Research: multi-pair schedules cannot beat the nine-stage template (Theorem S, no new κ)

[themilkman216]

Research note: a proved scheduling bound and a priced negative result. No κ claim; the selected record and upstream/ are unchanged. Only research/multipair-schedule-bound/ is added.

Two B_K conventions have never been varied: one bank pair per local-word invocation (format-limits-corrected rule-table row 13) and the pairwise exchange ρ (row 5). The format-limits ceiling theorem explicitly does not cover non-B_K schedules. This package closes that gap.

• Theorem S (written proof, checked). Any schedule of shear-type local words with s stages over K bank pairs needs N ≥ 2K + ⌈2K/(s−1)⌉ local-word invocations. This holds for any number of pairs per stage, mixed orientations, arbitrary block-diagonal bridges, any pair-level signed ρ and port-level calls. It generalises research/cross-shear-stage-bound.
• Consequence. Per port, D/W ≤ 1 − (cst/v)·s/(s−1), so every schedule is dominated port for port by the B-template at the same s. The proof tracks row-space intersections with X and Y and charges each stage's rank to progress or "waste".
• Attainment. At odd s, parallel borrowing chains attain the bound. Seven schedules replay exactly to the signed exchange, including rank-2v stages that mix forward and backward words. B_4 reproduces the (Sch) idle windows.
• Even s. Exhaustive GF(2) searches give the K = 2 frontier exactly equal to the bound. At (K = 3, s = 4), the smallest even case, the bound is not attained: an unrestricted equality search over all 2,051 stage shapes finds no schedule.
• Priced on PR Conditional κ = 0.001679754956125544: fine source-prime padding (PR1786) at the free epsilon -> 1 point, with gauge-free interiors and the radix-distributed union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1800's T74 words. The best attained schedule reproduces the T74 complex exponent 0.001679754956125544521335 exactly and does not beat it. s = 7 hits the hosting wall; s = 11/13/15 lose 1.40/3.71/6.46% of b. The only open point, s = 8 if the bound were attained there, is worth at most +0.26% (window-aligned idle) or +0.70% (packing ceiling).
• Multi-pair invocations share nothing. Pairs active in the same stage carry independent data, so each keeps its own centre path and helper network (Lemma Q excludes the fused unit). This differs from the temporal centre-sharing PRs Finite centre-phase fanout across two shared-source panels #1708/Certify shared reverse centre phase with a charged precision guard #1748/Planning result: exact overhead budgets for centre-path sharing on the actual selected profiles (no new kappa) #1791/Research: selected adjacent centre-owner census #1829/Add exact rank-one center-difference sharing mechanism #1838.

Reproduce

python3 -B research/multipair-schedule-bound/verify.py
python3.12 scripts/research_package.py --package research/multipair-schedule-bound

Expected final markers: PASS_MULTIPAIR_SCHEDULE_BOUND and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result. The verifier takes about 30 s and uses the standard library only. Both pass locally on Windows with CPython 3.11 and 3.12. Hosted CI is not claimed.

The verifier checks:

• the proof-step inequalities on 10,697 random stages over GF(2), GF(3) and GF(5), with two false-strengthening controls that must fail;
• the exact chain replays, byte-equal to the committed output;
• the exhaustive K = 2 frontier;
• recomputed exact prices for 14 schedule families, using the retained moment engine and complex-local-word-ceiling/ceiling.py.

Three negative controls must be rejected: a flipped chain sign, a perturbed T74 extract, and a below-bound schedule. The longer K = 3 searches have committed outputs and commands in the README; they are not rerun by verify.py.

Scope and inputs

• inputs/T74-WORDS.json is an extract of PR Conditional κ = 0.001679754956125544: fine source-prime padding (PR1786) at the free epsilon -> 1 point, with gauge-free interiors and the radix-distributed union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1800's provider registry, with its source commit, Git blob and SHA256 pinned. It holds the two local words' per-call profiles, helper widths and call split. The verifier recomposes PR Conditional κ = 0.001679754956125544: fine source-prime padding (PR1786) at the free epsilon -> 1 point, with gauge-free interiors and the radix-distributed union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1800's published global profile from it exactly. The T74 construction itself is not re-verified here.
• Not covered:
  • even-s attainment beyond (K = 3, s = 4);
  • ρ through a third bank class (the "two unpaid coordinates" lever);
  • non-shear local words;
  • a frame-checker replay of mixed-orientation stages (legality is argued from the contract).
• Card coverage-law-change, scope changed: this tests the schedule freedom, not two unpaid coordinates. TASK.json records the reason and two structured open questions.
• Overlap: find_overlap.py was refreshed before submission. Nothing on record covers multi-pair or mixed-orientation schedules.

Credit

The work builds on:

• cross-shear-stage-bound and the archived borrowing-chain campaign;
• format-limits-corrected (rule table, (Sch)/(Dfc), Lemma Q);
• complex-local-word-ceiling;
• PR Conditional κ = 0.001679754956125544: fine source-prime padding (PR1786) at the free epsilon -> 1 point, with gauge-free interiors and the radix-distributed union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1800 (jeremykothe2) and its T74 lineage (PR1625 mdumitrean, PR1661 midhatn, PR1319/PR1297 rohanarun and others) for the priced words.

Prepared by themilkman216 with substantial Claude (Anthropic) assistance; the investigation ran as parallel fresh-context research branches. Apache-2.0. No independent human review is claimed.

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