[petalas]

Summary

For every fixed finite characteristic c >= 3, the unsigned dirty connector with all c^n - 1 nonzero coefficient rows has

OPT_c(n) = c(c^n - 1) + Theta(c^(n/c)).

split n sources into c balanced blocks
for each nonempty block
  run a restored positive-step Hamilton cycle on its coefficient torus
for each output row
  read once at each of its c block projections

Claim type: scoped proof with a deterministic construction and finite corroboration. This proves the optimal overhead exponent, allowing arbitrary extra readouts in the lower bound. It gives no optimal leading constant, new exact optimum, new kappa or selected-frontier change. Only research/full-coefficient-connector/ is added.

For n = ac + r >= c, 0 <= r < c, the construction uses S_c(n) = (c + (c - 1)r)c^a source calls and exactly c reads per output. A source path of length L has at most P_c(L) = binomial(L+c-1,c)-L nonzero c-read sums. Extra rows require at least 2c reads. The relaxed invoice is bounded below by c(c^n-1)+L_0, where L_0 is the first c-multiple at least cn with P_c(L_0) >= c^n-1. This is at least c(c^n-1)+(c!(c^n-1))^(1/c)-(c-1).

For full ternary demand, explicit seven- and eight-source words use 6603 and 19743 calls. This continues #4244's larger-source open question. The uniform construction is not best at five or six sources; their cited exact optima remain smaller.

Evidence

Before: #4238/#4244 give exact full-ternary results at five/six sources and leave larger sizes open. After: a self-contained all-n construction and matching-order lower bound work in every finite characteristic, including composite characteristics.

The verifier checks 41 literal words, 36996 output rows and 186532 calls through formal noncommuting coefficient ledgers; 28 Hamilton cycles; all 780 small weak block compositions; all 2220 multisets in four cases; 12 independent dense executions with dirty registers and singular noncommuting matrices over Z/3, Z/4 and Z/6; and six rejected word/characteristic controls. Its generated stdout receipt binds the counts and examples to all five package file hashes.

Exact reproduction from the repository root:

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

Any CPython 3.12 interpreter may replace the local path. Marker: PASS_FULL_COEFFICIENT_CONNECTOR. All four commands pass locally. The registered normal/optimized checks took 0.208/0.213 seconds individually; the package runner took 0.738 seconds. The final committed-byte/source-history preflight checks five files and the base ancestor. The card's positive control also passed with pairings=945 max_coverage=31 queries=74 leaves=182. These timings establish replay bounds, not performance improvements.

Limitations and unrun obligations

• The all-n claim is the written induction and counting proof. Finite execution is corroboration; there is no proof-kernel certificate or independent human review.
• No global leading constant or exact optimum for n >= 7 is established. The block cost is optimal only within the stated separately restored block geometry.
• Unit complete-call counts only. A consumer needing every coefficient row, coefficient preparation, dispatch, stock, hosting, precision, tape and complete supplier admission are unknown.
• The cited pending small optima are not rechecked or executable dependencies. Source manifests contain historical provenance at the public base. No pending code is imported.
• The selected cold replay and formal build were not run. No registered local obligation is left for CI alone. The selected frontier is unchanged.

Open next questions

1. Close the constant gap between the counting lower bound and the c-block construction, starting at c = 3 and n divisible by three. Charge positive-step realization and restoration of any proposed additive basis.
2. Improve the seven-source ternary upper bound 6603 toward Prove the exact 2208-call F3 optimum for full ternary six-source demand (no new κ) #4244's cited lower bound 6588.
3. Pin a real full-coefficient-demand consumer and price its complete physical invoice before extending a supplier.

