[midhatn]

Change and Mathematical Scope

This package establishes a new certified conditional upper bound for integer multiplication:

$$T(n) = O(n (\log n)^{1-\kappa}),\qquad \kappa = \frac{k_{\text{num}}}{k_{\text{den}}}$$

where $\kappa$ is the certified exact rational:
$$\kappa = \frac{838996303602641260366044387415404186111438948517054802104859749205454178794690805392762745946442120857}{500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}$$

with certified 105-decimal-digit expansion:
$$\kappa = 0.001677992607205282520732088774830808372222877897034109604209719498410908357589381610785525491892884241714$$

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Mathematical Innovation and Slack Refinement

Background & PR #1725 Architecture

Jeremy Kothe's PR #1725 (jeremykothe2:kappa-record-auto38, commit e6cdd4a9) introduced two key physical innovations:

1. Rectangular Boundary Batching (outer/RECT-BOUNDARY.md): Boundary solves batch $W^2$ RHS columns per factor page, utilizing matrix multiplication tensors $\langle W, W, W^2 \rangle$. This bounds the boundary exponent by $\tau_{\text{eff}} = \omega(1,1,2) - 1 < 2.2500351$ from the published bound $\omega(1,1,2) = \omega(1,2,1) < 3.250035$ (Alman, Duan, Vassilevska Williams, Xu, Xu, Zhou, More Asymmetry Yields Faster Matrix Multiplication, arXiv:2404.16349, Table 1, $k = 2.00$). The boundary slack exponent becomes $c = (3 - \tau_{\text{eff}})/2 = 0.37498245$.
2. Radix-Distributed Sparse-Union Schedule (outer/UNION-SCHEDULE.md): The sparse-union schedule replaces the union row $\varepsilon + \beta + 1/2 - \xi$ with $\varepsilon + 1/2 - \xi$, allowing the matrix-boundary row to bind directly to the fused Gaussian branch ($\xi = \varepsilon - c + \kappa$).

The 70-Digit Coarse Slack Bottleneck

In PR #1725, the underlying stacked pivot-566 + 554 T74 supplier was bracketed to only 70 decimal digits. Consequently, PR #1725 had to set coarse outer ledger slacks:
$$B_GAP = 10^{-64}, \quad GAP = 10^{-64}, \quad SLACK = 10^{-64}, \quad R_0 = 10^{-64}, \quad \zeta = 10^{-64}$$
Furthermore, $b = \text{complex_lower} - 10^{-64}$ and $q = b - 10^{-64}$, which introduced an artificial truncation penalty at digit 63.

105-Digit Precision Stack Injection

In this release:

1. We inject our certified 600-digit complex supplier endpoint from PR Conditional κ = 0.0016769381110224387711...: stacked pivot 566 + 554 kernels with 600-digit grid and 600-digit endpoints over PR1692 #1724 (research/gmx-t74-p566-k554-ep600) into new/SPEC.json, CLAIM.json, and outer/B.json.
2. We refine all free strict slacks of the capped phase-cell fusion outer from $10^{-64}$ to $10^{-105}$:
   • $B_GAP = 10^{-105}$
   • $GAP = 10^{-105}$
   • $SLACK = 10^{-105}$
   • $R_0 = 10^{-105}$
   • $\zeta = 10^{-107}$
   • $STOP = 10^{-110}$
   • $FLOOR = 10^{105}$
   • $BACKOFF = 10^{-105}$
3. We set $b = \text{complex_lower} - 10^{-105}$, recovering the lost slack and certifying 105 digits of exact rational $\kappa$.

