[petalas]

Summary

Scoped proof with finite corroboration: for every odd prime p, the signed two-source dirty connector serving all p²−1 nonzero coefficient rows has exact unit-call optimum 2p²+2p−4. In particular, F5 needs exactly 56 calls, comprising eight source calls and 48 readouts.

h = (p-1)/2
source path = (+B0)^h (+B1)^h (-B0)^h (-B1)^h
for each nonzero coefficient row v
  express v as P-Q between two square-perimeter prefixes
  read +A_v at P and -A_v at Q

The lower bound covers every legal word, including arbitrary extra readouts. A source path shorter than 2(p−1) misses at least 2p requested rows under paired readout; their additional calls exceed the saved source calls. The proof is uniform in compatible map dimensions and permits singular noncommuting maps and arbitrary incoming dirt.

Only research/signed-prime-connector/ is added. The prefix/difference method retains Ross Richardson's #943/#934 credit; conorholds's #2208 supplies signed query-model context, Riquelme Soares's #4189 supplies unsigned F3 context, and 1337isnot1337's #2266 treats a different GF4 scalar-source alphabet. The new result is the square construction and its matching all-word lower bound. Produced by Nick Petalas. Apache-2.0.

Evidence

Before: the cited connector interfaces provide prefix methods, binary-mask and unsigned results, and a distinct GF4 scalar-source result. After: a self-contained proof determines the signed full-coefficient two-source optimum over every odd prime field.

Local CPython 3.12.15 on macOS arm64 passes:

• Literal compiled words and independent perimeter difference coverage for p=3,5,7,11,13, giving 20,56,108,260,360 full calls.
• Complete signed source-word censuses at p=3 below length four and p=5 below length eight. They visit 85 and 21,845 words; 9 and 641 close, including the empty word. Minimum missing-row counts are 6 and 10. Every closed path satisfies the integer-range and excess inequalities.
• Ten dense executions with dirty data, singular source maps and noncommuting readout maps; ten rejected semantic/domain controls.
• Normal and optimized registered checks, metadata/pins, committed bytes/source history, official export integrity and an optimized Git-free export replay. The verifier emits a generated receipt bound to all five package-file SHA256 values.

The two registered checks took 0.041 and 0.043 seconds locally; the package runner took 0.292 seconds. These are timeout checks, not performance-improvement claims. No registered local check is left for CI alone.

From the repository root:

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

Any CPython 3.12 executable may replace the local interpreter path. Expected scientific marker:

PASS_SIGNED_PRIME_CONNECTOR p5_calls=56 primes=5 dense_cases=10 controls=10

Provenance is pinned to public base 624de9dc66858af0945e5f8a0942488d62280d43; predecessor code and results are not replay dependencies.

Limitations and unrun obligations

This is a scoped abstract-call theorem, not a complete multiplication theorem or a new κ. The written proof covers every odd prime; the executable fixture list is finite and exhaustive short paths are checked only at p=3,5. Unequal tariffs, composite characteristic, extension-field coefficients and additional scalar letters are outside the theorem.

A concrete recursive consumer, complete primitive implementations, admission, preparation, dispatch, physical routing, frames, stock, precision and tape costs remain unknown. No selected cold replay, formal proof-kernel build or independent human review is supplied. The selected frontier and frozen sources are unchanged.

Open next questions

1. Determine the exact weighted optimum, starting with the short-path/readout trade at p=5.
2. Specify and analyze composite-characteristic or extension-field coefficient alphabets.
3. Identify a real caller for full F_p² demand and price its complete physical invoice before further construction work.

Related open work

A fresh check of open contributions since the advance CLAIM found no additional PR studying the same signed two-source full prime-field demand.

• Research: high-arity connector toggle cost floor #2249 studies charged high-arity source toggles for six binary sources; this result uses two sources over odd prime fields and only signed singleton calls. No identical optimum is compared.
• Research: seven-source weight-at-most-four connector boundary (L<=18 excluded, 216-call floor) (no new kappa) #4079 studies seven-source binary weight-at-most-four demand; it has a different output set and characteristic. No shared numerical count is claimed.
• Add five-source arity-four connector cost bound #3909 prices arity-four source calls for five binary sources; its operation and dispatch/addition ledger differ from this signed unit-call alphabet.
• research: audit minimal joint-output connector charge #3897 prices a joint-output operation over F2; this theorem retains separate signed readouts and has no joint-output letter.

CI notes

Hosted CI passed on head 449552df73e231f4d6784c3472303f4e992d2701 in run 39856522933. The plan check passed in 3m0s and the registered package replay passed in 1m10s. The run reports the same exact head and overall success. All registered commands and publication checks also passed locally. No CI failure or unrun registered local obligation remains.

Merge danger

Two-way door. The scope is five new files in one isolated research package and its explicit replay registration. Production code and the selected result are unchanged.