Copied retained centers and two-stage assembly for conditional kappa = 3.84569e-5

[icekylinx]

This extends PR #32 at 0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 with one new commit, retaining the #10 → #18 → #24 → #32 ancestry. It also adopts the attributed two-stage and corner contributions from Paureel and PR #29/#31/#33, alongside the retained PR #21, PR #23 and RaD interfaces described below.

Under the retained multiplication framework and those analytic/tape interfaces,

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{384569}{10000000000}=3.84569\times10^{-5}. $$

Construction

The copied retained-center schedule supplies each scatter from a transformed copy while the original role proceeds directly to cleanup. In both invocation orientations, the charged ranks change from (r,h) to (r,h-r), preserving scalar actions and auxiliary endpoints. This reduces the two-stage rank budget from Wm-N+2L to Wm-N+L. The rank-r copy and the separate rank-one endpoint correction remain fully charged.

The bit network uses (25,23), retaining the verified producers, positive labels and compatible carrier assignment. The PR #31 corner method, parameterized in PR #33, gives the data profile 11 singletons; 21,15,481. Its 47 rational pivots and 315 weight-independent zero-minor certificates support the contiguous blocks in one controlled basis.

The complex network uses (28,28) with mixed dyadic centers and disjoint-center count 19. The copied-center schedule transfers to forward/inverse phase residuals. The two-stage endpoint uses a paid inverse phase on a copy and fixed sign corrections, with the translated auxiliary endpoints preserved.

The strict component savings are 384599/10000000000 (bit) and 717/10000000 (complex). Their certified moment gaps exceed 2.10899506e-12 and 8.52696172e-9, respectively.

The assembly adapts Zhihao Chen (jacklightChen)'s PR #23 composition of RaD (hipotures)'s completed-child semantic precision, arbitrary-coordinate routing, phase-cell Gaussian inverse and bulk resampling. Translated complex endpoints follow Zhihao Chen's PR #21. The actual scalar gate charge includes the copied centers and gives C1=1; nested adapters pay the product of both row stocks, bounded by p^2000. All 47 strict constraints and seven assembly margins hold, with final absorption gap greater than 4.0746e-11.

Contribution sources and prior PRs

My preceding contributions under icekylinx form the direct implementation lineage:

PR	Contribution	Conditional exponent saving
#10	Contiguous projector batching, simultaneous controlled bases, mixed-width recursion and whole complex residuals; corrected Gaussian input enclosure while retaining P=34p.	1.2299998e-7
#18	Partial-swap frames and removal of the source rank penalty; binary retained totals, compatible carrier reuse and positive frames; selected (25,23,57) basis.	1.884586e-6
#24	Complementary auxiliary endpoint gauges, selected (32,30,40) basis, general nondegenerate binary phase residuals and updated precision guard.	5.98615e-6
#32	A1/A5 block refinements, fixed I+J middle-axis profiles certified by bounded-minor CRT, mixed low-loss complex centers, complete data-front batching and instantiation of the attributed semantic/bulk assembly.	1.2523415e-5
This PR	Copied retained-center scheduling in both orientations, two-stage complex endpoint transfer, selected (25,23)/(28,28) networks and instantiation of the attributed semantic/bulk assembly.	3.84569e-5

The adopted contributions are distinct from these new finite refinements:

Source	Contributor	Role in this construction
Original multiplication manuscript and repository framework	OpenAI; Douglas Colkitt; underlying analytic work of David Harvey and Joris van der Hoeven	Multiplication reduction, analytic recovery and the fixed-tape framework.
#2, #4	Bortlesboat; dleen	Alignment, retained totals and stage sharing.
#3, #5, #6, #13	eumemic	Complex producers, Gaussian resampling, aligned circuits/centers and nonzero auxiliary source-frame context.
#7, #16	Zhihao Chen (jacklightChen)	Retained finite producers and nested-basis context.
#21	Zhihao Chen (jacklightChen)	Translated complex endpoints, used here with the new mixed-center network.
#20, pinned RaD sources	RaD project (hipotures)	Completed-child semantic precision, arbitrary-coordinate routing, phase-cell Gaussian inverse and bulk locality/tape transfer.
#23	Zhihao Chen (jacklightChen)	Compatibility with the retained framework, literal scalar charge, product row reserves and strict semantic/bulk parameterization, adapted here to the selected networks.
Two-stage source	Aurel Prosz (Paureel)	Two-stage topology and paid copied-stream endpoint correction.
Linked two-stage work	Swapnil Jain	Two-stage development credited by PR #29; no source code imported from this repository.
#29	Zhihao Chen (jacklightChen)	Unequal two-stage compatibility, common basis, gauges and physical accounting.
#31	Rohan Arun	Corner incidence method, symbolic rank partitions and exact rational pivots.
#33	Dominik Scholz	Parameterized corner generator, specialized here to (25,23).

Baker–Harman–Pintz supplies the prime-interval theorem used by the adopted packing argument. Historical AI-assistance disclosures are preserved. The table records construction dependencies and contribution scope; independently published exponent savings are not added together.

Proof and attribution

• notes/copied-centers-note.tex: selected construction and assembly, supplied as LaTeX source.
• scripts/copied_centers_producer.py: new producer and exact corner reconstruction.
• certificates/copied-centers-network.json: recursive moments, rank budgets, semantic precision, row accounting and assembly.
• references/copied-centers/ and references/semantic-bulk/: necessary pinned proof sources, original notices and licenses.

The result remains conditional on the retained analytic/tape interfaces, constructive setup and eventual thresholds. Finite checks do not formally verify the complete theorem. The inherited patches/batched-23.patch remains the #10 baseline; this extension is presented in its standalone proof.

The new copied-center schedule, complex transfer and selected composition are contributed by icekylinx, with substantial OpenAI GPT-6 Astra and Codex assistance. Original attribution is retained, including Paureel at c82d09eb4781b68435e3fa3eb6beea72fa900fab, PR #29 at 9d963275075fa98f1da821e27757b238dafd6b3c, the PR #31 method at 02f68f95369c39955c5ffcfddbd75cd83e26b305, and PR #33 at a499a345c040aa15b68e984a0904eadc485c7a7e. The retained PR #21/#23 and RaD sources remain pinned in SOURCES.json. The repository retains Apache-2.0; imported RaD sources retain CC0. Exploratory variants, duplicate snapshots and large raw graph dumps are excluded.

Validation

• Matched the unchanged h25/h23 bit producer counts and histograms to their previously verified certificate.
• Recomputed all 47 rational pivots and verified all 315 zero-minor partitions by 630 exact integer-rank calculations. The complete corner profile matches the selected certificate.
• Rebuilt the new h28d19 mixed-center complex producer and matched every carrier count and histogram entry; exact scalar and frame checks passed.
• Reconstructed both child lists and certified their moments, copied-center rank budgets, semantic induction, product row reserves, 47 strict constraints and seven assembly margins.
• Checked the final source diff for whitespace errors. Proof sources are included; no PDF compilation was performed this round.

Unchanged previous-round producer and regression checks were not rerun.