Padded triple covers with sequential dirty reuse: conditional κ = 4.13596233702e-4

[eumemic]

κ = 4.13596233702e-4 = 206798116851/500000000000000.

κ ≈ 413596233702/10¹⁵ > 2⁻¹².

Under the retained multiplication, weighted-child and fixed-tape interfaces, this finite conditional witness supports T(n) = O(n(log n)^(1−κ)). It improves our fully verified #131 claim by 24.50015% and exceeds #135's current numerical candidate 3.62108222954e-4. The complex side binds.

Two extra dimensions let three completed local cores reuse one physical scratch bank in every stage of icekylinx's Cayley cover. Each core completes its arbitrary-dirty cancellation before the next starts. The stage offset cancels from the completed auxiliary residual, including in the reversed middle stage. All nine local words per three-vertex cell remain paid. The two added dimensions cost a width-two finishing child on each data bank.

Per three-vertex cell	Complex	Bit
local / cover dimension	24 / 72	23 / 69
persistent roles	91,935	97,224
recursive rank	6,612,144	6,702,384
deficit	7,176	6,072
largest child	66	66
strict coarse saving	413939067/10¹²	22550837/(5·10¹⁰)

The complex supplier retains #117's immutable DAG and #131's 2,108 compensated birth-cut reuses. Its 28,705 virtual roles occupy 26,597 physical slots. Exact containment descent additionally permits degenerate operation frames and source gauges under #130's generalized Clifford theorem. The full physical forward word, actual reverse chronology, exact source/readout coefficients and arbitrary-dirty dependency cuts are independently replayed; all 12,160,614 expanded scalar operations per invocation stay charged. The scalar reserve continues to use the larger virtual-role inventory.

For each physical source gauge σ, three sequential auxiliary residuals occupy separate active factors. Their final tail is defined by the exact operator equation Tail = F_total (R3 R2 R1)^−1, so phases cancel algebraically. Its rank is 3 dim σ. Rank-zero tails remain actual paid Clifford adapters. The certificate adds an explicit supplemental adapter charge to the full finite router and recomputes the semantic guard. The full complex orthogonal group has 2,556 bits; actual global role stock has 2,571 bits. The row coefficient is 30,729 with retained degree 160,000.

The bit construction uses the unchanged source-bound PR97 physical word. Its new tails remain weighted split-projector children, with one common inherited diagonal weight across each complete shared use. Three-element low-residue orbits preserve every high-matrix batch. Conditioning on an orbit representative is charged by a factor three in the bad-class bound. Every nonzero edge—including the new data finishes—pays the complete 32·69² local-ring fallback on fraction 10⁻¹⁶. The stopped ordinary bit saving is 11265104579/25000000000000 = 0.00045060418316. Internal borrowed rows and all atom adapters retain their restoration and subordinate-cost contracts.

Validation at current head c205b7d: fresh regeneration of the physical local compiler, gauges and reuse pairs passes; its full literal forward/reflected audit passes. Exact complex geometry checks all 542 source-frame types in all three stages and all 2,024 actual ports. The bit supplier reconstructs the retained physical ledger and checks weighted identities, rational grouping, both contaminated moments and the next coarse grid point. Independent review of the complex geometry, weighted bit composition and finite coset batching found no remaining blocker under the stated contracts. Exact final assembly passes all 47 strict constraints, seven margins, and next-grid rejection checks. All 33 package corruption controls pass, including in the publication checkout. The complete read-only aggregate passes against the frozen package: local compiler and reflection receipts reproduce byte for byte, both padded ledgers and exact assembly pass, all 33 controls pass, and the source snapshot remains unchanged. All 45 current-head CI checks pass, including the complete focused verifier on Python 3.11, 3.13 and 3.14.

python3 research/cover-local-reuse/verify.py

The verifier binds the complete source closure, regenerates the local word and receipts in a temporary copy, checks byte equality, reconstructs both padded inventories and moments, runs corruption controls and requires an unchanged source snapshot. The focused workflow runs Python 3.11, 3.13 and 3.14. Concise construction arguments are in PADDED-GEOMETRY.md and BIT-PADDED.md alongside the code and certificates.

This remains a conditional finite witness. General Clifford normal forms, the common generic basis, uniform weighted local-ring compilation, ordered affine passes, row borrowing, analytic recovery and fixed-tape transfer remain explicit inherited proof contracts. Finite checks are not a formal verification of those all-size interfaces or a practical speedup claim.

Credit: icekylinx with OpenAI GPT-6 Astra/Codex assistance for #130's three-stage cover, arbitrary-subspace Clifford theorem, weighted bit compiler, fallback and row contracts, and the #104/#115 lineage; eumemic with Anthropic Claude assistance for #117's scalar DAG; James Chang/jamesyc for #124's compensated birth-cut reuse; an664 for completed-core sharing; Avi Eisenberg/ikeboy for the physical compiler and #132's cover-pairing stimulus; Zhihao Chen/jacklightChen and Swapnil Jain for the retained PR97 bit word; Rohan Arun, Aurel Prosz, RaD and all inherited contributors. Our arbitrary-frame/gauge descent, sequential grouping in every stage, padded triple covers and independent checks were prepared for eumemic with OpenAI Codex assistance. Original licenses, source-specific notices and assistance disclosures are preserved.

[Mister-Meeseeks]

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

Contribution: Pad to 72 with local copies and sequential banks.

Review and limits: Exact-head CI evidence; unpaired sequential banks, rank-zero adapters, scalar costs and rare-class factor are retained.

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: c205b7d9df0ab933c4b7392ea6625c05dbd5e4c0. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.