[petalas]

Summary

Scoped algebraic proof with exact finite corroboration: a square-zero binary coupling N of rank q>0 needs exactly the smallest even number of rank-one idempotent reflection calls greater than q. The count is q+1 for odd q and q+2 for even q, with arbitrary free binary scalar mixing and any finite restored dirty workspace. The zero coupling costs zero.

rank-three square-zero coupling
  factorwise compiler: 6 reflection calls
  joint compiler:      4 reflection calls, proved optimal

This extends ywz65's merged #2716 implementation by jointly compiling the coupling. Its product identity, dirty compiler and predecessor lineage retain credit. The new package was produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed. No new kappa or selected-frontier change.

Advance CLAIM.

Evidence

• Before, Research: arbitrary binary couplings preserve the dihedral rank tariff #2716 supplies two address calls per nilpotent rank-one factor, giving 2q for a rank-q square-zero coupling. After, explicit joint projectors attain the new count. Trace parity and a rank-saturation contradiction prove the matching lower bound uniformly, including every finite restored dirty workspace.
• Local CPython 3.12.15 normal and optimized registered checks pass. Direct runs each took 0.27 seconds, below their 120-second limits.
• The independent shortest-projector-sum search covers all 66,066 binary matrices through four roles. All 343 square-zero matrices match the formula, including all 210 rank-two matrices with exact minimum four.
• Complete formal endpoints and a separate faithful address-action interpreter check 343 small fixtures and 2,678 arbitrary-input basis columns. Eight conjugated fixtures cover ranks one through eight with two dirty spectators, another 176 columns and 48 omission controls. Four boundary controls also pass.
• Metadata/source pins, committed-byte/source-history preflight, warning-strict static inspection, seven-file committed export and optimized Git-free replay pass on head eaea286b055724895adb4ca949b5ac0495be5ff8. The verifier emits a generated receipt bound to package and input hashes and writes no files.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/square-zero-idempotent-compiler
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/square-zero-idempotent-compiler/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/square-zero-idempotent-compiler/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/square-zero-idempotent-compiler --check-source-history

Substitute another CPython 3.12 path elsewhere. Expected markers are PASS_SQUARE_ZERO_IDEMPOTENT_COMPILER rank3=4 previous=6 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The exact endpoint is I+(1+[g])N in full formal F2[C2], with one fixed nonidentity involution label and binary rank-one idempotent paid couplings. Other address labels, clean/discarded workspace and nonconstant free maps are excluded. The universal statement is the written proof; bounded enumeration corroborates it.

No native reflection supplier, scalar conversion/routing/frame/hosting/precision/tape/stock invoice, recursive child pricing or complete multiplication theorem is supplied. Four formal calls do not by themselves prove a lower physical total cost. Lean, Docker and the selected full cold replay were not run; all registered package commands ran locally. No predecessor executable or census was replayed. Canonical main remains 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7, at the submission fetch.

Open next questions

Can the joint compiler improve a fully charged native boundary? Pin a literal rank-three caller and compare all scalar basis changes, reflection work, frames and movement against the six-call factorwise compiler. A native reflection contract and complete invoice remain the blockers.

Related open work

• Prove the UT3(F4) central floor survives nilpotent calls (no new κ) #4849 proves a UT3(F4) central-target floor after admitting nilpotent calls. Its address group, endpoint and paid alphabet differ; it does not determine the fixed-involution square-zero compilation minimum. No numerical comparison or additional computation of that predecessor is asserted.
• Prove exact signed monotone unitriangular tariffs (no new κ) #4918 proves signed monotone unitriangular tariffs in a related formal rank-one-idempotent register model. Its noninvolutory address labels and diagonal targets differ; it computes no square-zero coupling compilation minimum.
• Prove exact orbit weights for ordered idempotents (no new κ) #4915 counts simultaneous centralizers and orbits of ordered idempotents. This PR minimizes the number of idempotent summands of a square-zero matrix; no centralizer or orbit population is compared.
• Prove exact dirty-fanout tariffs over unital rings (no new κ) #4916 prices dirty fanout with unit demands over unital rings. Here the coefficient 1+[g] is nonzero square-zero and hence nonunit, so its unit-demand theorem does not price this compiler; no shared minimum is compared.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T23:38:30Z were refreshed. Nine other new open submissions were inspected for scope; the nearby entries above are added without replaying their computations. No new submission in that snapshot asserts this exact compilation count, and no disagreement in a shared computed quantity was found.

Scoped live open-PR and handoff checks, full main README searches and distinctive merged searches were repeated immediately before committing. No competing claim to this compilation count was found. Remote snapshots and sparse local coverage cannot exclude unpublished work. David Arias's merged #2738 already covers finite-dihedral arbitrary-coupling tariffs; that different candidate was discarded before this result was pursued.

CI notes

Both hosted checks passed on contribution head eaea286b055724895adb4ca949b5ac0495be5ff8 in run 39873649539: plan in 2m56s and the registered research/square-zero-idempotent-compiler replay in 1m28s. Run metadata confirms that exact head, completed status and success. The replay executes both warning-strict normal and optimized registered commands. These automated checks corroborate the scoped package; independent human and formal proof-kernel review remain unclaimed.

Both registered scientific commands and all required local preflights passed. The historical catalogue baseline command fails with KeyError: 'bit_lower' on the selected schema; the current selected-baseline metadata command passes. Neither is a mathematical premise of this package.

Merge danger

Door: two-way. Blast radius: isolated.

Only six files in one new opt-in research package are added. Shared implementations, frozen predecessors and the selected frontier are unchanged.