Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12

[icekylinx]

This extends my PR #115 at cf60442983c153a3a0f5272ea4713f99eba13c86, with one new commit. The #10 → #18 → #24 → #32 → merged #36 → #104 → #115 proof lineage and attributed ordinary-leaf, semantic/bulk and fixed-tape interfaces are retained.

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

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{3146011}{10000000000}=3.146011\times10^{-4}. $$

Construction

The global architecture uses three invocation banks on a regular Cayley cover in dimension 3h-2. Each local port retains its source line while the three active spaces exchange its complement with two separate blocks. The actual data frames agree at both stage boundaries. Three signed shears complete the interchange, so there are no positive-width data connectors or copied rank-one endpoint corrections. All three local words, arbitrary dirty cleanup, copied centers and auxiliary exteriors remain charged. The per-vertex rank deficit is 2v-3ell, with 2v+3R persistent roles.

The complex supplier adopts eumemic's PR #117 h24 positive DAG, with 91,770 additions, 71,185 compatible carrier links and 28,705 auxiliary roles. A new Clifford frame theorem applies to every binary subspace, including degenerate intersections and partial gauges. Exact representatives use preimage Lagrangians; actual rank-zero adapters reconcile frames before scalar gates. The local verifier rebuilds full backward intersections and 4,599 source gauges, with readouts executed in reverse selection order.

The complex cover is O(70,2). Its full order, finite routers and scalar operations remain in the semantic certificate. The per-vertex inventory has 90,163 roles, rank 6,309,018 and deficit 2,392. The strict complex saving is 786999/2500000000, with moment gap exceeding 2.3047018455e-10. Exact denominator-21 decoding and the completed-child C1=1 contract are retained.

The bit supplier uses the unchanged Zhihao Chen (jacklightChen) PR #97 physical ledger based on Swapnil Jain's witness. Its dimension is 67. A finite GL_67(Z/q^w) cover batches high matrix digits by fixed low-residue types. Unit-lower factorization leaves variable pivot units inside weighted reversed children. Nongeneric classes receive a full local-ring Gaussian fallback of at most 32m^2 proper children; its entire extra moment is charged on a fraction 10^-16 of every ideal edge. The coarse saving is 747/2000000, with certified contaminated gap exceeding 6.3397366396e-7; stopping gives ordinary bit saving 3731649599/10000000000000.

The bit selector stock occupies O(w log e) original address digits and is restored by an explicit outer ordinary-interchange wrapper. Its uniform batching, weighted leaves, fallback, atom adapters, borrowed-row restoration and binary padding are included in the proof. The complex and conservative legacy reserves use external stock p^160000, with coefficient 68,505. All 47 strict constraints and seven assembly margins hold, with final absorption gap exceeding 3.4173577301e-12.

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
#36	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
#104	Opposite-bank factorization, stopped product-ring interchange, aligned rational centers, exact odd-denominator grid and selected semantic/bulk composition.	7.79476e-5
#115	Pair-first cube producer, compatible backward frame enlargement, partial source gauges with reverse readouts, and stopped whole-projector composition of the PR #97 bit word.	9.26336e-5
This PR	Arbitrary-subspace Clifford frames, three-stage Cayley covers, weighted local-ring bit compiler, charged rare-class fallback, internal row borrowing and exact assembly.	3.146011e-4

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 used by the retained ordinary leaf.
#33	Dominik Scholz	Parameterized corner generator, specialized to (25,23) in the retained ordinary leaf.

The selected local words additionally rely on these pinned contributions:

Source	Contributor	Role in this construction
#97, e15350b66e4bf796153a68c240a6cc1b788abe3e	Zhihao Chen (jacklightChen)	Actual deferred bit word, reflected physical ledger, commuting fan-order clarification, gauged exits and signed complex endpoint correction.
Pinned round-seven witness	Swapnil Jain	Deferred readouts, V leaves, lifted frames and the frozen witness underlying PR #97.
#62, ad0f25ff7b23cff7f08ad237c2254e6ecf74257e	Avi Eisenberg (ikeboy)	Cyclic interval contractions and core-aware pair primitives used by the new cube producer.
#117, cbb05ce504d571546d9b7794c186a613c659c3bf	eumemic	Selected 91,770-addition positive h24 complex DAG and exact support/label replay; original Claude assistance disclosure retained.

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/three-stage-cover-note.tex: selected cover construction, weighted compiler, row restoration and final assembly, supplied as LaTeX source.
• notes/general-clifford-frames.tex: exact representatives, nested transitions, arbitrary source gauges and complete reverse/complement words.
• scripts/three_stage_cover_producer.py: pinned PR Deferred bit word and a 28,705-role complex producer under stopped-product accounting: conditional κ = 1.044939e-4 #117 DAG reconstruction, generalized frames and independent physical-word checks.
• scripts/three_stage_cover_network.py and certificates/three-stage-cover-network.json: full inventories, contaminated bit moment, finite complex group/router charge and exact assembly.
• references/three-stage-cover/pr117/: original positive DAG, replay, source pins, notices and license.

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 finite group is specified mathematically and is not materialized by the verifier; uniform q-adic batching and row borrowing are written proof dependencies. Earlier submitted code, source-specific licenses and upstream review records are preserved.

The generalized Clifford theorem, three-stage geometry, weighted cover, rare-class compiler and internal-row construction are contributed by icekylinx, with substantial OpenAI GPT-6 Astra assistance. Repository integration uses Codex assistance. Adopted authorship and assistance disclosures remain in SOURCES.json, NOTICE and the preserved snapshots. Existing Apache-2.0 and imported CC0 terms apply to their respective sources. Unselected reclaim branches, alternative covers, research motifs, large generated graphs and PDFs are excluded.

Validation

• Verified the supplied base archive and consumed dependency hashes.
• Rebuilt the new PR Deferred bit word and a 28,705-role complex producer under stopped-product accounting: conditional κ = 1.044939e-4 #117 local DAG, matching, full subspace intersections, physical word and source gauges. Every selected profile field matched.
• Independently checked scalar supports and signs, divisor 21, old-frame containment, full dependency/carrier intersections, actual frame transitions, center closure, untouched source roles, reverse target chains and positive-rank accounting.
• Reconstructed the retained bit local ranks from pinned physical paths, then rebuilt both three-stage inventories and certified strict moments, full rare-class charge, stopped saving and subordinate adapter/row tolls.
• Recomputed the complete finite complex group/router charge, exact semantic guard, external row reserve, 47 strict inequalities and seven assembly margins.
• Refreshed shared source hashes and dependent joint-dual certificate digests; recomputed joint arithmetic and confirmed unchanged mathematical fields and prior replay payload.
• Checked staged source bindings and deterministic certificate regeneration. Proof sources are included; no PDF compilation was performed.

Unchanged prior producer and full historical suites were not rerun. The incremental command is make three-stage-cover-verify; its scope is documented in docs/three-stage-cover.md.