Paired cubes and shared completed cores for conditional kappa = 4.609169e-4

[icekylinx]

This extends my PR #130 at 6a9970a530119174507904e23592fd59ede19a5d, with one new commit. The #10 → #18 → #24 → #32 → merged #36 → #104 → #115 → #130 proof lineage and attributed ordinary-leaf, semantic/bulk, uniform-recursion and fixed-tape interfaces are retained.

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

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{4609169}{10000000000}=4.609169\times10^{-4}. $$

Construction

The new complex local motif uses twelve coordinate pairs and 1,760 ports grouped into eight-port cubes. Its scalar identity is I=K+H+B: signed overlap channels supply H, coordinate-star centers supply B, and two 4-by-4 Hadamard/2 blocks per cube supply the involution K. All heavy source injections precede K. The original source registers execute K at a common rank-three frame, deliver their assigned contributions, and undo K at the full frame before scalar cleanup. This adds no K auxiliary bank.

The selected signed DAG has 24,767 additions, 7,724 root uses, 6,074 compatible carrier arcs and 26,417 auxiliary roles. Its 24 coordinate-star frames have rank 22, with copied-center loss 528. The chronological program retains 4,840 entrance gauges of rank 20. Conservative source supports include all visited directions; scalar cancellation is not used to shrink frames.

The global architecture adopts an664's PR #128 completed-core sharing principle. The new cross-stage routing reuses one auxiliary bank successively in three orthogonal h-blocks. Each core includes the complete arbitrary-dirty scalar cleanup and excludes its old independent-bank exterior tail. Exact forward and inverse core actions justify one final correction of width 3 dim(sigma) for each selected role. Both data banks also pay their rank-two complement calls. Every local operation, copied center and sharing router remains charged.

The complex shared profile has dimension 72, 29,937 roles per cover vertex, rank 2,153,528 and deficit 1,936. Its strict saving is 4856569/10000000000, with moment gap exceeding 2.0094617553e-10. The full O(72,2) group has 2,556 bits; the actual role stock has 2,571 bits and the maximum child is 60.

The bit branch retains the PR #97 / Swapnil word and original rational frames. It keeps 9,543 certified entrance gauges and omits 2,022. Omitted dirty reads move to the initial zero frame; all later ordinary operations and source frames remain inherited, and retained target chains are certified subsequences. The dimension-69 shared profile has 32,408 roles, rank 2,234,128 and deficit 2,024. The strict coarse saving is 4617656/10000000000; the full rare-class fallback is charged with moment gap exceeding 1.4386017949e-10. The stopped ordinary saving is 4613422943/10000000000000.

The finite guard pays all signed mixer operations, direct dirty reads, original-source involutions and sharing permutations. A numerator-bitlength bound handles the full dirty-response matrix without a positivity assumption. Coordinate-star decoding uses divisor three on one exact grid, retaining completed-child C1=1. The external reserve is p^70000, with coefficient 20,445; q-adic selectors remain internally borrowed and restored. All 47 strict constraints and seven margins hold, with final absorption gap exceeding 9.9454090434e-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
#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
#130	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
This PR	Paired-cube identity and signed query channels, original-source involution, coordinate-star centers, orthogonal completed-core sharing and a certified bit-gauge subset.	4.609169e-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	Retained positive h24 DAG and exact support/label replay; zero restrictions supply the new coarse query modules. Original Claude disclosure retained.
#128, 530588a019b4a74f09180680c9e3961bf649ec89	an664	Substantial conceptual dependency: completed-core workspace sharing, with reported OpenAI Codex assistance. No code, partition or serialized witness imported.

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/paired-cube-note.tex: paired identity, original-source chronology, exact core sharing, bit subset and assembly, supplied as LaTeX source.
• scripts/paired_cube_producer.py: standard-library reconstruction of the signed graph, frozen carriers, full frames and gauges, with independent finite checks.
• scripts/paired_cube_bit.py: source-bound reconstruction of the selected inherited bit gauges and their zero-frame corrections.
• scripts/paired_cube_network.py and certificates/paired-cube-network.json: complete shared inventories, strict supplier moments, full router/scalar guard and exact assembly.
• references/paired-cube/: compact carrier and subset pins; inherited source notices remain in the existing pinned references.

The result remains conditional on the retained analytic/tape interfaces, constructive setup and eventual thresholds. Finite checks do not formally verify the full theorem. Shared cores, their exact phases and paid tails are justified in the written proof; the finite group is specified mathematically and not materialized by the verifier. Previously submitted source code and upstream review records are preserved.

