Fused paired cubes and paid balanced transfer: conditional κ = 5.93970203079492e-4

[chafreaky]

Change and mathematical scope

This conditional finite witness reaches

$$ T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{593970203079492}{10^{18}}=0.000593970203079492. $$

It composes same-singleton broadcast fusion with PR161's physical complex frame descent/reuse, selects 1,296 improving operation-frame descents in its p12 bit word, refines the complete paid moments and atom toll, and applies the paid balanced positional transfer. This is approximately 1.04% above PR161's 0.0005878747.

The exact PR161 prerequisite at d14e29157bc905be1ced0776dd893d0714013f3a is carried over main d1d6c070f5a8c684727ee7ec35d930f9ebfa9758; those changes are not claimed here. The new package is research/paired-cube-balanced-161/ and its dedicated Python 3.11/3.13/3.14 CI workflow. Existing repository verification groups are preserved.

• Complex: 13,041 physical auxiliaries, 15,681 total persistent roles, 2,640 compensated handoffs, maximum child 20, deficit 1,320. The scalar bill conservatively retains 15,681 pre-reuse roles.
• Bit: 23,368 auxiliaries, 26,888 persistent roles, 1,296 changed frames, unchanged scalar word and gauges, maximum child 60, deficit 1,936. All 888 new integer Gram determinants are nonzero and below the retained prime lower bound 2^80.
• Exact suppliers: complex 297370240363723/(5*10^17); bit worst-case coarse envelope 37165860768041/62500000000000000; paid atom exponent 594323213359226694617/10^24.

The complete finite group, scalar, routing, denominator and row charges remain explicit. The source-normalized 47-constraint body retains every geometric and stopping inequality. 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 the #131/#143 physical-frame and reuse lineage); @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 atom-toll re-instantiation; @jacklightChen (Zhihao Chen, #23/#29/#97) and Swapnil Jain for the retained semantic/bit lineage; @rohanarun (#100/#103) for the paid positional layout; and our #141 for the retained exact arithmetic body. @eumemic's #154 isolated snapshot technique is used for the full gate, without adopting its unrelated source-pin/workflow changes. Detailed compiler lineage and assistance disclosures remain in NOTICE and the prerequisite sources. This contribution was prepared by Chafik Boukhalfa with substantial OpenAI Codex assistance.

Validation

Complete local repository gate: PASS on head e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997. All 14 native verification groups plus the new finite package passed in 280.83 seconds (4 minutes 41 seconds). Every group used an isolated snapshot with a tree identical to the published commit; input bytes remained unchanged before and after verification.

All 48/48 GitHub checks passed on the same head, including all three finite-certificate jobs (Python 3.11, 3.13 and 3.14). The complete local gate and remote CI checks are complete; the PR is ready for review.

Passed:

• python3 -B research/paired-cube-balanced-161/verify.py --write, followed by a complete source-pin self-check: both exact physical words, all formal source/target/dirty columns, frame chains and nondegeneracy, partner chronology, complete rank recounts, paid moments, full bridge, 47 strict inequalities, seven margins and 19 assembly mutation controls.
• Independent exact full-group/scalar/row and all-47 arithmetic audit, with 18 adverse controls.
• Independent exact Gram audit covering all 1,296 changed bit frames: 888 distinct bases, maximum determinant 73 bits and ambient determinant 77 bits. Every originally permitted prime remains valid for the new frames.
• git diff --check.

The selected source closure pins 1,678 files. The reproducible read-only command is:

python3 -B research/paired-cube-balanced-161/verify.py

The full gate uses all 14 native make verify-* groups plus the new package, each in an isolated clone of one frozen snapshot, with exact recipe coverage, before/after source hashes, eight group workers and JOBS=1:

python3 -B research/paired-cube-balanced-161/verify_parallel.py --output /new/path/verification --jobs 8

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 these 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: Fused singleton broadcasts and paid balanced transfer.

Review and limits: Fresh complete finite verifier passed, including both words, frames, moments, bridge and strict assembly. Analytic and all-size interfaces remain inherited.

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: e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.