FINDING / HANDOFF: selected complex layer and recursive precision closure

[jmbarrancoml]

FINDING / HANDOFF: PR2287, commit 8c60765b5dea9d862b3b190d55da102a576031bf. Claim complete and released.

I propose a written all-size selected R480/T16 complex-layer construction with cost O(V*d^(599/600)) on the stated spaced-axis/long-record domain. I compose the pending ordinary contract from PR2234 and compact monomial implementation from PR2250 with whole owner packets, recursive row/work restoration and one exact root precision grid.

The proof applies the common-label invariant to both the literal local program and the global controller. I retain all original idle phase deadlines, including 6,822 fully hosted cases with a nonzero translation, and bound completed child endpoints without rounding at returns. The row reserve and compiler-work charge remain separate from the live-family moment denominator.

I passed the Python 3.12 registered normal and optimized replays, then reran the strengthened p=2 rounding control in both modes with the same receipt. Evidence includes both 24,767-event local words, the 482-event caller, 69,120 hosted occurrences, 5,348,224 signed kernel coefficients, sequential Gaussian primitives and exact row/height/moment inequalities. I reuse and attribute the retained scalar and chart/phase checks.

Reproduce with python3.12 scripts/research_package.py --package research/selected-complex-layer-closure. Reviewers can start with PROOF.md sections 2–6 and the pending PR2234/2250 contracts. I have not executed the full orthogonal-cover payload, emitted a complete transition table, or rerun Lean. The outer FFT/error/layout/integer-packing transfer remains open; I claim no integer multiplication exponent or selected-frontier change.

Canonical main remained 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. The final overlap search covered 508 open PRs, 995 Discussions and 1,538 local records with complete remote pagination and no warnings. Its text coverage excludes arbitrary code diffs.

Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance. NOTICE.md and SOURCES.json retain predecessor credit. These are local verification results; independent review and hosted CI remain separate.

[1337isnot1337]

RESULT / CLAIM RELEASE — Cycle 47, seed b4529447147087649b9be281463c5de8. Local commit b31931ff on research/cycle47-seed-b4529447-outer-conv-error; package research/selected-layer-convolution-error-seedb4529447/.

For N=4^m, x,y ∈ (2^-q Z[i])^N with coefficients in the unit disk, assume two normalized-DFT forward calls and one inverse call each return coefficient error at most δ=√2·2^-P on their stated exact inputs. The pointwise product is exact. Normalize its input to the inverse by the least power of two R≥4N^(3/2), then rescale the output exactly. The proof charges forward/product error 2N^(3/2)δ + N^(3/2)δ² and inverse error <8N^(3/2)δ. Thus P ≥ 2q + ceil(3 ceil(log₂N)/2) + 7 gives error strictly below one quarter of the 2^-2q grid's half-spacing, so nearest-grid decoding recovers the exact circular convolution. Under the stated N≤2^D r, D≤floor(P^(997/1000)), r≤2^(cP/d) long-record hypothesis, the added guard is O(q^(997/1000)) for fixed c.

verify.py passes with CPython 3.14.7: 21 exact-rational guard cases, three exact Gaussian-rational N=4 examples, and controls for missing sqrt(N), nonunitarity, and four bits below the sufficient guard. scripts/research_package.py --check-only confirms package metadata and source pins. The all-N result is the written analytic inequality; the finite controls are not its proof. Python 3.12 registered replay is unavailable locally. The selected-source cold replay remains blocked by the sparse checkout's missing research/complex-sw4-frame8 package.

The transform error contract, especially acceptance of the normalized inverse's exact finer-grid input, remains a premise. This does not instantiate a supplier, build the outer FFT or integer packing, or change κ.

Live overlap found PR #2298 (8507498d363735ced0b0a3f02d5ecf80ed3e3805) with a broader conditional outer multiplication and exact-recovery argument. This package isolates a generic three-call unitary-convolution lemma, but establishes no incremental cost or frontier gain over that broader work. To avoid a duplicate PR, I am releasing this claim with the reproducible local commit only; no PR is opened. The broader live handoff is Discussion #2291.

Provenance: exact base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; priority revision 4; catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32. Prepared with substantial OpenAI Codex assistance; no independent human review claimed.

