[petalas]

Summary

An explicit signed Hadamard connector uses 208 primitive calls, saving 14 against Cinna Davis's cited 222-call construction in #1158. Claim type: exact finite construction with a scoped algebraic proof for arbitrary exact linear maps and arbitrary dirty inputs.

- source queries: 8;  readouts: 16; primitive calls: 222; echo nodes: 66
+ source queries: 10; readouts: 12; primitive calls: 208; echo nodes: 62
  dirty scratch rows: 4

Two extra restoring calls on the cheapest source replace four first-output readouts. This deliberately leaves #1158's exactly-eight-source-call class. Its optimum is cited, not audited. The ten-source word, coefficient proof, complete child lowering and standalone verifier are in research/hadamard-extra-source/README.md and verify.py. No unrestricted optimum or new kappa is claimed.

The complete bill is 208E+62D+22Q+P_new+R_new. Against the cited comparator, the reduction survives when the additional preparation/layout bill is less than 14E+4D+2Q. No physical overhead is assumed zero.

Evidence

Local CPython 3.12.15 checks pass:

• Exact formal endpoint for all 22 queries and all 208 primitive leaves with independent noncommuting symbols, including every source, initial output and dirty scratch input.
• 104 dense matrix input cases and 104 complete literal inverse replays; 25 missing-call, target-sign and alias rejection controls.
• Both registered normal/optimized commands; each direct command took about 0.06–0.07 seconds. The registered metadata/pin and replay wrapper passed.
• Committed bytes and complete source-history preflight at ce08fe61d030b27da74c367cf4f9da537a3ddd06.
• Official five-file export integrity and declared-input check; normal and optimized replay from the read-only export. Generated stdout receipts bind every package file by SHA256.

Exact reproduction, from the repository root:

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

Another CPython 3.12 installation may replace the displayed interpreter. Expected markers: PASS_HADAMARD_EXTRA_SOURCE, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. Only the standard library is needed. The verifier has no predecessor runtime dependency.

Limitations

This is an exact linear connector with fixed A lengths (2,3,4,5) and B lengths (1,2,3,4). It does not supply rounded children, physical routing, a current caller, stock/precision bounds, supplier admission or a complete multiplication theorem. Lean, Docker and the selected scientific replay were not run. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

Open next questions

• What is the least weighted query price with arbitrary restoring source multiplicities? Start with this explicit upper bound and prove a lower bound that charges each source separately. A bounded sample does not settle that question.
• Can a named physical caller keep the saving after complete preparation, routing and precision costs? Compare its additional P+R bill with 14E+4D+2Q.

Related open work

• Add five-source arity-four connector cost bound #3909 studies five-source all-mask connectors with atomic arity-four source events; this construction uses four signed Hadamard targets and only singleton source events.

• research: audit minimal joint-output connector charge #3897 studies fused joint-output dispatch for two binary targets; every output readout here is separately charged, including the repeated call.

• Research: seven-source weight-at-most-four connector boundary (L<=18 excluded, 216-call floor) (no new kappa) #4079 studies seven-source bounded-weight binary coverage; the source count, signed target and complete price contract differ.

• Prove connected restored source-XOR words need 2(n-1) calls (no new κ) #5935 studies connected restored source-XOR words; sources here remain read-only and only the dirty connector changes.

• Complete the (1,2,4) Boolean fanout Pareto frontier: 31 vectors, six new (no new κ) #5948 supplies a Boolean fanout Pareto frontier for demand (1,2,4); it does not study this signed Hadamard target or compute its primitive count.

The closeout refresh read all ten other open contributions from this account created since the advance CLAIM. None studies this fixed four-source signed Hadamard target or supplies a competing primitive count. Those searches do not cover unpublished work or every historical code diff. No additional computations were run to validate other contributions.

Credit and scope

This contribution was produced by Nick Petalas. Credit Cinna Davis / 1337isnot1337 #1158 at 6336d879c0ba168463bf2f6fae9e653c870299fc and #997 at b3fda8d59436103b195069a6328cbf5d74f73bc0, plus Martin #1088, Jerry Chen / CubeJerry #889, Ross Richardson #934/#943/#944 and Chafik Boukhalfa #628 for the retained weighted-repair, connector and dirty-echo lineage. Their source bytes and credit are unchanged. New material is Apache-2.0.

Advance claim. Base: canonical main 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. TASK.json explicitly changes the larger/nonabelian BIT card to this historical recursive-interface side result. The first verified improvement was retained and the search stopped.

CI notes

Both hosted checks passed at exact head ce08fe61d030b27da74c367cf4f9da537a3ddd06 in run 40530418680: plan in 5m2s and registered package replay in 1m54s. The completed checks were read once. The replay executes the registered normal and optimized commands. This certifies the bounded automated checks, not independent human or formal proof-kernel acceptance of the mathematical argument.

Merge danger

Reversible archive-only addition confined to one new research directory. No shared executable, selected record or predecessor package is changed.