[petalas]

Summary

Prove that Cinna Davis's PR2271 nine-target parity boundary needs at least thirteen unit row XORs, even with arbitrary CNOT ordering, target feedback and any finite restored dirty workspace. Combined with its credited thirteen-call construction, this settles the global optimum in that row-XOR alphabet.

- 13 is optimal within the one-anchor tour class (PR2271).
+ No program of at most 12 arbitrary row CNOTs meets the same boundary,
+ even with restored dirty auxiliaries and target feedback.

Claim type: scoped written mathematical proof with exact finite lemma corroboration. The new result is the lower bound. The predecessor construction is cited, without importing or replaying its verifier or certificate. No new kappa or selected-frontier change is claimed. The contribution was produced by Nick Petalas.

Advance CLAIM. Scientific base is canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. The card's changed scope is recorded as a local algebraic side result.

Evidence

The written proof counts the nine mandatory target writes. Three surplus writes leave only one temporarily changed source and at most one twice-written target. A pure intermediate of that target can only be the original single source coordinate. The remaining source itinerary cannot supply all three required parities. The argument covers arbitrary finite dirty workspace; it does not enumerate twelve-gate words.

Local CPython 3.12.15 checks pass:

• All 168 GL3 binary bases and 1,008 one-write transitions.
• All 55,987 fixed-source words through six gates, with 380,713 prefix visits.
• All 12,288 four-write word/offset cases; 2,720 restore the source, and their eight pure target intermediates have the required single-coordinate form.
• All 680 bounded surplus allocations and 219 relaxed closed itineraries.
• Eight rejected interpreter inputs, three semantic discriminators and a four-write positive boundary control.
• Warning-strict normal and optimized checks, timed separately at 0.1864 and 0.1877 seconds. Their generated six-file hash-bound receipts agree exactly.
• Registered replay, metadata/source pins, source compilation, function-size inspection, whitespace and committed-byte/source-history preflight all pass at d54576b126e39033e97d4ae34fd51ef9ff524281.
• An optimized standalone six-file copy without Git or an executable search path produces the identical receipt.

Reproduce from the repository root, using the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/global-thirteen-add-fanout/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/global-thirteen-add-fanout/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/global-thirteen-add-fanout
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/global-thirteen-add-fanout --check-source-history

Other machines may substitute their CPython 3.12 path. Expected markers are PASS_GLOBAL_THIRTEEN_ADD_LOWER_BOUND, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The universal statement rests on the written proof. Finite checks corroborate its lemmas and are not a proof-kernel certificate or exhaustive twelve-gate census. All initial rows are independent arbitrary dirt; clean rows, free basis changes, nonlinear or input-dependent programs and bit gates mixing different row coordinates are outside the theorem.

No physical caller, native implementation, movement/tape bound, precision/stock ledger, complete recursive invoice or supplier admission is provided. All five selected assumptions remain. No predecessor scientific suite, selected cold replay, Lean or Docker build was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Determine whether one initially clean, zero-return row changes this exact minimum, with initialization and restoration charged.
• Locate an actual supplier occurrence and price its complete source references, common frames, routing, native work and restoration.

Related open work

• PR6427 gives exact fixed-weight mask-tour tariffs, with one anchor and one write per target. This changes the allowed program class to arbitrary CNOTs and proves the lower bound for PR2271's original mixed-weight boundary; no additional computations were run to audit or compare that PR's numerical fixtures.

The completion refresh inspected all 14 open account contributions created since the advance CLAIM, including this PR. The thirteen others are PR6572 through PR6576 and PR6578 through PR6585. None studies PR2271's nine-target boundary or computes its global thirteen-call minimum. No additional scientific computation was run to audit another contribution or compare unrelated quantities.

The earlier weighted pair-tour PR4899 and the paid clean-row tariff PR6254 have different restricted alphabets and are merged. Live open/account PRs, handoff claims, reservations and main/merged object and general-form searches were checked before commitment. Sparse notes, unpublished work and arbitrary historical diffs remain coverage limits.

CI notes

Both hosted checks passed at exact head d54576b126e39033e97d4ae34fd51ef9ff524281 in Registered research packages run 41617608388: plan and registered package replay.

The completed check list was read once. Hosted replay executes both registered warning-strict normal and optimized commands. The local scientific, package, committed-byte/source-history and standalone checks listed above also passed. No failed or waived check remains. CI certifies execution of the finite lemma checks; independent human and formal proof-kernel acceptance of the written proof remain unclaimed.

Merge danger

Door: two-way. Blast radius: one research package. Adds six files only; shared code, frozen predecessors, selected certificates and workflows are unchanged.