Coarse-threshold and coordinate refinement: conditional κ = 5.2062354654791e-5

[huxint]

This updates the selected witness with a shifted coarse-cell boundary, compatible two-region carry exchanges, and a cyclic h25 coordinate flag used during cost pricing and in the final word.

κ = 52062354654791 / 10^18 = 5.2062354654791e-5, with strict bit saving 26032532642421 / (5×10^17).

The complete pinned PR82 profile at 3410b940aa26e5876202152dfa7c4f66451ee22c is excluded at the new bit saving. The exponent improvement over that profile is 0.0568021906%. Newer community submissions report larger values; this is a scoped historical comparison, not a latest-record or global-optimality claim. The initial PR77 witness remains available at 83298467291d3bd4b53f76faaef60dbf585be171.

Paid quantity	Pinned PR82	Selected refinement
h23 auxiliary roles	27,075	27,043
h25 auxiliary roles	35,500	35,505
Physical width W	133,289,600	133,224,855
Complete recursive rank mass	76,639,673,100	76,602,444,725
Rank deficit	1,846,900	1,846,900

Selected choices

• Both axes retain the anchored [1,1,2] split, envelope ordering, future-compatible unit completion, and the inherited reversible frame compiler.
• At h23, coarse cells use columns when i+j < ng+1 and rows otherwise: a two-position shift of the previous half-plane boundary. At h25, all coarse cells use rows. Both forms evaluate the same disjoint edge sum.
• Three passes of improving carry exchanges include 113 legal two-region cycles at h23 and 86 at h25. Each preserves cardinality, unique uses, eligible dependencies and the regional row-independence constraints. Open two-region paths were searched but none was selected.
• h23 uses the preserved PR82 final flag and original-coordinate oracle pricing. h25 uses i -> (i+1) mod 25 in both oracle pricing and the final word, with cost-ordered live and retired dependency candidates.
• The actual W requires 27 wire bits. The bridge derives coefficient 843 and degree gap 7007/25, retaining degree 2000 and suffix slope 8000. Stale width fields are rejected.

Every operand promotion, retained-center copy, source growth, endpoint correction and cleanup remains charged. The graph/compiler choices are discovery heuristics; acceptance uses the entire resulting physical word and recursive profile. All source triples, terminal descriptors, masks and scatter targets are relabeled together. Scatter multiplicities are checked exactly, including rejection of canceling but uncharged additional gates.

Material and validation

The separate research/balanced-split-reclamation/ package contains the portable generator, frozen sources, both words, actual profiles, exact certificates and a six-page paper. Historical repository files remain unchanged. A dedicated CI workflow is included.

Already reproduced on the refined construction: both raw words byte-for-byte; every source, target and arbitrary dirty basis vector in both orientations; physical transitions reconstructed from literal incidences; complete bounded-minor/CRT profiles with zero disagreements; full paid rank identity; independently evaluated rational moment enclosures; the complete PR82 exclusion; all 47 strict assembly inequalities and seven margins; adjacent bit/kappa grid rejection; and 17 failure-mode tests. All 58 source/artifact pins match both the working files and committed Git blobs. The PDF was compiled twice with resolved references and no overfull boxes, and every rendered page was inspected.

The inherited broad verification groups completed on the original submission tree. The initial make -j1 verify reached the positive-frame group and stopped because this host lacked Boost headers. After a checksum-verified task-local Boost installation and removal of stale temporary build output, verify-positive, verify-joint, verify-pair, and verify-tests passed sequentially: 552 test methods in 81 modules and 20 historical patch checks. This is an interrupted-and-resumed verification, not a claim of one uninterrupted run. On refined construction f277632, the combined focused make target passed all 17 tests and byte-for-byte regeneration, and a fresh root integration run passed 516 tests in 78 modules plus all 20 patch checks. Receipt-only commit 17c979e records these results without changing the source freeze. Complete provenance is in VALIDATION.json; GitHub CI status is separate from these local receipts.

make -C research/balanced-split-reclamation verify
make -j1 verify

Proof PDF · LaTeX source

Scope and credit

The base theorem, all-size residual/frame compiler, finite-alphabet tape model, analytic/semantic transfer, routing, prime setup and exact recovery remain inherited hypotheses. This is finite conditional mathematical work, not a full formal multiplication theorem, independent human expert review, or a practical benchmark.

Prepared for huxint with substantial OpenAI Codex assistance. Retain credit to eumemic (PR57/69), Avi Eisenberg (PR62), Rohan Garg (PR59), Rohan Arun (PR65/67/78/82), Dominik Scholz (PR63/68/76), Chafik Boukhalfa (PR60/71/79), Thomas DiFiore (PR74), Alejandro Zarzuelo Urdiales (PR70), and all inherited contributors and licenses. General carry exchanges, coordinate conjugacy and the inherited graph/compiler methods retain prior-work status. This submission supplies its explicit coarse/coordinate/cycle choices, complete reproduction and exact comparison.

[Mister-Meeseeks]

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

Contribution: Balanced splitting, carry cycles and coordinate flags.

Review and limits: Fresh rebuild and tests of the dedicated package; complete W and finite bridge are charged.

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: 17c979eb6dcd4c300e6263c66111f220a9e926e7. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.