[petalas]

Summary

Scoped algebraic construction with exact finite corroboration. Shared scalar conjugators lower the canonical rank-two clean square-zero compiler from sixteen to ten CNOTs, retaining three coordinate reflections and the exact clean return.

- specified clean bill: 3R + 16C + H
+ specified clean bill: 3R + 10C + H
  specified dirty bill: 4R +  4C

The new clean stream beats the specified dirty stream exactly when R > 6C+H, improving the previous R > 12C+H comparison. This answers the scalar-sharing handoff in Open next question 2 of JoyboyBrian's merged #5007. Its construction and reflection-call minimum are cited, not audited or replayed. The contribution was produced by Nick Petalas. No scalar-network optimum, new kappa or supplier admission is claimed.

For arbitrary data a,b,c,d and linear involution g in characteristic two, the full formal endpoint is (a+c+g(c), b+d+g(d), c, d, g(h)). A zero helper therefore returns exactly to zero. The written proof derives the new stream and independently tracks all five inputs. Every scalar addition and inverse operation is counted.

Credit JoyboyBrian #5007 for the clean construction and open question; Nick Petalas #4912 for the dirty optimum and compiler; ywz65 #2716 for the involution-product identity; Adam Gall #1911 and the retained register-model lineage. Only seven files in the new research/clean-rank-two-conjugators/ package are added.

Advance CLAIM.

Evidence

• The complete new endpoint passes coefficientwise over formal F2[C2]. A separate literal bit-array interpreter passes all 1,024 two-address states, including all 256 clean-helper states, plus 3,215 complete basis columns for all 119 involutory permutations on one through six addresses.
• All thirteen individual primitive omissions reject. Controls also expose the arbitrary-dirty-helper substitution, a noninvolutory address substitution and a changed noncommuting order. The finite checks corroborate the written uniform identity.
• Both warning-strict registered commands pass locally under CPython 3.12.15 on macOS arm64, standard library only. Timed direct commands took 0.0452 s normal and 0.0432 s optimized, rounded upward. These are check durations, not performance claims.
• The registered package runner, syntax inspection, exact committed-byte/source-history preflight, official ten-file committed export and export integrity pass on head b313bad5735ebaa9e303b9813a29d949917e2c76.
• Both direct commands pass from a read-only, Git-free export with empty PATH. All eleven exported files, including the export manifest, retain their bytes.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/clean-rank-two-conjugators/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/clean-rank-two-conjugators/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/clean-rank-two-conjugators
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/clean-rank-two-conjugators --check-source-history

Substitute another CPython 3.12 path elsewhere. Expected markers include PASS_CLEAN_RANK_TWO_CONJUGATORS reflections=3 CNOTs=10 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier writes no files and emits a generated receipt bound to all package and cited-input hashes.

Canonical main was refreshed before committing and remains de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. The selected source-binding orientation passes after sparse materialization. The overlap helper paginated 405 open PRs and 5,290 Discussion records; its sparse local coverage is incomplete. Distinctive searches of main's full README tree, merged PRs, relevant live bodies and fresh claims found no competing ten-CNOT result. Unpublished work and arbitrary historical diffs remain coverage limits.

Limitations and unrun obligations

The new result prices specified literal words. Neither scalar stream is claimed globally optimal. Reflection, CNOT and clean-resource prices must include their complete stationary implementation costs; none is supplied here. Common data-basis transport must be charged equally on both sides. A general dirty helper is not restored.

Native suppliers, physical routing/frames/tapes/hosting, clean-resource availability, recursive admission and the complete multiplication theorem remain open. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, selected scientific replay and predecessor executables were not run. Every registered local command passed.

Open next questions

Can a literal native rank-two caller supply complete prices satisfying R > 6C+H? First bind all data movement, inverse scalar maps, reflection work and clean reservation/return to actual operations. Missing native contracts and a complete caller invoice are the blockers. No further rank or scalar-optimality search is included.

Related open work

• Prove split determinants do not ensure radical reachability (no κ) #5854 studies split determinants and reachability over a multi-generator square-zero radical algebra. Its zero-residue target and paid alphabet differ from this fixed-involution literal compiler; no common minimum is compared.
• Prove a sharp asymmetric UT3 tariff gap: 54 versus 53 (no new κ) #5856 proves an asymmetric UT3 block-call tariff gap with two noncommuting address labels. This package counts CNOTs in a specified one-involution clean compiler; no common numerical quantity is compared.
• Prove exact all-role right-vector costs for small local products #5861 proves right-vector factorization costs over products of local algebras. Its paid factors and scalar-cost convention differ from the literal coordinate-reflection/CNOT stream here; it does not determine this compiler's scalar count.

Completion refresh inspected this account's open PRs created since the advance CLAIM, #5854 through #5862. The other submissions concern symplectic completion, Hermitian commuting repair, inverse norms, binary-header association and unitary-cycle conditioning. None computes this fixed-involution compiler's CNOT count, and no disagreement in a shared computed quantity was found. No predecessor or peer mathematical executable was run. Merged #5007 and #4912 remain the direct predecessors.

CI notes

On contribution commit b313bad5735ebaa9e303b9813a29d949917e2c76, run 40479306439 completed with plan failed; replay skipped. The only failed step is Onboarding helper tests.

Its command is python -B -m unittest discover -s tests -p 'test_research_*.py' -v. All ten failure/error records are in unchanged tests/test_research_current_assignment.py; the summary is FAILED (failures=8, errors=2, skipped=22) after 511 tests. The errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight failures concern missing expected current-assignment resolution/validation, including 0 != 16, expected ValueErrors not raised and empty resolved-path lists. This matches the known main de5ec2e7a6074068452f45b1a191b5c72832df4a failure. PR #5854's run 40478863245 independently shows the same ten failure names and the same eight-failure/two-error summary.

The registered package replay was skipped. Hosted replay has not certified this package; only the local scientific checks, registered normal/optimized commands, committed-byte/source-history preflight and Git-free export checks described above have run. Shared tests and implementation are left unchanged for the maintainer repair. No additional commit was made during this CI handoff.

Merge danger

Door: two-way.

Blast radius: isolated.

One new opt-in research package. Shared implementations, frozen predecessors and the selected result are unchanged.