Balanced-column carry exchanges: conditional κ = 5.1915547765237e-5

[DominikScholz]

This PR composes balanced coarse columns with envelope ordering and carried-signal exchanges to obtain the conditional bound

$$ T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{51915547765237}{10^{18}}=5.1915547765237\times10^{-5}. $$

Against the updated PR #74 at 5830ccfca8466aff613767222558e29b57fa8225, whose reported κ is 10366199626713/(2*10^17), the exact gain is 10568703959/125000000000000000, or 0.163125609609%. Against PR #76 the gain is 5527241253/31250000000000000, or 0.3418559067%. These compare pinned conditional claims, not globally optimal bounds. Bit saving is 3244890195589/62500000000000000. All inherited theorem hypotheses remain conditional.

Change

Replace each coarse two-group total by a balanced sum of column sums, using the anchored PR #69/#77 graph adapter. Compose that graph with the [1,1,2] anchored split, PR #74 envelope-node ordering, cover-core/reverse-node schedules, and three-pass carried-signal exchanges from PR #76.

Every summand remains present exactly once with disjoint support; support interning can reuse column sums in interval strips. After node renumbering, invalidate the adapter's node-ID support cache and verify again. An independent dense support audit checks the scalar/data-output identities without that cache. Frame containment, dirty wrappers, next-use scoring, literal XORs and all paid transitions remain checked. PR #77's separate cost-ordered clearing compiler is not imported.

Quantity	PR #76	This PR
h=23 auxiliary roles	27,256	27,128
h=25 auxiliary roles	35,720	35,598
Physical wires W	134,095,520	133,585,058
Bit saving	25870676537201/(5*10^17)	3244890195589/62500000000000000
Conditional κ	51738676045141/10^18	51915547765237/10^18
Assembly backoff	10^-12	unchanged

The complete paid recursive rank is 76,809,561,450, with deficit 1,846,900. Endpoint copies, promotions, clearing controls, data growth and the inherited complex layer remain charged. Width-derived bridge values are wire-bit logarithm 27, row coefficient 843, degree 2000, and degree gap 7007/25.

Verification

Ready for review following the completed focused finite checks below. Deterministic compiler regeneration and broad repository/CI review remain separate pending checks.

Passed: independent dense scalar/data-output audit; complete compiler input/dirty bases in both orientations; independent saved-word replay; exact frame/event reconstruction; regenerated complete fixed-basis profiles with zero CRT disagreements; independent rational moment bounds; all 47 strict assembly constraints and seven margins; and adjacent bit/κ-grid rejection.

Pending: fresh portable deterministic compiler regeneration, broad repository checks and CI review. This PR changes no CI configuration, workflow, matrix, gate, Makefile target or verification dependency. Finite checks do not prove the inherited all-size compiler, fixed finite-alphabet tape, analytic/semantic reduction, routing, prime-selection, precision or recovery interfaces. No unconditional multiplication theorem, global optimality or practical speedup is claimed.

The copied package independently passed the dense scalar audit and complete saved-word/profile/arithmetic validation after publication. Its current package receipt matches the original results exactly except the updated documentation-manifest binding.

The original validation receipt binds the frozen source/artifact manifest retained in provenance/frozen-manifest.json. All runtime sources, physical words, profiles and arithmetic are byte-identical to that checked freeze. Only proof/comparison metadata is updated, explicitly recorded by the current manifest.

git fetch https://github.com/CrocSwap/integer-mult-bounds.git 1bef94fd40a746452548c84a4a8f8834670a3113
mkdir -p research/round6-pr71/baseline
git archive 1bef94fd40a746452548c84a4a8f8834670a3113 | tar -x -C research/round6-pr71/baseline
python3 research/round9-balanced-columns/dense_audit.py
python3 research/round9-balanced-columns/validate_selected.py --bundle research/round9-balanced-columns/selected-both --output /tmp/balanced-columns-validation.json

Assert-enabled Python and C++17 are required; CXX may specify a compiler/SDK. The proof provides sequential regeneration commands.

Commit: acae4964aef41c69f3c57e72a45e76afc4d4934d. Proof and scope, Exact certificate, Completed finite receipt, Dense scalar audit, Source and word manifest, Original checked freeze, Exact comparison.

Attribution

eumemic's PR #69 supplies balanced coarse columns; huxint's PR #77 at 83298467291d3bd4b53f76faaef60dbf585be171 supplies the anchored column adapter; Thomas DiFiore's original PR #74 at 9ee3e9e1d451cd62e7d55a3c66e0280304dcb653 supplies envelope ordering. Chafik Boukhalfa's PR #71 supplies anchored splits and next-use scoring; Alejandro Zarzuelo Urdiales's PR #70 supplies exchanges; Rohan Arun's PR #65/#67 supplies schedules, profile costs and arithmetic; Dominik Scholz's PR #68/#76 supplies pending controls and the preceding composition. Avi Eisenberg's PR #62, eumemic's PR #57 and the complete predecessor chain retain credit. This composition was prepared by Dominik Scholz with substantial OpenAI GPT-6 Astra assistance. All original contributor notices and Apache-2.0 terms remain preserved.

[Mister-Meeseeks]

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

Contribution: Bounded column-carry variants of the same construction.

Review and limits: Fresh dense audit and selected-word validation against an exact #71 baseline; portable regenerated evidence distinguished from the original claim.

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: acae4964aef41c69f3c57e72a45e76afc4d4934d. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.