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Split centres and source-basis reuse for conditional κ = 0.001120864952258278 - #433

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This combines a new split-centre complex construction with source-centre reuse on the bit side, in one commit based on main at 12a4934d15006e8c10f9b2264f13f8f341be802b.

Under the retained compiler, compressed-window, arbitrary dirty-bank, restored-work, fixed-tape/common-grid, precision and analytic interfaces,

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{560432476129139}{500000000000000000} =0.001120864952258278. $$

The combined saving is approximately 6.9522% above #425. The bit supplier binds the new assembly.

Construction

On the complex side, cancel the coordinate-source contributions that appear with opposite signs in the centre and side decoders before lifting them to address operators. The 112 code-line targets then use eight rank-eight centres; the eight coordinate targets use rank-one identity copies. Their total copied rank remains 72. Removing 112 coordinate-source helpers and eight dedicated full-centre helpers reduces R from 680 to 560. The dirty response is recomputed from the new word, and both orientations include literal source/helper cleanup and exact heterogeneous-centre phases.

On the bit side, start from #426's R767 word. Form its nine retained sums with an invertible basis change on the source rows, copy them sequentially, and apply the literal inverse before continuing. This removes nine helpers. Compose the resulting R758 word with the compressed dimension-64 geometry from #386, replaying the actual reflected copy chronology and all new bank owners. The changed prefix needs 223 bits and is charged at 256 bits; the old 192-bit allowance is not reused.

Selected supplier Complex Bit
Ambient dimension 73 64
Auxiliary rows 560 758
Live recursion stock 12,720 14,502
Physical stock 13,298 14,511
Rank deficit 2,496 2,496
Certified coarse saving 0.001183986549206329 0.001122122700265139220566

Both exact moments include the retained bad-branch allowance. Six fixed bootstrap levels and all 47 strict outer inequalities give the displayed κ; the next grid point and invalid-parameter controls are rejected. The result is calculated from the two emitted programs, not by adding independently reported gains.

Contribution sources and prior PRs

Source Contributor Retained contribution
#425, e11ccea1d092beb1b01b4ee4ce49da478234254c Zhihao Chen, with OpenAI Codex Mutable-basis Hamming packets, common-frame scalar interface and complex verification.
#426, 88d552526893cd15325948bb1487e9a4d4bf32f6 ai-nanachi, with OpenAI Codex R767 bit word, 55 clean-reset reuses and 54 frame changes.
#386 Zhihao Chen, with OpenAI Codex Compressed nine-stage geometry, routes, ownership and finite accounting.
#370 Dylan Crow, with Anthropic Claude Hamming E8 bit producer, root table and record exporter.
E8 and retained framework Jacob Sussman and the credited upstream contributors E8 unit, frame/compiler, rational-pricing and analytic interfaces; original notices preserved.

My earlier contributions remain historical context for this work:

PR Contribution
#10 Batched recursive boundaries.
#18 Partial-swap frames and unequal tensor factors.
#24 Endpoint and gauge integration.
#32 Compatible constructions and certificates.
#36 Structured semantic/bulk assembly.
#104 Stopped product-ring integration.
#115 Selected partial gauges.
#130 General frames and three-stage covers.
#144 Paired cubes, original-source chronology and shared completed cores.

The new split-centre construction, source-basis composition and integration are contributed by icekylinx, with substantial OpenAI Codex assistance. Earlier Pro exploration used GPT-6 Astra. Existing authorship, licenses and source-specific AI disclosures are preserved in NOTICE.md, notices/ and SOURCES.json.

Proof and validation

The selected package is research/split-centres-global/:

  • PROOF.md: split-centre cancellation, arbitrary-dirt lifting, source-centre/reflected chronology and conditional assembly.
  • CERTIFICATE.json: both component bounds, fresh finite invoice, bootstrap cutoffs and all 47 strict outer margins.
  • EXPECTED.json, SOURCES.json, MANIFEST.json: exact result, source pins and byte-level bindings.
  • verify.py: standard-library reproduction; dedicated CI covers Python 3.11 and 3.14.

The clean-directory incremental replay passed locally on Python 3.14.8:

  • All 800 complex and 998 bit local columns; both orientations, arbitrary dirty restoration and private-work erasure.
  • All 7,782 unbanked global bit columns, 54,576 bit bank assignments, and the complex owner/padding/cover decoder.
  • The 248 changed complex Gaussian pairs, with 2,670 unchanged verified pairs retained by exact source hashes; all 240 endpoint adapters.
  • The changed bit nested-frame/reflection checks, both fresh finite invoices, two rational moment engines, strict assembly and mutation controls.
python3 -B research/split-centres-global/verify.py /tmp/split-centres-check

To regenerate every selected Gaussian phase pair, add --replay-phases and use another fresh output directory. The default reuses phase evidence only when the regenerated canonical word and full recipe hashes match. The unchanged historical full suite was not rerun; no PDF was built. CI for this commit is not represented as passed before it runs.

This remains a finite conditional construction. The inherited all-size interfaces remain assumptions; no unchanged legacy gx/gxcore acceptance, new Lean proof, global optimality or unconditional multiplication theorem is claimed. The main selected certificate is left for maintainer review.

@icekylinx
icekylinx force-pushed the research/split-centres-global-kappa branch from 2992f23 to 6d92445 Compare October 11, 2026 12:05

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Target: PR #433, head 6d92445
(base 12a4934, fetched 2026-10-11T12:13:32Z).

Verification

Independent third-hardware replay of this head, from a fresh detached
checkout at the staged SHA.

