Paid clones on skip-prefix strips: conditional κ = 4.529040672e-5

[chafreaky]

This improves the conditional exponent from PR #53 by 0.686876%, and from PR #36 by 17.769260%:

$$ T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{141532521}{3125000000000}=4.529040672\times10^{-5}>2^{-15}. $$

The result remains conditional on OpenAI's base theorem and the inherited analytic and fixed finite-alphabet multitape interfaces.

Construction

Starting from PR #53's skip-prefix graphs, add 253 whole-chain clones at h=23 and 353 at h=25. Each added gate recomputes an existing scalar sum from disjoint sources and replaces one complete continuation chain. Two previously unused carrier capacities retain its inputs. All 606 added gates and 1,212 added carrier links are included in the cost.

Quantity	PR #53	This PR
Roles at h=23	32,946	32,693
Roles at h=25	43,409	43,056
Physical wires W	160,799,739	159,592,676

Every clone retains its parent's original core/cover envelope. The common I+J bases, all 4,073,300 data pairs, copied centers, paid endpoint corrections, complex network and assembly interfaces remain unchanged. The exact bit saving is 1132311451/25000000000000.

This branch builds on #53 at 3ffd4021995c959ac02d12920e0279ae97dd03c7. The new implementation and proof are in research/skip-clones/.

Verification

Completed on the published construction:

• Exact scalar identities, dense global supports and independent recount of every selected carrier use.
• Fresh fixed-profile CRT replay and complete dirty-scratch basis checks in both orientations.
• Exact rational moment bounds, next-grid rejection, 47 strict assembly constraints and seven positive margins.
• 11 focused tests and 65 adversarial rejection cases, including malformed partitions, partial chains, provider collisions, invalid counts and stale source pins.

Latest main (d55bcd6) is integrated at 1d1bb12. Fresh make -j1 verify-research verify-skip verify-tests passed: 28 recovered research audits, complete PR #53/#54 producer replays, 20 + 11 focused tests, 450 integrated tests, and 20 historical patch checks. Regeneration was byte-clean. All 152 upstream-added files, all 239 pinned source entries, and the numerical certificate are unchanged. Both the new upstream research CI group and the existing skip CI group remain, across Python 3.11/3.13/3.14, with both formal jobs preserved. The integration receipt distinguishes these fresh scoped checks from prior complete runs.

The earlier full make verify passed after integrating main 64ed4cb, at tested merge 03aff8b: 277 integrated tests, all focused and complete producer replays, and 20 historical patch checks, with byte-identical regeneration. An independent review confirmed preserved main checks, credits, and numerical certificates. That full-run record remains intact; no fresh full-suite or local Lean result is claimed for the latest integration. All 23 GitHub CI jobs passed on 1d1bb12 (run): all seven Python groups on three versions, plus both inherited formal packages.

The original construction also passed its full suite at 7210de7 (218 integrated tests and 18 patch checks).

make skip-clones-verify
make verify

The local proof states the argument and the inherited obligations requiring expert review. This is not a Lean proof or a claim of global optimality.

Attribution

• Avi Eisenberg / ikeboy, with Anthropic Claude assistance (Skip-prefix strips: conditional κ = 4498144/10^11 > 2^-15 #53): skip-prefix strips.
• RaD / hipotures, with OpenAI Codex assistance (Enlarged positive-frame networks: conditional kappa = 4.171211781e-5 > 2^-15 #51): source-partition and whole-chain cloning.
• Chafik Boukhalfa, with OpenAI Codex assistance (Changed graphs and fixed bases: conditional κ = 4.09953e-5 > 2^-15 #43, Weighted matching and exact data recovery: conditional κ = 4.10062945e-5 #46, Reordered exclusion sums: conditional κ = 4.11862541e-5 #48 and this PR): original-envelope specialization, composition and independent checks.
• Rohan Arun, with Anthropic Claude assistance (Weighted carrier matching: conditional κ = 4.100629254e-5 > 2^-15 #44, Climbed orders on reordered exclusion sums for a conditional κ = 4.123863984e-5 bound #49): weighted matching and certificate-scaffold components.

All inherited notices and credits remain, including icekylinx, Dominik Scholz, James Chang, Zhihao Chen, Paureel, Swapnil Jain, eumemic, Douglas Colkitt, OpenAI, David Harvey and Joris van der Hoeven.