Balanced coarse sums: conditional κ = 5.136837484e-5

[eumemic]

Conditional exponent saving: κ = 5.136837484e-5 = 1284209371/25000000000000.

This checked construction is now superseded by later frontier candidates; the comparisons below record its context at submission and do not claim the current lead.

The interval-strip producer currently builds each four-edge coarse sum as a chain. Reassociate it into two column pair sums and their total, so existing interning shares those intermediates with strip computations. With the inherited joint frame compiler and descending-rank reclamation, this gives the conditional witness κ = 5136837484/10¹⁴ > 2⁻¹⁵ (exactly 1284209371/25000000000000).

The bound is 0.6530% above #68's reported candidate (5.1035127217697e-5) and 0.6679% above #64. These comparisons are to the reported bounds checked at submission, not a global optimality claim. The exact bit saving is 513710137/10^13.

Quantity	Rank-first interval baseline (#63/#64)	Balanced coarse columns
Auxiliary roles, h=23	27,918	27,698
Auxiliary roles, h=25	36,586	36,300
Physical wires W	137,151,806	136,139,300
Total child rank	78,860,441,550	78,278,250,600

The deficit remains 1,846,900 and maximum child width remains 529. Every copied center, endpoint correction, clearing XOR, and inherited data and complex contribution remains charged. The scalar change preserves disjoint additions and the original common-point envelopes. Physical words and actual contiguous fixed-I+J profiles are regenerated from the changed graph; a lower role count alone is not used to infer κ.

Passed on construction 91aa1f17e6e3fc063241686a41a34ddd0dc24c50, based on current main 230b5ea:

• Independent dense global scalar coefficients, complete output inventory, and every core/cover.
• Both freshly generated words reproduce byte for byte; full input/target/dirty basis replay in both orientations and every physical frame incidence pass.
• Actual fixed-basis transition profiles and CRT controls have zero disagreements.
• Exact rational recurrence, exclusion of the next bit-saving grid point, all 47 strict assembly inequalities, seven margins, and eventual cutoffs pass.
• The complete pinned predecessor network is excluded at the new bit saving.
• make balanced-coarse-verify passes in the clean current-main worktree; a dedicated CI group is added without removing existing checks.

Composite full local verification passes. The initial make verify run and the resumed remaining targets together cover every verification group, including all 78 isolated test modules and all 20 patch-application checks. The macOS OpenMP issue was resolved with the installed libomp runtime and a temporary compiler adapter; the positive-skip exact profiles were checked using checksum-verified official Boost 1.88 headers staged outside the repository. The final positive-skip build used a fresh temporary directory because its generator intentionally refuses to overwrite a previous build.

Commit 6817afe refreshes only the inherited joint-dual manifest's Makefile/NOTICE hashes and the two derived certificate hashes. Mathematical content, physical words and profiles are unchanged. The complete joint verification was rerun successfully, including fresh word regeneration, exact profiles, all 47 assembly conditions, seven margins, ten tests and the independent PR60/61 arithmetic audit. Positive-skip's exact profiles, certificate and four tests also pass.

Before this metadata fix, GitHub CI had 36 passing checks and only the three joint jobs failing on the stale Makefile pin. CI for updated head 6817afe is complete: all 39 checks pass, including the three joint jobs. This PR remains a draft.

Reproduce with make balanced-coarse-verify. Start with the concise README, exact certificate, and independent scalar audit. The package also retains an independently replayed rank-first baseline with pinned sources for comparison.

This remains conditional on the inherited base theorem, all-size framed-word/compiler transfer, fixed finite-alphabet tape layout, analytic, routing, setup, precision and recovery interfaces. It is a finite checked construction, not a complete formalization or measured runtime improvement. The unchanged data geometry and its ten rational recoveries are inherited; the full repository run exercises the retained checks.

The new coarse-column reassociation and integration were prepared by eumemic with OpenAI Codex assistance. The surrounding graph is Avi Eisenberg / ikeboy's #62 (Anthropic Claude assistance), joint frame compilation is eumemic's #57, and rank-first reclamation is Chafik Boukhalfa's #60 (OpenAI Codex assistance). Dominik Scholz's #63 and rfu08's #64 provide the concurrent composition and verification context. All inherited contributor credits, AI attributions, licenses and notices are retained.