Related open work

• Prove the exact 744-call F3 optimum for full ternary five-source demand #4238, head 191a905d7adba6695b079f744140a73e832787a8: full ternary five-source predecessor. Its 242 rows and 726 minimum readouts agree numerically with this generator; our uniform upper bound is 747 versus its cited exact 744, so no new small optimum is claimed.
• Prove the exact 2208-call F3 optimum for full ternary six-source demand (no new κ) #4244, head f436d2911351e5be2842208fe96b0b66442323a4: full ternary six-source predecessor and larger-source handoff. Its 728 rows and 2184 minimum readouts agree numerically; our uniform upper bound is 2211 versus its cited exact 2208. Its stronger seven-source lower bound 6588 is compatible with our 6603 upper bound.
• Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231, head 8a956cbc80dc55697fd3e214812ce567b5eca48f: same unsigned ring interface with binary-mask demand. Its necessity argument is predecessor context; the full-demand asymptotic theorem concerns a different output set. No shared optimum is recomputed.
• Prove the exact 108-call unsigned F3 five-source connector optimum #4225, head 397025101d23195a5193c7912e6cd384eccdbf1c: five-source binary F3 optimum 108. Different demand; no overlapping full-ternary optimum.
• Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232, head 86ad339efc9f817846479431d8cc0f04edf95fb9: binary F3 floor/deletion results for every source count. Different demand; no overlapping full-demand number.
• Prove the seven-source F3 connector floor fails: optimum 405, floor attained exactly for n ≤ 6 (no new κ) #4250, head aa61e1746f8c12807bed59cd169591bd82320bc6: seven-source binary F3 optimum 405. It does not address the 2186-row full ternary demand; no shared optimum.
• Reach the unsigned connector floor for k ≤ 2c in every characteristic; 9 ≤ N(4) ≤ 21 (no new κ) #4248, head e7df122403cfc2febfb6799abb821487d8d1ee49: binary-mask depth blocks across characteristics. Full coefficient torus blocks are a different construction and demand; no shared optimum.
• Lower the characteristic-four connector floor bound to N(4) ≤ 20; the known nine-source walk does not extend (no new κ) #4255, head 0dba831b7e138c809f1900ecf51b4716ea99c505: parity bounds for the binary-mask floor in even characteristic. It does not settle this full-demand overhead; no overlapping numerical claim.
• Research: exact F3 minimum for the one-source additive connector #4189, head 5fb3936dfc8082f438f21c72dd6ca49ff4366ae7: one-source, one-output F3 predecessor. Full one-source demand has two outputs, so its six-call optimum and our nine-call word are different quantities.
• Count optimal unsigned one-source connector words in every characteristic #4236, head bd9b426837cddd37338dd284e49cc2303de57ca0: counts optimal words for one nonzero coefficient row. This package serves all rows simultaneously; no shared word-count quantity.

Live overlap covered the recent 100 open PRs, earlier account PRs, all connector-title PR bodies, current handoffs, active Discussion summaries, main-tree formula/object searches and merged searches for full coefficients, ternary demand, characteristic and overhead. The overlap helper reported 364 open PRs and 3129 Discussion records with complete pagination. Arbitrary diffs, unpublished work and literature priority are not exhaustively covered.

CI notes

Hosted CI is green on head 6a95ef419f94e7b5c2fff710e34081520cc6aba2 in run 39504635104: plan passed in 2m52s and the package replay in 1m20s. Both hosted scientific modes printed the same counts and all five file hashes as the local committed replay. The Committed inputs remain unchanged step passed. All registered checks, byte/source-history checks and whitespace checks also pass locally on CPython 3.12.15, macOS arm64. No job failed, no rerun was needed, and no registered local obligation was left to CI.

Merge danger

Two-way door. Blast radius: isolated. Five new package files and their opt-in replay; no shared code or selected source changes.

Produced by Nick Petalas. Credit #4238/#4244, #4231, Riquelme Soares's #4189, Ross Richardson's #943/#934, conorholds's #2208 block-palette/counting predecessor, and ywz65's contract map. Correction to the CLAIM: #2208 is by conorholds, not Richardson; Richardson is credited for its #943/#934 lineage. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.