[ToikY]

Change and mathematical scope

For the two independent rank-56 BIT IDLE fields from #7001, any aligned cover using tracked-coordinate flips and grade-preserving binary maps must satisfy sum_i V_i r_i ≥ 112V. With the existing raw child tariff, aligned children of rank ≤56 cannot save cost. Any positive volume at smaller rank strictly increases that cost. At the fixed moonshot target 0.0040075385047236, the concrete four rank-28 child construction adds approximately 0.3065621907390684 V, even with ordinary routing and tape granted free. This is a scoped proof and finite control suite. NO VERIFIED GAIN in κ. The independently replayed complete comparator remains #7161 at 0.003644890067940.

Task and overlap

Refines the paired IDLE component in the campaign. Base e545235. The registered manifest pins four canonical source files by path and SHA-256. The bound concerns the aligned two-field contract and arbitrary child volumes. #7163 studies the full-fibre rank-one setting. The distinct result here is the unequal-volume rank-mass inequality for a concrete retained pair. The separate navigation commit reuses the already replayed #7161 docs to let hosted source checks locate the selected artifact.

Validation

From a clean checkout with CPython 3.12, run:

python3.12 -B scripts/research_package.py --package research/paired-bit-axis-volume-bound

Local registered replay passed, including 230 exact finite binary columns and missing-half, cross-grade and changed-input controls. The all-width argument and directed prices are in the package proof. Hosted status is pending at submission.

Credit

The #7001 authors supplied the retained BIT caller and its routing tariff. #6726, #6834 and #6980 supplied the delayed-cover lineage. #673 and #7163 supplied related restrictions on independent-field compression. Clear1nsight supplied the replayed complete benchmark in #7161. Codex and subagents materially assisted the proof, exact checks and independent audit.

Open next questions

A grade-changing candidate needs a literal lowered route with a positive selected-reference cross-grade minor after fully charging transport and restoration. An aligned cover, or a route that only conjugates the same aligned action and restores the frame, cannot provide this gain. The package includes the aggregate threshold and its separately priced scope.