  • Verifier exit 0 in both modes. Default mode reuses hash-matched phase
    evidence and passes 13/13 steps. Full mode (--replay-phases) regenerates
    every selected Gaussian phase pair and passes 13/13 steps with 2918/2918
    exact phase edges.
  • A second independent replay on separate hardware also returns verifier
    exit 0, 13/13 steps PASS, 12.33 s wall.
  • kappa = 560432476129139/500000000000000000 = 0.001120864952258278.
    The fraction is exact to the last of 18 decimals and identical to
    EXPECTED.json, CERTIFICATE.json and this PR's body.
  • Binding component: bit. The bit lower bound sits below the complex lower
    bound by exact cross-multiplication, so the bit moment bound is what caps
    kappa.
  • Recomputed gain over the #425 baseline is 6.9522005058117555%, matching
    this PR's 6.9522%.
  • Digit-exact: an independent closed-form re-derivation (stdlib fractions,
    no package code) reproduces the same fraction on the 1e-18 grid. The same
    code reproduces the earlier published 203592481059369/200000000000000000
    exactly, so the formula is not fitted to this PR.

Negative controls

All rejected. The package assembly controls next_grid, zero_eta,
zero_beta and unit_beta each raise. So do 8 bit mutations, 7 complex
mutations and all 18 bridge-shear omissions. In an independent perturbed
family, dropping the binding bound by one 1e-18 tick makes the claimed
kappa fail at the first constraint. The accepted margin is 0.585 of one
tick, so the certificate is tight rather than slack-passing.

Limits

The claim is conditional as this PR states: the all-size compiler,
compressed-window, correlated dirty-bank, restored-work, tape and common-grid,
precision and analytic interfaces remain assumptions, and no Lean-kernel
proof or unconditional multiplication theorem is claimed or verified here.

AI-assistance disclosure: prepared with AI assistance under human oversight.

Mister-Meeseeks pushed a commit that referenced this pull request Oct 12, 2026
…, wording)

- check_extended_labels.py: a source must convert exactly once (entry adapter
  only from ('pre', i)); identity adapters pinned to frame 0 / FULL; wrappers
  keyed by the start frame; unknown events rejected; new mutant (duplicated
  source entry adapter); adapter mutants now also run on the anchor.
- code/check_adapters.py (new): #433's data-adapter identities on all 512
  columns of every port (H6), 0 failures on all four words.
- verify.py --instances: confirms #484's own checks passed on inputs equal to
  #484 EXPECTED.json, runs both instance checks; #484 files it imports are
  pinned in MANIFEST (external_files).
- PROOF/README: explicit H2 bridge and C-linearity, H6, exact statement of
  #484's endpoint imprecision (mid-stream adapters are identities), "every
  input" qualified by its inherited facts, semiring wording.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Mister-Meeseeks pushed a commit that referenced this pull request Oct 12, 2026
- Model: units are physical streams (family, cover class) as in #442/#458
  and the template's route_semantics. New LemmaA/Streams.lean:
  streamOwner_injective(_of_disjoint), sweep_unique, lemmaA_streams (one
  owner per stream from disjoint footprints on shared families); plus
  lemmaA_piece and lemmaA_async (Lemma A feeding #505's window variant).
- code/check_owners.py rewritten for stream units (data wires alone,
  disjoint footprints on shared families, centre copies in distinct private
  slots, atomic calls; 3 mutants): all four #484 words pass, including the
  32 gauge80 / 16 kernel79 slot-sharing calls and the anchor.
- code/check_windows.py (new, independent): every hosted block inside its
  pending child's open window, blocks disjoint, family-group windows
  disjoint (2 mutants); all four words and #505's emitted async template.
- verify.py: runs Lemma G's check_extended_labels.py ((F) is read in the
  extended labels), fails on a missing LEMMA-A.json, checks I-helper counts
  against EXPECTED, prints the owner and window results; --async T.json.gz.
- Docs: unit model stated, coverage per word, #505 replay results and
  what is not re-run, commutative-semiring wording, credits #433/#442/#500.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
@Mister-Meeseeks

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Thank you @icekylinx. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Split complex centres and bit source-basis reuse.

Review and limits: A structural improvement in both suppliers, with dirty response, reflected chronology and larger prefix charge. Retain mutable-source/compiler and analytic hypotheses.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 6d9244518175caf9e0089f92486959bc2c317273. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.

thakrarsagar added a commit to thakrarsagar/integer-mult-bounds that referenced this pull request Oct 14, 2026
- The producer's lower_child check (signed Z4 tableau plus column 0) is
  complete but not independent; K2 is a control on this package's own
  comparator, not a gap in the producer's check. Lemma G's I-tr and
  check_adapters.py ran on four other words.
- Scope is record level for CLIFFORD-RECIPES.json; identity adapters are
  866 per orientation (352 entry + 514 exit).
- Gate-order equivalence holds on all 224 adapters (counted in RESULT).
- B3 accepts only the selected record or checkpoints/frontier1094.json;
  B8 uses a path pin in the source manifest.
- Tests (26): mutated child/adapter records rejected by evaluate(),
  receipts cover each pair once, measured digit bound on the 16-gate
  identities and a beta = 2K collision, I5/I6/I7 and fixture-pin
  negatives, calibration reproduced.
- Credits: HyperplaneEnd Lean (jeremykothe2, CrocSwap#593), Lemma G (DreamingOfClouds),
  CrocSwap#433 H6 identities, PR425 (jacklightChen), CrocSwap#1029/CrocSwap#662 engines.
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