[ToikY]

Change and mathematical scope

NO VERIFIED GAIN. The κ change is exactly 0%. The strongest independently reproduced comparable witness remains #7161 at κ = 0.003644890067940, head e3f8d39.

The exact paired #7001 BIT endpoint and retained native, QUERY and IDLE tariffs cannot reach a nonnegative signed moment at κ = 0.0040075385047236 by changing the number or stage allocation of LOCALs. The proof allows arbitrary invertible scalar bases and auxiliary roles initialized and restored to zero. Independent stage ranks, the two common root coordinates, and concave mask packing bound every native count. The strongest signed upper is −2.3610606020114406897430. Higher multiplicities have upper −28.3066209237471478769601. A different endpoint, operator, per-native bill or recursive proof remains open.

Task and overlap

Changed bit-new-register-model scope under #7111. Canonical base e545235. SOURCES.json pins the literal paired caller, invoice and interval engine. #7160 supplies the 36-LOCAL QUERY-rank proof. This package extends it to arbitrary counts and stage multiplicities and charges the IDLE masks. #6984, #6800 and #6825 supply related rank arguments with different endpoint scopes.

A separate documentation-only prerequisite reuses the exact selected-replay navigation from #7161 commit bb36fc6, following #7147/#7156. The canonical navigation test failed before applying that blob and all six targeted status tests passed with it. The selected record is untouched.

Validation

From the exact checkout with CPython 3.12 (tested 3.12.14):

python3.12 -I -B research/bit-paired-schedule-bound/verify.py
python3.12 -B scripts/research_package.py --package research/bit-paired-schedule-bound
python3.12 -B scripts/check_research_commit.py --package research/bit-paired-schedule-bound --check-modes

The replay reconstructs both original invoices, checks 81 opposite-arm endpoint intersections, 108,832 grouped price cells and 272 exact one-slot prices, and compares mask packing against 96 small brute-force fixtures. A changed caller byte fails authentication. An independent dynamic program confirmed the packing optimum used by the greedy construction. No new physical caller or supplier is submitted.

Credit and next question

Credit DreamingOfClouds, CubeJerry, zxv and the retained #7001 lineage for the paired controller, Nick Petalas #6800 and #6825 for related rank methods, #6984 for the fixed-window audit, #7160 for independent-stage saturation, and Rohan Arun and Clear1nsight for the current comparison and navigation repair. OpenAI Codex and subagents materially assisted the proof and independent checks.

The next supplier test changes the native debit or endpoint and reprices every new operation before attempting the all-width operator and tape admission.