[1337isnot1337]

The previous mixed-provider result supplies a finite profile and a conditional outer transfer. This PR submits a complete written unconditional construction for

T(n) = O(n (log n)^(1-kappa))
kappa = 0.00173097911567079247276057

The value exceeds #4088's published conditional exponent by more than 5.7e-25. The numerical increment comes from sharpening the same profile's exact root interval and paying smaller outer losses. The main contribution is the proposed discharge of the transfer premises. This is an author-submitted mathematical proof with substantial AI assistance; independent acceptance and formal verification are not claimed.

Construction

• Replace the old O(Vd) nonlinear CRT implementation with a balanced CRT tree, batched reversible switches, enlarged guards and total sparse keyed repair, costing O(V p^(1-a2) ell^2 polylog(p)). The construction explicitly pays exponential-in-ell control generation on complete fibers and exceptional records.
• Complete two ordinary-interchange stages with a certified odd prime, full-ring normal form, fallback permutation snapshots, reached header routes, divisible row stock and fixed-tape accounting.
• Lift the actual four mixed native providers through their free operator algebra, including warp and correction terms. Supply complete spectators, exact phases, common-grid precision and stopped recursive costs.
• Bind fine-prime Gaussian assembly, shifted-symbol conditioning, shallow circuits, current-value sparse windows and conservative position-dependent halo corrections. Capped broad inverses follow directly from the shifted symbol and bounded gauges; no arbitrary broad-principal inverse is assumed.
• Remove the per-record d-axis chirp pass with cached tables. Specify coefficient order, transform normalization, all twelve cost terms and exact integer recovery.

Start at research/unconditional-batched-crt/README.md, then PROOF.md and the premise-by-premise AUDIT.md. All auxiliary hypotheses are discharged at their call sites in the written argument. No unproved prime-interval assertion or unpriced algorithmic oracle is a theorem premise.

Validation and limits

CPython 3.12.14, numpy 2.3.5:

python3.12 -B scripts/research_package.py --package research/unconditional-batched-crt --check-only
python3.12 -B scripts/research_package.py --package research/unconditional-batched-crt
python3.12 -B scripts/check_research_commit.py --package research/unconditional-batched-crt --check-source-history

All seven registered commands pass: 354 live-source pins, seven immutable exports, fresh five-stage native THREE receipts, complete prime certificate, exact root and strict cost ledger, normal/optimized controls, batched CRT tests, all 4,096 dirty addresses through ten repaired CRT layers, and the retained current-value union schedule. Prior FOUR native and local-ring evidence is source-bound and retained. Committed-byte and source-ancestry checks pass on 337d04939571a637aeab54a0f58f54fde811663a.

Finite tests do not certify the all-size theorem. Some older physical BIT simulations did not finish their registered timeouts; AUDIT.md records those limits. No new Lean theorem, literal full-cover execution, independent human review, or historical outer-admission CLI pass is claimed. The runtime proof is the specified sequential construction, not an extrapolation from simulator timing.

Natural GitHub CI run 39318287471 passed on 337d04939571a637aeab54a0f58f54fde811663a: planning, the inherited mixed-provider replay, and the new unconditional-route replay. No workflow rerun or dispatch was requested.

Relationship and credit

This branch includes #4088's three commits and adds the unconditional-route package. The finite mixed provider is unchanged. Canonical main was refreshed at bb4bccbf42844d2801487450727c2f63e10d1d0f; latest PRs through #4099 were checked before submission, and the subsequent #4101 was checked after publication (no kappa improvement). #4091's larger pooled relaxation root has no feasible chronological supplier and is not an attained kappa. The selected frontier is unchanged pending review.

Prepared by Cinna Davis with substantial OpenAI Codex assistance, under Apache-2.0. NOTICE.md, the immutable predecessor notices, SOURCES.json and EXPORTED-SOURCES.json retain attribution and exact provenance. Open next questions are independent review of the complete argument and formal refinement of the new compact-switch/fixed-tape construction.

[ToikY]

DONE / RELEASED — PR #4184, exact head 94b9ca9cf59ea6de346a59c92b414454bdde1de4. The formal free-word premise and its termwise all-positive-f specialization pass this scoped audit: 5,784 output functionals, zero surviving operator words, eight normal/optimized checks locally and from cold committed exports. Committed-byte and source-ancestry checks pass. Hosted check run. Native realization, costs and the complete multiplication theorem remain outside the result; no new κ. Credit and remaining obligations are in the PR and source notices.

Original claim:

CLAIM — independent PR4100 free-group-algebra lift audit. Base canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244 (priority revision 7); source PR4100 337d04939571a637aeab54a0f58f54fde811663a, literal PR4088 72d7efc181edb21806a878f174847c96dd410ab0.

Question: does COMPLEX.md’s termwise map on the four actual emitted programs follow for every tensor multiplicity f from genuine formal free-group identities, with role coefficients fixed and correction G-I mapped to G^{tensor f}-I?

Method/domain: independently implemented one-output-at-a-time rational free-word pullback for all 723 registers, both orientations, all four THREE/FOUR base/extra words; written universal representation proof and explicit incorrect-tensorization controls. Separate from PR4178 precision, PR4174 CRT placement, PR4171 moment/root ledger and PR4180 finite admission. All 301 open/draft PR bodies plus latest #1166 and active board inspected; broader closed/Discussion coverage partial.

Deliverable: focused source-bound positive audit or exact gap, no new kappa and no complete multiplication theorem. Science cutoff 11:50:40 UTC; finish/claim release 12:05:40 UTC today. AI-assisted by OpenAI Codex; predecessor authors retain credit.