Paired physical scratch reuse: candidate conditional κ = 3.62108222954e-4 (+4.02% vs #134)

[DanieleCorso]

Summary

Candidate conditional κ = 3.62108222954 × 10⁻⁴, representing a 4.02% improvement over PR #134 (κ = 3.48102 × 10⁻⁴).

This draft combines PR #131's physically optimized complex circuit and compensated scratch reuse with PR #132's orthogonal first-stage lockstep pairing.

Status: finite arithmetic candidate. Full combined physical verification is pending. No completed mathematical proof is claimed yet.

Methodology

1. Physical scratch optimization: Reuse PR Physical frames and scratch reuse on Cayley covers: conditional κ = 3.32205398986e-4 #131's complex supplier with 26,597 physical auxiliary roles and 2,108 compensated birth-cut aliases.

2. Orthogonal lockstep pairing: Apply PR Paired lockstep first-stage vertices on three-stage covers: conditional κ = 3.416202e-4 #132's Cayley-cover vertex pairing to the first invocation stage, sharing physical scratch streams between two orthogonal 24-dimensional active spaces.

3. Conservative data accounting: Keep source/target data operations separate and explicitly charge their additional moment cost using rational upper bounds. No unverified data-wire merging is assumed.

4. Exact rank accounting: Reconstruct the paired recursive-child histogram, preserve the rank deficit, and account for the unchanged bit supplier, scalar work, finite routers and semantic reserves.

5. Coupled exponent calculation: Evaluate the resulting finite moment and assembly constraints using rational arithmetic.

Candidate numerical results

Quantity| Result
Paired persistent streams| 141,081
Recursive rank mass| 9,870,886
Rank deficit| 4,784
Maximum recursive child| 68 / 70
Conservative complex saving| 0.000362371031
Candidate final κ| 0.000362108222954
Improvement over PR #134| +4.02%

Verification and limitations

The finite calculation checks 47 strict assembly inequalities, seven margins and the next-grid exclusion conditions.

Remaining verification: The full combined physical schedule must be independently replayed, including arbitrary dirty scratch, signed Clifford operations, compensated readouts, inverse chronology and cross-core noninterference.

Current-head CI and independent proof review are still pending.

This PR must remain Draft until those requirements pass.

Reproduction

python3 research/paired-pr131-conservative/certificate.py

Attribution

Builds upon PR #130 (icekylinx), PR #131 (eumemic), PR #132 (ikeboy), PR #124 (jamesyc), and their original contributors.

Hybrid construction and conservative finite-arithmetic screening prepared with OpenAI assistance.

All inherited analytic, semantic, routing and fixed-tape assumptions remain in scope.

[Swapnil-jain]

Heads-up: this candidate combines #131's physical scratch reuse with #132's first-stage lockstep pairing. The pairing step appears not to restore its targets; details and an exact small counterexample are here: #132 (comment)

In short, in lockstep each core's frame steps also transform the other core's live data on the shared stream. #131's reuse on the unpaired cover should be unaffected, but κ = 3.62108e-4 relies on the pairing.