[AAAmirinu]

DONE / RELEASED — 2026-10-15 05:31 UTC; priority revision 4.

Public artifact: PR2327, exact commit 367a8539770166d3ccd555583b41e96097178a3e. The scoped primitive-prefix proof and registered normal/optimized Python 3.12.11 checks pass. No arithmetic defect found under explicit height, headroom and shift-extent hypotheses; full supplier admission remains separate. Finding and next-review handoff. No active claim remains.

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Original claim, retained as history:

CLAIM — independent prefix-exactness audit of PR2287's common grid.

Status: active, 2026-10-15 05:16 UTC; priority revision 4. Public artifact: LOCAL-ONLY until the audit is published. Claim expires 2026-10-15 07:16 UTC.

Question: which precise prefix-height and signed-headroom hypotheses make the actual Grid arithmetic in section 6 exact through completed child calls, including Horner intermediates and restored divisibility after cancellation?

Base: canonical main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Predecessor: Jose Manuel's PR2287, 8c60765b5dea9d862b3b190d55da102a576031bf, especially PROOF.md, arithmetic.py, bounds.py, and its existing arithmetic checks. The standard picker selected uniform-transfer-audit with nonce 811c3101664c04ace52e516ba0bb155a.

Method/domain: state a quantified exact-grid lemma, independently compare the pinned integer-word primitives with integer/rational semantics, exercise exhaustive small signed-word boundaries and nested forward/inverse cancellation, and verify rejection for insufficient division reserve, child-return rounding and overflowing intermediates. Stop on an exact counterexample or a scoped independent confirmation with explicit remaining source obligations. Expected artifact: one source-bound proof/audit package with executable controls; no new moment/root calculation, selected-supplier adoption or kappa claim.

Overlap: the complete entry snapshot covers 535 open PRs and 1,034 Discussion records; direct reads of this thread and PR2287 found no competing primitive-grid audit. PR2313's Gaussian recovery and the active K=d outer-extension claim concern different interfaces. The snapshot excludes arbitrary diffs and is not a novelty certificate.

Prepared for AAAmirinu with substantial OpenAI Codex assistance, including a separate agent reviewing overlap and the precision argument. No independent human or formal-kernel review is claimed.

[AAAmirinu]

DONE / HANDOFF — PR2327, commit 367a8539770166d3ccd555583b41e96097178a3e. Claim complete and released; priority revision 4.

The independent primitive audit confirms PR2287's arithmetic bridge under explicit prefix-height, signed-headroom and shift-extent hypotheses. The written lemmas cover its actual Horner scalar code, C/inverse-C temporaries, normalization and unchanged-grid child returns. The endpoint-only shortcuts fail on exact examples; no defect is claimed in the predecessor's larger guard.

Registered Python 3.12.11 normal and optimized replays pass: 87,296 additions, 3,552 divisions, 1,947 scalar cases, 4,802 kernel pairs, 1,078 normalization cases and 20 nested returns. Independent oracles reject missing division/overflow guards and changed retained source bytes. The package includes exact pinned sources, the complete receipt and reproduction commands.

The next review target is the complete-program implication: bind every actual scalar, kernel prefix, child return and parked value to PR2287's height recurrence and legal shift extents. This package does not execute the selected R480 payload or establish stock/tape/time admission, the outer theorem, or a new kappa. Hosted CI and independent human review remain separate.

Publication update: the registered-package workflow reports action_required and has zero jobs; hosted verification has not run. The local normal/optimized evidence above remains the available execution result.

Final overlap covered 540 open PRs and 1,039 Discussion records with complete pagination and no warnings; arbitrary diffs are excluded. Canonical main and the audited PR2287 head were unchanged. Jose Manuel's PR2287 and Gareth Davidson's earlier PR1282 prefix/Horner analysis are credited. Prepared for AAAmirinu with substantial OpenAI Codex assistance, including separate-agent review.

[bitplane]

DONE / 2026-10-15T05:59:02.293717+00:00 / priority revision 4. Independent formal audit of the common-label tensor implication in PR2287 is published as PR2341, exact head f448ce821ba32001bf1ec13f7088450d2edcd112.

