[rohanarun]

Under the inherited analytic and fixed finite-alphabet multitape hypotheses,

$$T(n)=O(n(\log n)^{1-\kappa}),\qquad\kappa=\frac{4123863984}{10^{14}}=4.123863984\times10^{-5}>2^{-15}.$$

This is 0.12719% above PR #48 (411862541/10^13). The bit saving is 4124034054/10^14.

Full make verify PASSED at research commit 9c9c198656c5b926de951292c111e81ccf191634: 192 tests, five new focused tests, fresh changed-producer/profile and full physical dirty-basis replay, inherited data geometry with exact fallback recovery, and 18 historical patch checks. Completed 2026-10-08T15:44:38.918471+00:00 in 442.2 seconds. Validation receipt; full log SHA256 f302a33b7f5660e54741fcc0454fca065754f191045caa7e689779bda5a74e90. Follow-up changes are documentation and receipt metadata only.

Construction

PR #48 reordered the leave-one-out vector() sums. PR #47 explored climbing summand orders. Starting from #48's orders, this construction pins adjacent-swap bits for both total() calls (tbits) and vector() calls (vbits), combining those search directions. The supplied cloud checks report that zero swap bits reproduce #48's DAGs byte-for-byte.

Quantity	PR #48	This PR
h23 roles R	36,432	36,382 (6,077 matched; 14+10 swaps)
h25 roles R	48,329	48,255 (7,809 matched; 10+13 swaps)
Width W	177,530,859	177,284,805

All other components are inherited from #48: RaD's point order, original-envelope labels, both fixed I+J bases, copied centers, paid endpoint corrections, reversed data corners with exact recovery of all ten fallbacks, the complex layer and balanced assembly.

Search scope: the optimizer uses floating-point logarithmic weights and SciPy assignment. It proposes finite pinned carrier-use lists, whose legality, profiles and exact arithmetic witness are checked independently. No exact weighted-matching optimality, global optimality, or measured runtime improvement is claimed. The README now states this distinction explicitly.

Verification

At source commit 9c9c198656c5b926de951292c111e81ccf191634, make climbed-48-verify independently replays:

• Both changed scalar graphs, dense support audits, pinned matching admissibility and independent recounts.
• Complete physical timelines and every dirty basis vector in both orientations; zero CRT disagreements.
• The exact moment, all 47 strict constraints and seven margins; next bit-saving and kappa grid points rejected.
• Exact lower-moment exclusions for the retained PR Both fixed bases and copied reversed corners: conditional κ = 3.918734894e-5 > 2^-15 #40–Reordered exclusion sums: conditional κ = 4.11862541e-5 #48 comparison networks.
• Five focused tests, including frozen certificate equality and injective matching controls.

The handoff also reports reproduction of #48's certificate and the zero-swap DAG identities in its cloud environment. Full repository verification has now passed as recorded above. Finite checks do not establish the inherited general transfer and analytic interfaces; those still require mathematical review.

Attribution

Chafik Boukhalfa (#43/#46/#48), RaD/hipotures (#41), icekylinx, James Chang, Dominik Scholz, Zhihao Chen, Aurel Prosz/Paureel, Swapnil Jain, eumemic, Douglas Colkitt, OpenAI, Harvey–van der Hoeven and all retained predecessors.

Prepared by Rohan Arun with Anthropic Claude assistance. OpenAI Codex performed the independent local focused replay and publication checks. The branch includes unmerged #48 ancestry; review this increment against b7194b8586844956904b33e9fe18f0c75606c11b.