Partial source gauges and cube producers for conditional kappa = 9.26336e-5

[icekylinx]

This extends my PR #104 at 948ce1510df750f4c18b96bdaef436a86f8bf834, with one new commit. The #10 → #18 → #24 → #32 → merged #36 → #104 proof lineage and attributed ordinary-leaf, two-stage and semantic/bulk interfaces are retained. PR #104 is the direct predecessor and is not yet merged.

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

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{7237}{78125000}=9.26336\times10^{-5}. $$

Construction

The (23,23) bit network applies the stopped whole-projector compiler from #104 to the actual deferred physical word of Zhihao Chen (jacklightChen)'s #97, based on Swapnil Jain's pinned witness. Every projector increment is one reversed child of its full rank. The strict coarse saving is 15513/125000000; stopping at the inherited atom exponent 1/1000 gives ordinary bit saving 1240183559/10000000000000.

The new (24,24) complex producer combines cyclic intervals and pair-first cube assembly. Its actual DAG has 146,284 additions, 120,097 compatible carrier links and 34,307 auxiliary roles. Frames are enlarged backwards through dependencies and carrier links, keeping a nondegenerate extension of each old frame inside the already selected successor frames.

The physical word selects 6,341 untouched roles for partial source gauges. Each gauge lies in the first source frame and all required target frames. Readouts execute in reverse selection order, giving actual nested target chains. A first transition of rank r becomes rank r-dim(sigma) and the exterior grows to m-h+dim(sigma); all target-data transitions and the paid rank-one signed endpoint correction remain charged.

The strict complex saving is 92651/1000000000. The bit and complex moment gaps exceed 1.1417e-10 and 4.1393e-11. Exact rational centers retain divisor 21 and the completed-child C1=1 contract. The larger maximum child widths, 528 and 574, require product row stock p^40000, with row coefficient 15,761. All 47 strict constraints and seven assembly margins hold; the final absorption gap exceeds 3.9406e-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
#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
This PR	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

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 new construction also adopts 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.

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/partial-gauge-note.tex: selected construction, chronological frame/gauge proof and assembly, supplied as LaTeX source.
• scripts/partial_gauge_bit.py: source-bound reconstruction of the pinned physical bit ranks and whole-projector child list.
• scripts/partial_gauge_producer.py: regeneration and independent finite checks of the new complex DAG, frames and readout schedule.
• certificates/partial-gauge-network.json: strict moments, stopped savings, semantic charge, product row stock and exact assembly.
• references/partial-gauge/: compact pinned witness, original proof sources, 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 #10 patch and upstream community review records are preserved; this extension is presented in its standalone proof.

The new cube integration, compatible frame enlargement, partial source-gauge construction and composition are contributed by icekylinx, with substantial OpenAI GPT-6 Astra and Codex assistance. Source pins and contribution scope are recorded in SOURCES.json and NOTICE. Existing Apache-2.0 and imported CC0 terms, author credits and historical assistance disclosures are retained. Research alternatives, large generated graphs, duplicate handoffs and PDFs are excluded.

Validation

• Checked the pinned PR Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97 source hashes; reconstructed auxiliary, source-data and target-data ranks from the frozen physical paths; matched its ledger and rebuilt every whole-projector multiplicity.
• Rebuilt the selected h24 complex producer, matching, backward frame enlargement, physical word and partial gauges. All saved counts and histograms matched.
• Independently checked exact scalar supports and signed decoder, frame nondegeneracy and containment, carrier nesting, center closure, untouched selected roles, reverse readout chains and positive-rank transition counts. Zero-rank no-ops were compared through deterministic construction; their multiplicity does not enter the moment.
• Recomputed both rational moments, the stopped ordinary-bit saving, exact denominator-21 semantic charge, product row reserve, 47 strict constraints and seven assembly margins.
• Refreshed the joint-dual manifest's changed shared-file hashes and both dependent certificate digests. Recomputed joint arithmetic and checked that its mathematical fields and prior replay payload remain unchanged.
• Checked staged source hashes, deterministic certificate regeneration and the final diff. No PDF compilation was performed.

Unchanged previous-round and upstream full producer/complete-basis suites were not rerun. The reproduction command is make partial-gauge-verify; the incremental scope is documented in docs/partial-gauge.md.