[petalas]

Summary

Prove an exact local routing optimum and construct an attaining gather. For h identical completed kernels on aligned K-bit neighborhoods, with c shared source/target neighborhoods, the minimum moved support is 2K(h-c) and the minimum coordinate-swap count is K(h-c). The construction fixes the overlap and swaps the two difference sets.

 h=3, Q=3, K=128G, source blocks {0,2,3}, target blocks {2,3,4}
- ordered gather: 4K moved coordinates, shortest coordinate-swap word 3K
+ new assignment: 2K moved coordinates, K coordinate swaps, both optimal

Claim type: scoped mathematical proof with finite exact corroboration. The arbitrary-input completed operator, whole-row padding boundary and restoration before the next coarse-dependent diagonal are preserved under the stated source contracts. This is a small constructive step toward PR6070's global-H installation question. No new kappa or complete supplier invoice is claimed.

This contribution was produced by Nick Petalas. Credit zxv's PR6070 for spectator placement and the open installation question, the retained arbitrary sparse extension/support router, Vlado Vojdanovski / PR1673, Jeremy Kothe / PR1758, Chafik Boukhalfa / PR1786, PR4359 and the selected P104/helper/native/compiler lineage. Elementary permutation support and cycle facts are standard. The new contribution is this boundary's optimal assignment, constructor and application; the predecessor theorem is cited, not audited or replayed.

Evidence

• Independent enumeration of all 5,913 permutations through seven neighborhood positions, 695,482 subset images and all 4,706 equal-size source/target pairs confirms both optima.
• 3,878 complete exact Gaussian basis columns, 93 nonsymmetric whole-neighborhood fixtures, three complete dirty padded-row fixtures, 13 domain/capacity rejections and two semantic controls pass.
• The encoded K=17152 fixture enumerates moved support 68608 to 34304 and shortest coordinate swaps 51456 to 17152. It checks the finite K guard and reservoir/suffix capacity; it does not execute a production tensor or export an actual call.
• Warning-strict normal and optimized direct checks pass locally in 1.550s and 1.558s. The complete registered normal/optimized replay passes in 3.578s, well below both 120s command limits. These durations establish replay feasibility, not a speed claim.
• All eight committed package files pass byte/mode and declared source-history preflight at a2e2d4da922b64c6b137fe74e9b0fe845c60dc24. Official export integrity and declared-input coverage pass. Optimized replay from a read-only Git-free export with an empty PATH passes; all nine exported files, including the manifest, remain unchanged.
• AST/syntax, JSON and whitespace checks pass. Ruff, mypy and pyright are unavailable locally. No required scientific command is left solely to CI.

From the repository root, using CPython 3.12.15 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/optimal-spectator-gather/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/optimal-spectator-gather/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/optimal-spectator-gather
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/optimal-spectator-gather --check-source-history --check-modes

Another CPython 3.12 executable can replace the local path. Expected markers are PASS_OPTIMAL_SPECTATOR_GATHER, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Every direct receipt is generated on stdout and binds the package hashes; no mathematical receipt is hand-written.

Limitations and unrun obligations

The theorem permits reassignment among identical kernels and assumes a common K-grid. Fixed ordered jobs, distinguishable operators, unaligned neighborhoods, CNOT costs, adjacent tape movement and arbitrary linear charts have different contracts. The proof covers every finite size; the executable planner and literal interpreter have explicit resource caps.

A physical support-router lowering, literal production global-H event, complete stock/callee binding, rounded padding-disposal proof, supplier/native/compiler admission and outer multiplication theorem remain open. The selected conditional kappa remains .003125804 and all five framework assumptions remain. Full selected cold replay, Lean, Docker, independent human and formal proof-kernel review are unclaimed.

Canonical main was refreshed before committing and remained a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8. The advance CLAIM matches the delivered scope. Overlap checks cover live open titles/bodies, handoffs, main README formulas, targeted merged work and recent account source usage. Some merged searches reached their result caps; sparse notes, arbitrary diffs and unpublished work remain limits.

Open next questions

1. Export one literal global-H call with aligned source/target neighborhoods and compute its overlap. Lower the new involution through the paid router, retaining actual complete stock, protected suffix, caller and returned endpoint. That physical installation is missing.
2. Specify the chunk-preservation contract for unaligned neighborhoods, then determine the exact assignment/completion optimum. The current aligned proof does not supply that interface.

Related open work

• Audit #4359 BIT row stock and its #4470 atom change (no new κ) #4612 studies the support-router width/stock interface in the selected BIT chain; this result optimizes the supplied coordinate permutation and does not recheck its bounds.
• Audit uniform Beneš compiler with charged sequential arena scans #4192 studies a charged uniform Beneš permutation compiler; this package chooses an optimal permutation at a different boundary and does not implement or price that compiler.

No open PR specifically optimizing this aligned spectator gather's moved support was identified in the refreshed and targeted searches. Merged PR6070 is the cited predecessor. No extra computation was run to compare another contributor's result, and no competing numerical value is asserted.

The completion refresh read the fifteen other account PRs still open and created since the advance CLAIM. None studies this aligned spectator gather optimum or reports a competing value for its instances. No extra computations were run to check another contribution. The adjacent cyclic-reversal and coordinate-cache results concern different endpoints and cost quantities.

CI notes

Both hosted checks passed at exact head a2e2d4da922b64c6b137fe74e9b0fe845c60dc24 in run 40913736761: plan in 4m29s and replay (research/optimal-spectator-gather) in 2m2s. The completed check list was read once; run metadata confirms the successful conclusion and matching head.

Hosted replay certifies the registered normal and optimized finite checks at this head. The listed local scientific, publication and read-only export checks also pass. The universal proof and inherited physical/caller obligations retain the limitations above; independent human and formal proof-kernel acceptance remain unclaimed. No CI failure remains to report.

Merge Danger

Door: two-way. Blast radius: isolated research package. Eight new opt-in package files; no shared source, frozen predecessor or selected-frontier changes.