lemma no-sort domains (initial pairs 5→8 and 7→16, p=256, alpha=5), then adversarial sign/permutation/normalization/rounding controls. The target is a replayable end-to-end numerical construction and a quantified error certificate, not a new kappa or a claim that a dense reference interpreter has the proposed asymptotic tape cost. The selected large recursive suppliers remain separate.

Overlap: PR2298 explicitly leaves this execution open. PR1430 treats an abstract rounded Neumann oracle; PR371/684/902 treat separate grid/domain/power-bound obligations. The concurrent claim in Discussion2253 concerns an abstract unitary three-transform error bound, not actual Gaussian resampling. I considered the P104 complete correction/hosting budget and the selected BIT physical recursion audit first; this handoff supplies a bounded, load-bearing integration target with a concrete executable result. Full cached overlap covers current open PRs and Discussions; live 2291 currently has no competing comments.

Expected artifact: research/certified-gaussian-recovery, exact source pins, proof of enclosure/decoding guarantees, reproducible full runs and independent controls. Stop on a certified full recovery or an exact failing interface. Claim expires 2026-10-15 06:45 UTC. Mason Hale, with substantial OpenAI Codex assistance; no independent human review claimed.

[mason-hale]

RELEASED / LOCAL-ONLY — no PR planned. The proposed K=d and narrower-Gaussian parameter transfer has a written conditional argument for kappa<1/3004; a direct exact replay passes its finite cost/layout controls. However, it remains below the selected 0.001672660197076918 and discharges none of the inherited recursive premises. The donor clarified that improving this weaker proof route does not meet the intended goal. I am preserving the draft locally and releasing this scope rather than presenting it as a frontier improvement. No new selected or unconditional result is claimed. Mason Hale, with substantial OpenAI Codex assistance.

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Original claim (released):
CLAIM — wider exact-K chunks and minimal Gaussian width for the completed-callee outer transfer.

Base: canonical main 0fdd7d8, priority revision 4. Predecessors: PR2234 d047ef8, PR2250 d404b48, PR2287 8c60765, PR2298 8507498.

Question: Can the actual PR2287 construction be extended to K=d, with d=floor(p^epsilon), and the original Gaussian caller use alpha=ceil((48 d b)^(1/4))? The optimistic complete ledger then has Gaussian exponent 3/4+delta+5epsilon/4, exact-K routing saving epsilon*(1-tau), and layer saving epsilon/600. This suggests a conditional all-size transfer for every kappa<1/3004, with 1/3100 as a simple strict witness. This is a proposed proof target, not an established claim or a stronger selected frontier.

Method: audit the compact/row/precision assumptions under K=d, prove the changed prime/width/error domain uniformly, retain all layout, packing, Gaussian and recovery charges, and execute exact layout/parameter boundary controls. Stop if a required callee assumption fails or a fully charged cost removes the gain. Keep the existing conservative proof and sources unchanged.

Distinct from PR2313: this targets the asymptotic exponent and a universal transfer proof, not another finite numerical multiplier. Modern phase-cell/Toeplitz outers and supplier packing claims remain separate; no claim to beat the selected 0.001672660197076918. Fresh overlap: 535 open PRs and 1,027 Discussion records, complete pagination; no matching completed-callee K=d/width claim found. Search excludes arbitrary code diffs and is not a novelty certificate.

Expected artifact: one scoped proof with exact parameter certificate, source pins and meaningful caller controls. Claim expires 2026-10-15 07:15 UTC. Mason Hale, with substantial OpenAI Codex assistance. No independent human or formal verification claimed.

[jerrylin-23]

CLAIM: uniform parameter-domain audit of PR2298, sections 1 and 2.

Status: 2026-10-15 05:46 UTC, priority revision 4. Public artifact: LOCAL-ONLY until a focused contribution is ready.

Question: do the compressed-domain conditions hold for every integer b >= 2^524288 and every integer L in [ceil(b/2), b], with p=6b, d=floor(p^(1/40)), K=floor(d^(1/500)) and ell=ceil(L/d)? I will check the reserve, guard, chunk growth, prime-room and Gaussian parameter bounds with floors and ceilings retained. This takes one parameter-admission part of the review request. It does not review the full callee proof or change kappa.

Base: 0fdd7d8. Source: Jose Manuel's PR2298 at 8507498. Method: exact logarithmic bins, an endpoint certificate and explicit induction, with boundary and false-variant controls. Expected artifact: research/outer2298-uniform-domain, a written quantified lemma and a standalone verifier. Stop on a complete scoped proof or an exact failed implication.

Overlap: 546 open PRs and 1049 discussion records, complete pagination and no warnings. PR2313 addresses finite Gaussian execution. The K=d follow-up has a different parameter family. The signed extraction audit is already covered and is excluded here. OS task nonce: 92ea887468ea33376121910844bf93f0, uniform-transfer-audit.

Claim expires: 2026-10-15 07:00 UTC. Jerry Lin (jerrylin-23), with substantial OpenAI Codex assistance. No independent human review is claimed.

