Certify extended paired cubes and bit frames (κ=5.94272360424954e-4)

[chafreaky]

Change and mathematical scope

This conditional finite witness reaches

$$ T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{297136180212477}{5\cdot10^{17}}=0.000594272360424954. $$

It composes same-singleton broadcast fusion with PR162's extended carrier closure and physical frame/reuse construction, replaces 1,704 rational operation frames in PR161's p12 bit word, pays the newly optimized atom toll, and applies the retained paid balanced positional transfer. This improves our verified PR163 saving by approximately 0.0509%.

The exact PR162 prerequisite at 1e8140fbb56e276bf80e5ec0621f6faa2b0c1f88, including PR161 at d14e29157bc905be1ced0776dd893d0714013f3a, is carried over main d1d6c070f5a8c684727ee7ec35d930f9ebfa9758. Those prerequisite changes are credited. The new package is research/paired-cube-plateau-162/ and its Python 3.11/3.13/3.14 CI workflow. Existing verification groups and the inherited PR162 checks are preserved.

• Complex: 12,995 physical auxiliaries, 15,635 total persistent roles, 2,640 compensated handoffs, 6,907 carrier arcs and 33,700 signed mixer operations. The scalar bill retains all 15,635 pre-reuse roles. Maximum child 20; deficit 1,320.
• Bit: 23,368 auxiliaries and 26,888 persistent roles. The scalar word, gauges and role stock remain unchanged. Exactly 1,440 operation frames shrink and 264 enlarge; every replacement satisfies value containment, nondegeneracy and both chronological role chains. The exact replay retains early center reads and partner deliveries immediately after their specified roots.
• Prime compatibility: all 1,176 distinct new integer Gram determinants factor into powers of 2, 3 and 5. The verifier checks each factor identity and a positive residual below 2^80; every residual is one. Some determinants reach 103 bits, so the proof uses factorization rather than a bound on the whole determinant. Every retained prime remains valid with the same bad-class allowance.
• Exact suppliers: complex 119103190497/(2*10^14); bit worst-case coarse envelope 118991327467969/(2*10^17); paid atom exponent 297312865030563829533/(5*10^23).

The full finite group, scalar, routing, denominator and restored-row charges are recomputed from the selected word. All 47 strict inequalities and seven margins remain. The general all-size Clifford/tensor, uniform weighted local-ring compilation, restored-row, paid-layout, recovery and analytic/fixed-tape interfaces remain source-specified assumptions. This is not an unconditional multiplication theorem or a practical runtime claim.

Credits: @icekylinx (#144/#130); @eumemic (#117/#155/#157/#161 and #131/#143 frame/reuse lineage); @DaysSky (#162 extended carrier closure and exact full-intersection compiler); @GamingPuzzled (Joel Pulikkan, #160 searched-bit-module composition); @jamesyc (#124 birth-read reuse and #34 balanced assembly); @an664 (#128 completed-core sharing); @gupt1156 (#148) and @geckods (Abhinav Ramachandran, #158) for paid atom-toll re-instantiation; @jacklightChen (Zhihao Chen, #23/#29/#97) and Swapnil Jain for retained semantic/bit lineage; @rohanarun (#100/#103) for paid positional layout; and our #141/#163 for the retained arithmetic and complete chronological replay. @eumemic's #154 isolated snapshot technique supplies the full native gate. Detailed lineage remains in NOTICE and the prerequisite sources. Prepared by Chafik Boukhalfa with substantial OpenAI Codex assistance.

Validation

Complete verification: PASS on head 7518fed2688baf25c7c32bae32674f3334b517da. All 14 native verification groups plus the selected finite package passed in 282.080 seconds (4 minutes 42 seconds), with exact snapshot/published-tree equality and unchanged input bytes. All 50/50 GitHub checks passed on the same head, including both inherited PR162 checks. The PR is ready for review.

• Authoring certificate followed by the complete read-only verify.py: both exact physical words, every formal source/target/dirty column, frame containment and chronology, complete rank recounts, paid moments, full bridge, 47 strict inequalities, seven margins and 19 assembly mutation controls.
• Independent full-group/scalar/row rebind, all 47 inequalities and seven margins, both complete-profile moment intervals and 18 adverse controls. The prior atom is insufficient for the new supplier; the newly selected atom has strictly positive toll gap, and its preceding 10^-24 grid point fails.
• Independent exact audit of all 1,704 replacement frames, 5,688 affected chain edges and all 1,176 Gram factorizations.
• Source closure covering 1,694 files; git diff --check.

Read-only selected verification:

python3 -B research/paired-cube-plateau-162/verify.py

The full native gate schedules all 14 make verify-* groups plus this selected package in separate clones of one frozen source snapshot, checks exact recipe coverage and before/after source hashes, and limits nested jobs to one:

python3 -B research/paired-cube-plateau-162/verify_parallel.py --output /new/path/verification --jobs 8

The two inherited PR162 extended-links checks are outside the native Makefile recipe and are additionally required through their preserved remote CI workflow. The complete 50-check remote inventory has passed.

The next bit grid exclusion concerns the fixed worst-case bad-class envelope, not the unknown true bad fraction. The next final kappa grid point is rejected at the selected certified supplies and parameters; no global optimum is claimed.

[Mister-Meeseeks]

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

Contribution: Extended carrier links and bidirectional bit-frame refinement.

Review and limits: Fresh full finite verifier passed. Larger determinants are admitted by their factor witnesses, not by claiming every determinant is below the prime threshold.

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: 7518fed2688baf25c7c32bae32674f3334b517da. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.