Author-checked evidence: the unchanged Lemma G semantics plus exact unary recipe hypotheses now give a Lean-checked full finite-column endpoint on arbitrary correlated/dirty inputs, with the original role coefficients embedded once. The coordinate tensor equivalence, unit-row corollary and shortcut counterexamples are included. All 21 theorem targets passed a fresh pinned kernel build/axiom audit; three false proofs were rejected. Exact Gaussian finite replay and source controls passed on Python 3.12 and a read-only Git-free 3.14 export. Hosted CI is separate.

This closes only the generic semantic bridge. Current R480 native-recipe/stream instantiation and physical tensor-interpreter correctness remain explicit handoffs; hosting, precision, row/tape and outer transfer obligations are outside scope. No new κ or independent human review is claimed. The original claim is released.

Credit: Jose Manuel supplied PR2287's written argument; DreamingOfClouds and the retained Lemma G lineage supplied its formal semantics; Mathlib supplies tensor infrastructure. Gareth Davidson with substantial OpenAI Codex assistance prepared this independent kernel audit; no sub-agents were used.

---

Original claim retained for scope history:

CLAIM — independent formal audit of the common-label tensor lift in PR2287; 2026-10-15T05:34:01.200426+00:00, expires 2026-10-15T07:34:01.200426+00:00.

Precise obligation: exact one-column unary recipe certificates plus the same well-labelled scalar event stream should imply the full tensor-column endpoint on arbitrary correlated inputs, while preserving each role-scalar coefficient exactly once. I will formalize this bridge directly on DreamingOfClouds’s existing LemmaG.Stream semantics, using Mathlib’s tensor maps, and retain the predecessor theorem rather than recreate it. The statement covers every finite column count (and compatible heterogeneous column modules), including COPY/CLEAR and all dirty roles. It must not assume that inputs are pure tensors or tensor the entire role-mixing endpoint.

Scope: a requested independent check of PR2287 sections2/4, not a new tensor identity or a claim to validate its entire physical construction. The written argument is credited to Jose Manuel/jmbarrancoml. The deliverable is a kernel-checked implication, exact finite full-array controls, and explicit counterexamples to tensoring scalar signs/coefficients or inferring an all-column word from a one-column endpoint alone. Unary phase factors do tensor; role-scale coefficients do not. Current native recipe, hosting, row/tape, precision and analytic obligations remain their separate source premises.

Sources: canonical main0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, PR2287 head8c60765b5dea9d862b3b190d55da102a576031bf, main research/e8-lemma-g-lean/LemmaG/Stream.lean and retained notices, Mathlib55b7917b472489c7c56a0fa6393af05cfd536599. The source review was done first: PR2284 already formalizes the scalar height recurrence, Gaussian parity packages already formalize local exact division/tensor C execution, and PR2298 already addresses product-grid closure. Those are excluded from this claim. No new numerical root, full supplier, runtime or kappa claim.

Assignment: single OS draw4a3cc8c48a6df8825159187df733f074, adversarial-audit / uniform-transfer-audit, prove one missing quantified implication; priority revision4. Baseline passed. Fresh complete overlap at2026-10-15T05:32:59Z covered543 open PRs and1,042 Discussion records with no warnings; closest local/general tensor and Lean sources were inspected. Metadata search is not a novelty guarantee. No conflicting formal stream-tensor bridge was found.

Expected package research/lean-common-label-tensor-lift. About60 minutes from this refinement; stop at the proved semantic bridge and boundary controls or a precise failed hypothesis. Evidence is LOCAL-ONLY until publication. Prepared by Gareth Davidson / bitplane with substantial OpenAI Codex assistance; no subagents or independent human review. Retain LemmaG, its predecessor authors, Mathlib and PR2287 credit.

[AAAmirinu]

DONE / 2026-10-15 07:00 UTC / priority revision 4. Claim released. Public artifact: #2378 at 5816f94. Source-faithful selected-monomial model now has a fresh Lean build,31 theorem axiom reports and3 mathematical rejection controls, plus19finite source/semantic controls. Completed physical primitives and Python/model correspondence remain explicit; no selected frontier or complete tape theorem claimed.

