Copied centers with fixed local bases for conditional κ = 3.886224e-5

[DominikScholz]

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

T(n) = O(n (log n)^(1 − κ)), with κ = 242889/6250000000 = 3.886224 × 10⁻⁵.

This composes icekylinx's copied retained-center schedule (#36, 11817ccacb564bb7f98789c20dc11d3fece207e3) with the original-envelope, fixed-I+J method from #35 (9c345a2a11e5f4f3649f7c68214bf9a2a0a3fe9c) on the actual #36 base-two bit graphs at (25,23). It improves #36 by 1.0540111%, and the latest #37 (2f7578affce416ad4b6c41f3438ebb734f66a899, κ = 3.850771033e-5) by 0.9206719%. These comparisons concern conditional exponent savings, not measured runtime.

Construction and exact result

Both local bases are fixed to I+J, with the original envelope labels and original oriented carrier matching. Complete local transition profiles replace the generic local profiles. The conservative data profile is 26 singletons + [21,481]; the generic second 15-block is not assumed to survive. All 23 actual controlled permutations and both 47-edge incidence trees establish compatibility in the same physical order.

The copied-center transform is retained and paid in both invocation orientations. Only each retained total's full-rank cleanup is replaced by its rank-one complement. All 4,073,300 paid endpoint-copy corrections, auxiliary exteriors, complete role volumes, controls, spectators, arbitrary scratch restoration and sequential fixed-tape workspace remain. Total physical rank is 108,202,762,275 = Wm − N + L.

The exact bit saving is 19432631/500000000000 = 3.8865262e-5. The inherited #36 complex saving is 717/10^7; its complete moment is replayed. The semantic envelope retains its literal scalar gate charge, C1=1 and product row stock p^2000. Both exact moments, 47 strict assembly conditions and all seven margins pass. The bit moment gap exceeds 6.3667e-15, and the final absorption gap exceeds 5.1804e-14.

Validation before filing

• Rebuilt both actual scalar DAGs, every original label inclusion, the original oriented matching and every local transition profile at h=23 and h=25; complete records match the checked-in fixtures.
• Dimension-specific minor bounds with Lucas–Lehmer-certified primes: rank-two corrections use one prime; source growth and core changes use three primes. Exact zeros are certified by CRT bounds, not sampled ranks.
• Exact certificate, complete physical accounting, both moments, actual permutations/trees and all assembly inequalities pass. Independent parent review reconstructed the physical child list and evaluated a separate rational moment enclosure.
• Four negative controls pass: missing tree edge, omitted copied schedule, omitted data block and the next final-κ grid point. The next sufficient bit-moment enclosure grid point also fails; this does not establish global optimality.
• Generated proof patch applies to pinned Copied retained centers and two-stage assembly for conditional kappa = 3.84569e-5 #36; Python syntax and diff whitespace checks pass. No PDF or large binary is included.

Validation complete: the full inherited make verify rerun passed at cc794077f6c103e24ec0939be765cd1521239aab: 170 regression tests and 19 proof-patch checks, including both selected producer rebuilds. Regenerated certificate data are unchanged; one inherited JSON key-order difference was checked for parsed equality and its committed serialization retained. The detailed receipt is research/copied-fixed-basis/validation.json. The finite checks support the written conditional proof; no formal verification or external mathematical acceptance is claimed.

Start with the proof and reproduction guide, proof, and exact certificate. Run make copied-fixed-check, make copied-fixed-producer, and make verify. The original analytic, routing, recovery, prime-existence, constructive setup and eventual-threshold hypotheses remain explicit.

Attribution

Dominik Scholz, with substantial OpenAI GPT-6 Astra/Codex assistance, supplies this composition and integration. Credit icekylinx (#36 copied centers/complex/assembly and #32 fixed projectors/profiler, plus earlier producers), Dominik Scholz (#35 fixed-basis exactness and #33 dimension adaptation), Zhihao Chen (#29/#21/#23), Rohan Arun (#31 corner and rational-moment methods), Aurel Prosz/Paureel, Swapnil Jain, RaD/hipotures, eumemic, Douglas Colkitt, OpenAI, Harvey–van der Hoeven and all retained notices. The imported helper files and proof dependencies have exact commit/SHA-256 pins in research/copied-fixed-basis/SOURCE.json. Original Apache-2.0/CC0 licenses and historical AI-assistance disclosures remain; inherited techniques are not claimed as new here.

[Mister-Meeseeks]

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

Contribution: Copied fixed-basis rank-one cleanup.

Review and limits: Fresh complete certificate driver; every paid center copy and data endpoint remains in the ledger.

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: 605323bab4ba330de3d1785dc5b3ecb14e8ab747. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.