Executed terminal accumulation and bit reuse: conditional κ = 6.16027622e-4

[huxint]

Change and mathematical scope

This composes our #170 bit-output fusion and compensated lifetime reuse with #168's newer modules, refines the complex frames, and integrates #166's terminal compiler into an actual all-column-checked executor. The resulting conditional bound is

$$T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad \kappa=\frac{308013811}{500000000000}=6.16027622\times10^{-4}.$$

This is +0.4988% over #173's advertised 6.129700e-4, and +3.219% over #170. Concurrent #186 reports 0.000661885549259598, which is higher; this package is a separately checked contribution, not a current-record claim. These compare exponent savings, not measured runtimes. Stacked on #169 at 4895ba1104cedc7908b6d6ba64fdd6290e0ed7ae; #168's words/modules/arcs and the inherited assembly are prerequisites.

Supplier	Physical auxiliary roles	W	Certified local saving
Complex, with 66 target-accumulated terminals	12,135	14,775	0.000616407346
Bit, fused outputs and 1,760 compensated reuses	18,732	22,252	0.000646746861 coarse

The complex side binds. Each removed terminal preserves the independently valued targets and eliminates three width-18 and three width-4 children. The actual new executor retains the original dirty-response columns of every surviving role, executes the target pre/post shears and accumulation writes, and restores every remaining dirty coordinate. Its coefficient bound is recomputed through this changed word before the exact packed rational replay.

The bit construction is regenerated from frozen arcs; every operation/frame binding, aliased birth, and actual target-read chronology is checked. All 25,849 original and replacement rational bit frames receive exact Gram/factor witnesses, so every retained prime q > 2^80 remains valid. Discovery-time frame IDs were explicitly matched by a complete frame-record bijection to the frozen compiler's IDs before binding the release inputs.

The inherited all-size Clifford/shared-core, weighted local-ring, restored-row, analytic recovery, balanced-layout and fixed-tape interfaces remain assumptions. No unconditional theorem, global optimum or practical speedup is claimed. The main selected certificate is not replaced; this is an additive research package.

Validation

python3 -B research/paired-cube-terminal-reuse/verify.py

• Complete source reconstruction and certificate generation passed locally (198 s, standard-library Python).
• Both complex shear directions check every rational source, target and physical dirty column; the bit word checks every F2 column and the defining integer decoder.
• Missing compensation, stale reuse, missing partner delivery, omitted target pre/post shears and terminal writes are rejected; geometric mutation controls remain active.
• Both paid supplier moments, the full finite bridge, all 47 strict constraints and seven margins, and the next supplier/final grid-point rejections pass. Full group, original logical scalar stock, routing, common-grid precision, bit fallback and row reserves remain charged.
• The complete package verifier passed on a clean GitHub-hosted Python 3.13 checkout: https://github.com/huxint/integer-mult-bounds/actions/runs/37927509885. make paired-cube-verify also passed locally.
• Python 3.12 exposed only a gzip OS-header mismatch in temporary input hashes; the header is now normalized and all seven payloads were checked byte-for-byte against the Python 3.13+ form. The final Python 3.12 clean read-only reproduction passed (227 s), and the final Python 3.13 CI also passed: https://github.com/huxint/integer-mult-bounds/actions/runs/37928449316. The full historical make verify suite also passed in a separate checkout with Boost headers supplied locally.

Proof, source pins and receipts: research/paired-cube-terminal-reuse/{PROOF.md,SOURCE.json,certificate.json}.

Credits

The terminal identity/checker are @jamesyc's #166, following the Dumas–Grenet accumulation literature; the compiler identity is not claimed as new here. This adds its transformed executor, a new formal bound/replay, instantiation on #168 and complete supplier/assembly integration. The retained rational-column checker is @chafreaky's #163–#165 lineage as corrected/adapted in @jon314159's #171; @GamingPuzzled's #169 supplies the bit-column and prime routines. Original source files/notices are retained.

@icekylinx and contributors supply the paired-cube and shared-core framework; @eumemic supplies #168's modules, words and physical layer; @DaysSky supplies closure compilation; @an664 supplies core sharing; Zhihao Chen, Swapnil Jain, the RaD project and the paid-atom/phase-stop contributors retain their upstream credits. Full attribution is in NOTICE.

Prepared by huxint with substantial OpenAI Codex assistance. Apache-2.0.

[Mister-Meeseeks]

Thank you @huxint. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Terminal accumulator executed on every required column.

Review and limits: Exact-head Linux full-word/CRT/assembly CI passed. Local macOS stopped at RLIMIT_AS setup; record that portability failure, not a mathematical failure or fresh Linux replay.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: c6eea3e8a46bf122878b512590f9f91a20260800. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.