Shared bit cores and gauged scratch reuse: conditional κ = 1.29186205798e-4

[eumemic]

κ = 1.29186205798e-4 = 64593102899/500000000000000.

κ ≈ 129186205798/10¹⁵ > 2⁻¹³.

This is a verified historical checkpoint. Later three-stage-cover candidates PR130 and PR131 give larger conditional savings.

Under the retained stopped-product, endpoint, analytic and fixed-tape interfaces, this finite conditional witness supports T(n) = O(n(log n)^(1−κ)). This is 4.19234% above PR114's checked 7722755 checkpoint and 4.94411% above PR128's pinned 530588a claim. These comparisons refer to those exact checkpoints; the PR list was checked through #128 immediately before submission.

The improvement combines a new 161-group bit partition with physical operation frames, compensated birth-cut reuse and gauged completed-core sharing on the complex side. The stopped ordinary-bit saving now binds the final assembly.

Supplier	Shared bit	Shared complex
dimension	529	576
physical local scratch roles	28,866	26,597
completed-core groups	161	87
original streams W	15,567,734	12,821,030
total recursive rank	8,233,987,097	7,383,051,200
rank deficit	1,344,189	1,862,080
largest child	518	560
certified supplier saving	129310447/10¹² (coarse)	130696544/10¹²

The retained stopping fraction yields the actual ordinary-bit saving 129219596453/10¹⁵. Exact assembly pays the scalar work, adapters, all three row stocks and the remaining margins to obtain the stated κ.

The bit construction partitions all 1,771 triples of 23 coordinates into 161 common-center sunflower classes of eleven. Seven base classes are translated through Z/23Z; an exact orbit-orientation search and seven perfect matchings supply the witness. Under H = 9I−J, distinct triples in a class have annihilating rational projectors. Whole completed cores can therefore share each dirty scratch role, with a paid complement of rank 529 − 11*(23−dim σ). The package reconstructs and replays PR128's pinned minimal-V scalar schedule; it does not claim to regenerate the unpublished optimizer that selected that schedule. Every old internal/data/copy/endpoint child and 659,476,769 projector adapter calls remains paid. A single common rational basis is selected for the enlarged finite projector family, followed by a fixed odd execution prime avoiding its finite bad-prime set; the verification prime alone is not an execution-prime certificate.

The complex construction retains PR117's 91,770 additions and 71,185 carrier links, optimizes frames at physical operations and uses PR124's compensation to reuse 2,108 dead slots. The literal source/workspace word is undone in true reverse chronology and restores every physical scratch slot for arbitrary dirty inputs. Its 470 distinct initial frames include 2,598 nonzero-gauge slots. PR128's signed partition supplies 83 groups of 24 and four of eight. Each group-role pays the actual complement 576 − |J|*(24−dim σ), with exact Gaussian phases checked in both orientations, including alternating blocks. All core children and signed group exteriors remain charged.

The core scalar guard uses all 28,705 virtual roles, because aliased readouts still occur. The separate basis/phase reserve uses the 26,597 physical roles and pays 64m² wrapper atoms per group-role in each orientation. These are inherited ordinary-bit adapter calls, not constant-time tape instructions. The total scalar/semantic guard is 158614225569472, row coefficient 1644, and row degree 4000. Modular address denominators do not add Gaussian decoder divisions to the retained exact 21-grid.

Validation at current head 59fade1: the full local aggregate passes, and all 39 current-head CI checks pass, including focused verification on Python 3.11, 3.13 and 3.14. Regeneration covers the complete bit geometry and all-column replay of 1,076,483 events over 32,408 formal columns, the physically aliased complex compiler, literal forward/reflected audit, signed partition, physical gauges, exact certificate and all 33 outer corruption controls. All 47 strict constraints, seven margins and next-grid exclusions pass; the original source tree stays unchanged. The frozen closure binds 69 package files and 20 repository dependencies. Proof profiles contain exact integer/rational records and reproduce with full equality; the canonical reflection receipt is also checked byte for byte. The numerical claim, physical words and complete paid histograms are unchanged from submission.

Reproduce from the repository root:

python3 research/cyclic-deferred/verify.py
make verify

The read-only aggregate regenerates both suppliers, actual scalar/reflection ledgers, physical source gauges, phases, complete profiles, exact certificate and controls in a temporary copy. BITSHARING.md and SHARING.md give the allocation/operator arguments. Finite checks do not prove the inherited common-basis, normal-form, transparency, ordinary-bit transfer, precision, streaming, routing, analytic, prime, recovery or fixed-tape interfaces. This remains a conditional finite witness, with no practical runtime or global-optimality claim.

Credit: an664 with OpenAI Codex assistance for PR128's completed-core principle, signed partition and minimal-V package; Xiande Zhang and Gennian Ge for the underlying complex resolvable packing (JCD 18 (2010), 209–223, DOI 10.1002/jcd.20234); James Chang / jamesyc with OpenAI assistance for PR124's compensated birth-cut reuse; eumemic with Anthropic Claude assistance for PR117's scalar DAG; Avi Eisenberg / ikeboy with Anthropic Claude assistance for PR110's compiler; Swapnil Jain for round-seven bit words, frames and deferred readouts; icekylinx, Zhihao Chen, Rohan Arun, Aurel Prosz, RaD and all inherited contributors. Concurrent frame work in DanieleCorso's #126 and GamingPuzzled's #125 is credited. The physical-frame optimizer, new 161-group bit partition, generalized gauge/reuse composition and verification were prepared for eumemic with OpenAI Codex assistance. Original notices and licenses remain preserved.

[Mister-Meeseeks]

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

Contribution: Rational 9I-J sunflower geometry and source gauge.

Review and limits: Exact-head CI evidence; 161 groups and the complete tail are paid.

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: 59fade1f7790db61a56b69e0d3974c0a97b41f4d. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.