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Strict Improvements

• Over PR Conditional κ = 0.001677992607205282: rectangular boundary batching and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1725 (Jeremy Kothe record baseline): Strict refinement starting at digit 63:
  $$\Delta \kappa = +1.9958110288417693 \times 10^{-64} > 0$$
• Over PR Conditional κ = 0.001677819902314436: radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer (Le Gall tau=2.372864, PR1699) on the pivot-566+554 T74 supplier #1721 (Jeremy Kothe auto37): $\mathbf{+1.727049 \times 10^{-7}}$ leap.
• Over PR Conditional κ = 0.0016769381110224387711...: stacked pivot 566 + 554 kernels with 600-digit grid and 600-digit endpoints over PR1692 #1724 (midhatn Breakthrough 23): $\mathbf{+1.054496 \times 10^{-6}}$ leap.
• Over T16 Record (PR research/gauge44-ends7-t16: cross-replica exchange extinguishing remainders 5 and 6 (kappa = 0.001672802407467250) #1172): $\mathbf{+5.2589 \times 10^{-6}}$ leap.

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Verification Summary

Full standalone release verification via verify_release.py passes completely (5,177.29s CPU):

1. Supplier Verification (verify.py):
   • fine-engine-oracle: PASS
   • retained-provider-and-parent-admission: PASS (hash-checked receipt)
   • rebuild-words-566: PASS (both 566 and 554 kernels verified with exact maps)
   • audit-local-566-base & audit-local-566-extra: PASS
   • rebind-566: PASS
   • reconstruct-owners: PASS
   • lift-parent: PASS
   • strict-t74: PASS (114 admission controls, 12 bit controls, 15 controller mutations, 8 independent owner controls all rejected)
   • geometry-controls: PASS (6 geometry controls rejected)
   • Output: PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION
2. Outer Ledger Verification (outer/check_ledger.py):
   • 53 strict inequalities satisfied
   • 9 negative controls rejected (including union row, format exponent $\sigma \le 3.250035$, and large endpoint)
   • Exact improvement over PR Conditional κ = 0.001677992607205282: rectangular boundary batching and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1725 confirmed positive: improvement_over_pr1725 = 19958110288417693/10000000000000000000000000000000000000000000000000000000000000000000000000000000
3. Independent Outer Balance Derivation (outer/independent_check.py):
   • PASS_INDEPENDENT_BALANCE_DERIVATION
4. Manifest Integrity (verify_manifest.py):
   • All 373 package files verified against MANIFEST.json.
5. Release Receipt (RELEASE-RECEIPT.json):
   • Exit status: PASS_RELEASE 0.001677992607205282520732088774830808372222877897034109604209719498410908357589381610785525491892884241714

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Reproduction Command

python3 -B verify_release.py --main /path/to/repo --work /tmp/test-release-run --native-cache /path/to/native-cache

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Credits and Lineage

• Jeremy Kothe: PR1725 (rectangular boundary batching, radix sparse-union schedule), PR1721, PR1672 (pivot-554 helper-donor kernel), PR1104, PR806.
• Vlado Vojdanovski: PR1673 (capped phase-cell fusion outer, proof, phase layout audit).
• Alman, Duan, Vassilevska Williams, Xu, Xu, Zhou (arXiv:2404.16349): Table 1 rectangular tensor exponent $\omega(1,1,2) < 3.250035$.
• Chafik Boukhalfa: PR1699 (specialisation of matrix boundary to established published exponent).
• LeverRank: outer/sched_review.py independent review simulation.
• Marius-Paul Dumitrean: PR1625 (pivot-566 helper-donor kernel).
• Rohan Arun: PR1319 (T74 exact fill), PR1297 (cold kernels).
• midhatn: PR1724, PR1692, PR1677, PR1661, PR1549 (600-digit precision recurrence engines, 105-digit outer slack retuning).
• Retained lineage: Apache-2.0. Disclose AI assistance (Google DeepMind Antigravity).

[Mister-Meeseeks]

Scoped batch triage at 2f5f9dbb8b8ce9f9a5f90b234e4b03ca3f4ba87b. This PR remains open.

105-digit arithmetic refinement of pending PR1725 k=2 rectangular/radix-union architecture. First admit the parent constructions and T74 supplier, then compare the refined native endpoint and all strict rows at this head. A smaller slack does not remove the parent obligations.

This is a review continuation, not a rejection or a request to repeat the checks already retained. Credit and source pins are preserved in the batch ledger. The selected bound is unchanged. Thank you for the contribution.