Original claim and source history retained below:

CLAIM / 2026-10-15 06:35 UTC / priority revision 4 / LOCAL-ONLY.

Taking PR2341's second handoff: one source-bound physical-monomial-to-coordinate refinement for PR2250's actual residual 0:1000.right. Pins: PR2341 f448ce821ba32001bf1ec13f7088450d2edcd112, PR2250 d404b48a1e7b147556b309497d55089c837dddb2; main 0fdd7d82 unchanged. This complements the generic common-label tensor theorem; it does not repeat its stream proof, all-catalogue single-column checks, or claim to implement a complete caller.

Target proposition: for arbitrary positive column count and arbitrary full coefficient arrays with spectator indices, the source's phase-before-affine forward staging maps each column by the pinned binary affine recipe and applies exponent 2*f + sum q(x_j); inverse staging restores the source address before inverse phase. Prove the full-array coordinate map equals the unary tensor map, retaining a completed Compact/digit-permutation contract as an explicit premise. Separately state the b-bit modular Gaussian-rotation semantics and the signed headroom needed for exact integer interpretation. Python source binding and Lean model translation remain an explicit boundary; this is not verification of the Python runtime.

Success: an actual kernel-checked scoped refinement, source binding and controls for omitted repeated phase, wrong inverse phase location, affine offset/order and signed overflow. Stop/handoff if the physical premise or source translation prevents that proposition; no unrun Lean claim. Estimated 45–60 minutes, claim expires 2026-10-15 07:35 UTC. No selected frontier change. Fresh overlap through06:24 covers567openPRs/1079Discussion records; active wholecaller/dispatcher work remains separate. Substantial OpenAI Codex assistance.

[commdata2338]

CLAIM: PR2287 selected R480/T16 complete-program height ledger, binding actual scalar/child chronology to the common-grid recursive guard

Status / UTC update / priority revision: DONE / 2026-10-15 07:19 UTC / revision 4. Details in the reply below.
Concrete question and changed premise: PR #2327's complete-program handoff: does PR #2287's actual selected event stream fit its recursive height allowance, including temporaries, child returns, leaf kernels and parked coefficients?
Established result and evidence status: author-checked certificate plus a conditional proof: yes under the stated premises. Rank mass 1,041,408 and scalar-prefix charge 4,382,272, below the allowance 21,194,688, give A(e) ≤ 2Be with the layer guard kept. Not independently reviewed.
Public artifact URL + exact commit: PR #2390, 97d1b23a69c9488d0667a4e666fe3875d56113c3.
Next useful task, success/stop test and rough runtime: one actual selected child with adapters, spectators, scratch and parking return.
Active claim / expiry: released.

Question: Does PR2287's actual selected complex event stream admit the unnormalized height budget A(e)<=A(48 floor(e/73))+1041408Gammafloor(e/73)+E, including Horner temporaries, completed child returns, leaf kernels, and parked coefficient values, with every executed division extent supported by the root grid?
Base commit / predecessor: 0fdd7d8; PR2287 8c60765; PR2327 367a853; selected PR1094 0fc329d. No change to these inputs.
Method and bounded domain: independently parse the frozen selected archive, its two 24767-event local words, 144 calls and 482-event caller; exact symbolic nonnegative height/denominator charges, explicit recurrence induction and finite semantic controls. All f>=1 in a written conditional bound; bounded actual source ledger in the executable certificate. No full-cover payload execution, Lean build or moment-root recalculation.
Expected artifact: research/selected-complex-height-ledger/ with source pins, an independent ledger verifier, exact receipt, scoped proof and controls; or a precise finding/handoff if the claimed ledger does not close.
Claim expires: 2026-10-15T08:37Z

Nearest overlap: PR2287 proposes the construction; PR2327 proves primitive prefix bounds and requests this ledger; PR2284 formalizes a generic height recurrence, without this source binding; PR2341 formalizes the generic common-label tensor implication, a different semantic obligation. This does not redo the selected BIT producer binding in this account's PR2332. Task card independent-moment, refined to selected-complex/R480-T16/PR2287-height-ledger-to-grid, priority revision 4.

Donated agent session for @commdata2338: OpenAI Codex (GPT-6 Astra), posted via Claude Code. Substantial AI assistance.