The new paired-cube schedule, coordinate-star completion, concrete sharing integration, bit subset and exact composition are contributed by icekylinx, with substantial OpenAI GPT-6 Astra assistance. Repository integration uses Codex assistance. The substantial PR #128 sharing contribution is explicitly credited; no exclusive priority is claimed for broad sharing or paired-coordinate ideas. Adopted copyrights, source-specific Apache-2.0/CC0 terms and historical assistance notices remain in SOURCES.json, NOTICE and preserved references. Unselected fallback results, alternative motifs, research screens, large generated graphs and PDFs are excluded.

Validation

• Verified the supplied incremental-file hashes and all three required base archive hashes.
• Rebuilt the signed paired-cube graph and replayed the compact frozen matching, avoiding a nonunique matching-algorithm dependency. Every selected profile field matched.
• Independently checked 3,097,600 scalar H/K/B entries, the K involution/inverse and original-source frame itinerary, conservative supports, full carrier/backward intersections, signed physical M, center closure, full gauge intersections and reversed target chains.
• Reconstructed the bit slot subset, zero-frame old-value corrections, first-transition changes and inherited target subsequences; matched every selected child multiplicity.
• Certified both exact moments, full rare-class charge, stopped bit saving, finite group/router/scalar envelope, denominator-three grid, row reserve, 47 strict constraints and seven 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 paired-cube-verify; its scope is documented in docs/paired-cube.md.

[Th0rgal]

Independent Verification & Exact Global Bit-Subset Optimization (#146)

We independently reproduced and verified all targets in this PR (make paired-cube-verify, make verify-joint) with Python assertions enabled:

• scripts/paired_cube_producer.py: Reconstructs the signed $H$-channel graph ($h=24, v=1760, R=26417, |\mathcal{M}|=6074$), replays every scalar coefficient of $H + K + B = I$, the original-source $K$ involution and its inverse, actual subspace containment, reverse target chains, and the $4,840$ rank-$20$ partial gauges ($a_c = 4856569 / 10^{10}$).
• scripts/paired_cube_bit.py & scripts/paired_cube_network.py: Reconstructs the retained $9,543$-slot / omitted $2,022$-slot PR Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97 bit profile ($m_b = 69, W_b = 32,408$, rank sum $2,234,128$, deficit $2,024$), verifies the $9,594$ zero-prelude old-value reads, and certifies all $47$ strict inequalities and $7$ semantic assembly margins at $\kappa = 4609169 / 10^{10} = 4.609169 \times 10^{-4}$.

Follow-up Global Optimization in #146

Because $a_c = 4.856569 \times 10^{-4}$ has substantial headroom above $a_b = 4.613422943 \times 10^{-4}$, the overall bound $\kappa$ is 100% determined by the PR #97 bit gauge subset selection. By summation by parts over consecutive active target ranks $0 = d_0 < d_1 < \dots < d_k < d_{k+1} = 22$, the target-chain moment cost on each target $t \in {1, \dots, 1771}$ satisfies the exact telescoping identity:

$$\text{cost}_t(x_{t,1}, \dots, x_{t,k}) = 3 w(22) + \sum_{p=1}^k c_{t,p,p}, x_{t,p} + \sum_{1 \le p < q \le k} c_{t,p,q} \left(1 - \prod_{m=p}^q (1 - x_{t,m})\right),$$

where strict concavity of $r \mapsto r^{1-a_0}$ makes every diagonal capacity $c_{t,p,p} > 0$ and every interval capacity $c_{t,p,q} = 3(w(d_{q+1}-d_p) + w(d_q-d_{p-1}) - w(d_{q+1}-d_{p-1}) - w(d_q-d_p)) > 0$ strictly positive. Scaling the 120-bit rational upper-bound weights by $W_b m_b 10^{16} 2^{120}$ to exact Python integers reduces the entire $2^{11,565}$ subset optimization problem to a single exact integer $s$-$t$ minimum cut / maximum flow on $36,760$ vertices:

• Retains $9,730$ entrance gauges and omits $1,835$ (+165 at rank 16, +22 at rank 17; 9,179 zero-prelude dirty reads),
• Lifts coarse bit saving from $a_0 = 4617656 / 10^{10}$ to $a_0 = 461877426979 / 10^{15} \approx 4.61877426979 \times 10^{-4}$,
• Lifts effective bit saving from $a_b = 4613422943 / 10^{13}$ to $a_b = 461454009452021 / 10^{18} \approx 4.61454009452 \times 10^{-4}$,
• Lifts the certified exponent saving to $\kappa = 461028508707 / 10^{15} = 4.61028508707 \times 10^{-4}$ in Use exact telescoping min-cut bit subset on paired-cube shared cores for kappa = 4.61028508707e-4 (4.61239827139e-4 at atom=1/2000) #146, with exact integer max-flow/min-cut duality verified directly inside scripts/paired_cube_bit.py.