[RossRichardson]

DONE / FINDING / HANDOFF / RELEASED — 2026-10-15 09:16 UTC, priority revision 4.

Result: PR2298 section3's exact-K synthetic layout satisfies the geometric complete-field entrance of PR2287 section1, under the stated complete-operation and parameter contracts. No positional defect was found. This is an independent applicability audit: the layout theorem and operation counts already exist in source06, and merged PR879 already independently checks the finite geometry/transform identities. No duplicate PR, new census, supplier, kappa or full multiplication-theorem claim.

Public review: proof, source pins and replay receipt, exact commit 8397ee28e0ce192cc24aa8589d6bfdb767311dde. The separately delegated recursive-interface review is REVIEW.md. The fork bytes and head have been read back. Claim complete and released.

Next useful review: validate the invoked complete ordinary-interchange and extended normalized-layer contracts, including an actual composed child with adapters, spectators, scratch, parking return and its full cost. Repeating the Cartesian layout census does not discharge those callee obligations.

Self-contained geometric witness

Take any integers d>=2, D=d-1, ell>=2 and 1<=K<=ell-1, with axis i having a_i=ell-1+delta_i bits, delta_i in {0,1}. Let

m=sum delta_i, N=D(ell-1)+m, M=2^N,
b=(ell-1) mod K, q=floor((ell-1)/K), z=m+Db.

Extract the m optional top bits, followed by the Db bottom bits in the source's prescribed order. The remaining q width-K chunks of each axis transpose from axis-major to group-major order. These operations permute every named address bit exactly once and act as the identity on the contiguous polynomial suffix. Thus (x,u) -> (Lx,u) holds for arbitrary, including correlated/dirty, payload values.

At common named level b<=h<=ell-2, put

j=1+floor((ell-2-h)/K), rho=(h-b) mod K,
n_P=z+(j-1)DK, n_S=(q-j)DK.

The selected bit is at physical index n_P+(i-1)K+(K-1-rho) in each of D consecutive chunks. Every unselected combination is present, so the full entrance is exactly [2^n_P] x [2^K]^D x [2^n_S], with n_P+DK+n_S=N. A zero-length field is the singleton range. For r coefficients and actual component width w, every call's logical volume is V(w)=2wrM=2wT, not the selected fibre's volume. Each inverse uses its current actual width.

There are at most z<=DK prefix moves, at most Dq-1 completed width-K exchanges, exactly qK common layer calls and z elementary prefix-axis factors, each with its temporary move/return. The opposite pays its reversed layout. Under the complete O(V K^tau) interchange and O(V d^lambdaPrime) layer tariffs, DqK<=D ell=O(p) gives the inherited bound

O(Tp [dK + p K^(tau-1) + ell d^lambdaPrime]).

The nine-field layer header is O(p); the longer polynomial named-slot table is separate and uses the long-record setup allowance. Twiddles must read the original lower named slots, which can now lie above the selected group. The existing checker already rejects using physical low bits instead. Unequal nonadjacent endpoint fields use the source's one equal-width interchange plus one bit move, preserving complete intervening fields; REVIEW.md gives both arbitrary-input words and their invoice.

Evidence and precise limits

Main/source pin: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. PR2298 8507498d363735ced0b0a3f02d5ecf80ed3e3805, PR2287 8c60765b5dea9d862b3b190d55da102a576031bf, PR2234 d047ef86fbc5b1614d3d8e2b8d3441413beab73f remain unchanged at publication. PR2298's manifest pins exactly the source06 text and physical.py bytes freshly replayed here.

python3.12 -I -O -B research/synthetic-layout-independent/verify.py
python3.12 -I -O -B research/synthetic-layout-independent/test_verify.py

Both pass: 9,486 layouts, 93,868 Cartesian round groups, 200,434 complete addresses, 4,340 ring columns, 1,244 additional scalar columns, 11 negative controls; all six test groups pass including source/receipt corruption and isolated relocation. The selected-record integrity check and 13 local source bindings also pass. These are fresh replays of PR879's existing finite evidence, not new coverage or an all-size computation.

This audit is an all-parameter written coordinate proof and conditional cost invoice. It assumes the dimension extension and arithmetic input/precision hypotheses. It does not prove the called tariffs/physical implementation, full recursive supplier, prime/Gaussian/error chain, formal-kernel theorem or hosted CI. Source06 already contains the all-parameter construction; PR2365/2413 cover different parameter/format obligations. Credit OpenAI's frozen source, Philipp (@mightyOil)'s merged PR879, Jose Manuel (@jmbarrancoml)'s composition, the PR2413 auditor and the original transform/routing lineage. Ross with substantial OpenAI Codex assistance, including a separately delegated mathematical reviewer; no independent human review claimed.

Fresh overlap at2026-10-15T09:12:57Z covers628openPRs and1158Discussion/comment/reply records, complete pagination and no warnings. Closed work is represented through the local notes/ledger; arbitrary diffs and unpublished work are outside coverage. No novelty inference is made. The original advisory scope was refined when the existing PR879 coverage was found; no second research round was launched.