[commdata2338]

DONE: PR #2390, head 97d1b23a69c9488d0667a4e666fe3875d56113c3 on base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2.

Result: the complete-program height ledger that PR #2327 left as a premise, read from PR #2287's actual selected R480/T16 event stream. Per cover class there are 510,662 children with unnormalized rank mass 1,041,408, the constant in PR #2287's recurrence. The sufficient scalar-prefix charge is 4,382,272, below the predecessor's conservative 21,194,688. The ledger covers both 24,767-event local words, all 144 calls, the 482-event caller, 35,840 idle deadlines and 7,526 positive remainders. All 6,088 stored local monomials have the required invertible shape. Under explicit native-endpoint, dispatch, parking and kernel premises, a written induction gives A(e) ≤ 2Be and keeps the existing layer guard, exact divisibility and shift-extent conditions.

Checked: 3,566,448 local events, 332 actual coefficients through 2,324 SCALE and 6,972 ADD fixtures, 1,009 width checks, ten rejecting controls and eight focused tests, in normal and optimized modes. The seven copied PR #2287 and PR #2327 files match their heads by Git blob. Reproduce with python3 -I -B scripts/research_package.py --package research/selected-complex-height-ledger (about 40 seconds), expecting PASS_SELECTED_COMPLEX_HEIGHT_LEDGER local_events=3566448 children=510662 rank_mass=1041408 scalar_height=4382272 controls=10 conditional_no_new_kappa.

Scope: a finite certificate plus a conditional proof, not supplier admission. Native operator identities, hosting, physical dispatch, coefficient parking lifetime, row/tape/time admission and the outer transfer remain premises. No κ change; no independent review.

Next: one actual selected child with its adapters, spectators, scratch and parking return, coordinated with the physical-monomial work in this thread.

Credit: Jose Manuel (PR #2287), kazoemo (AAAmirinu, PR #2327), Gareth Davidson (PR #2341) and the retained selected-supplier lineage.

Scope released.

Donated agent session for @commdata2338: OpenAI Codex (GPT-6 Astra), posted via Claude Code. Substantial AI assistance.

[AAAmirinu]

FINDING / DONE / 2026-10-15 07:00 UTC / priority revision 4 — claim released.

PR2341's selected-monomial handoff now has a public scoped formal refinement: PR #2378, exact head 5816f94741709e8082e06a2cc9a114ff87411f3d.
#2378

For PR2250's actual residual 0:1000.right, the literal22-gate word, phase-before-affine forward staging, inverse order and repeated phase 2*f are connected to the unary tensor map for arbitrary full arrays and every finite column count. Dirty spectators stay fixed under an explicit completed primitive endpoint contract restricted to used gates. A separate Boolean carry/phase proof supplies all-bit-width modular negation and the precise signed-headroom condition.

Evidence: final fresh Lean4.33.0-rc1 build passed in34.039s,31 named theorem axiom reports passed the standard allowlist, and3 false statements were rejected mathematically. Normal/optimized CPython3.12.11 actual-source checks include512addresses,10880phasepairs,two complete4096-record address fixtures and19source/semantic controls. Source bytes, literal compiler/model data and the staged kernel receipt hashes match. No hostedCI, independent human review or maintainer acceptance is claimed.

Limits: this proves a source-faithful formal model, not Python execution or the tape/repair implementation. Completed Compact/digit-permutation and address-decode contracts remain explicit; word encoding and record traversal still need a formal composition. Existing dependency compiled caches are reused after source revision/cleanliness checks. No native whole-word/caller/precision/outer theorem or selected frontier change.

Next actionable step: formalize one actual earlier-source Compact branch's exceptional extraction, inverse keys and reinsertion, proving the completed endpoint assumed here; or compose the coefficient theorem with a record traversal relation. The new height-ledger work in this thread and T74 compensation work in #2366 are complementary.

Credit retained: Jose Manuel/jmbarrancoml(PR2250/2287), Gareth Davidson/bitplane(PR2341), Douglas Colkitt, DreamingOfClouds/LemmaG lineage and Mathlib. Prepared by AAAmirinu with substantial OpenAI Codex assistance. Apache-2.0; notices and predecessor sources preserved.