Fixed local bases in two stages for conditional κ = 1.6631776e-5

[DominikScholz]

Result

Under the retained analytic and fixed finite-alphabet multitape hypotheses,

T(n) = O(n (log n)^(1 − κ)), with κ = 16631776/10^12 = 1.6631776 × 10^-5.

This is a separate fixed-basis construction built on #33, preserving that PR and its generic-basis witness. Relative to #34 at 7fecbe3651e095fb0f450756afdeefd2a9ee80a2 (κ = 1.639226629e-5), it increases the exponent saving by approximately 1.46%. This comparison concerns κ, not a measured runtime improvement. Subsequent result: #36 by icekylinx at 11817ccacb564bb7f98789c20dc11d3fece207e3 now claims κ = 3.84569e-5 through copied retained centers. This PR preserves the separately reusable fixed-basis construction and evidence; it is no longer the leading bound.

Construction

Fix both local rational bases to I+J at dimensions (47,45) and use the original envelope labels and original oriented carrier matchings. Adapt icekylinx's #32 projector formulas and physical-transition profiler to these two factors. Dimension-specific bounded-minor inequalities certify the entire pivot profile exactly: rank-two corrections use one prime, and both rank-three source growth and rank-four core changes use three primes.

Two bipartite-tree arguments prove that the full data corners remain nonsingular in this fixed specialization. The first 43-block survives for every nonzero primal/dual coordinate choice, giving the conservative data profile 48 singletons +43+1933. The additional 37-block from #33 is not claimed here.

All physical roles and the paid rank-one copy correction remain. The combined rank is 24767869848810, with m=2115, W=11710626180, and deficit 104521890. The complete local histograms replace the earlier generic internal profiles without changing these totals.

Exact assembly

Quantity	Rational choice / certified margin
Bit saving	1663233/10^11
Complex saving	18/10^6
β	1/20
Headroom	1/10^12
Final κ	16631776/10^12
Bit moment gap	> 3.14e-14
Final absorption gap	> 7.49e-13

The semantic C1=1 guard, product stock degree 47000, arbitrary-scratch restoration, and inherited eventual analytic/setup/recovery thresholds remain. All 47 strict assembly conditions and seven margins pass. These finite certificates supplement a conditional written proof; they do not formally verify the full multiplication theorem.

Evidence and reproduction

• Complete proof: notes/fixed-basis-two-stage.tex.
• Exact witness: research/fixed-basis-two-stage/certificate.json.
• Full source reconstruction: make fixed-basis-two-stage-producer.
• Arithmetic and proof-patch check: make fixed-basis-two-stage-check.
• Repository checks: make verify.

Before filing, fresh reconstructions of both scalar DAGs, 663,929 original envelopes, 1,327,858 operand inclusions, complete original carrier records, and both full fixed-basis profiles passed. The two profiles cover 693,208 and 797,244 distinct matrices; all three-prime profiles agree. Independent mathematical review covered projector algebra, every transition case, minor bounds, tree restrictions, first-block identities and full accounting. The integrated exact certificate and generated proof patch also pass.

Validation complete: the integrated full make verify passed (183 regression tests, an additional 4-test targeted run, and 21 proof-patch checks), including fresh complete profile rebuilds at both dimensions. Regenerated tracked artifacts were unchanged. Independent Standards and Spec reviews are complete: a stale comment was corrected and standalone arithmetic-helper hashes are now enforced; corruption controls reject all three altered helper modules. No unresolved correctness or Spec findings remain. The detailed pinned record is research/fixed-basis-two-stage/validation.json. No PDF was generated.

Sources and attribution

• icekylinx, Structured projector blocks and bulk assembly for conditional kappa = 1.2523415e-5 #32, pinned at 0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52: fixed I+J projector formulas, transition profiler and bounded-minor exactness method; unchanged sources and hashes retained in references/fixed32/.
• Zhihao Chen (jacklightChen), Two-stage controlled corners for a conditional 15536/10^9 > 2^-16 bound #29, pinned at 9d963275075fa98f1da821e27757b238dafd6b3c: two-stage construction and finite interfaces; Aurel Prosz (Paureel) for the topology and paid copy correction; Swapnil Jain for linked two-stage development.
• Rohan Arun, Exact data-corner certificates for conditional κ = 1.5878574e-5 > 2^-16 #31, pinned at 02f68f95369c39955c5ffcfddbd75cd83e26b305: data-corner refinement and exact moment/assembly/cutoff modules, inherited through Compose smaller two-stage dimensions and data blocks for conditional κ = 1.638103206e-5 #33.
• Zhihao Chen, Translated partial-swap frames for a conditional 5499/10^9 > 2^-18 bound #21/Semantic precision and bulk resampling for a conditional 1099/10^8 > 2^-17 bound #23, RaD/hipotures, icekylinx's preceding producers, and every predecessor credited in NOTICE retain their attribution and licenses. Reversed two-stage basis and balanced assembly for conditional κ = 1.639226629e-5 #34 by jamesyc is the contemporaneous comparison, not an imported assembly dependency.

Dominik Scholz contributes the fixed-basis composition, dimension-specific exactness bounds, compatibility proof and regenerated evidence, with substantial OpenAI GPT-6 Astra/Codex assistance. The inherited dimension research also used Anthropic Claude Opus 5.5. Apache-2.0; original notices preserved.

[Mister-Meeseeks]

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

Contribution: Fixed I+J basis and a conservative 43-dimensional block.

Review and limits: Fresh own verifier and negative controls; no unsupported 37-dimensional 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: 9c345a2a11e5f4f3649f7c68214bf9a2a0a3fe9c. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.