[petalas]

Summary

For the unsigned F3 complete-call connector of #4225/#4238 with six immutable sources, all 728 nonzero ternary rows and one arbitrarily dirty restored connector, the exact optimum is 2208 calls, six above the 2202-call operator-count floor. This answers #4238's six-source open question. The claim type is a scoped proof with an explicit finite witness. No new κ and no selected-frontier change. Only research/f3-ternary-six-source/ is added.

• Deletion lemma, averaging form (proof). Deleting source j (its B_j calls and every A_v with v_j ≠ 0) from a legal n-source word leaves a legal (n−1)-source word. Summing the surviving extra call triples over j gives Σ_j E(W_j) ≤ (n−1)E(W), so e(n) ≥ ⌈n·e(n−1)/(n−1)⌉.
• Lower bound 2208. With Prove the exact 744-call F3 optimum for full ternary five-source demand #4238's five-source value e(5) = 1 (cited, not re-run), e(6) ≥ 2. No census at n = 6 is needed.
• Construction 2208. The 24-event path B5 B4 B0 B1 B0 B3 B2 B4 B3 B1 B0 B1 B4 B5 B0 B1 B0 B2 B5 B3 B2 B1 B0 B1 (B0, B1 six times, others three) has every nonzero row of F3^6 as a triple sum of boundary vectors, so 24 + 3·728 = 2208.
• Larger n (lower bounds only). Combining the recurrence with the counting bound gives at least 3, 4, 8, 14 extra triples at n = 7, 8, 9, 10, i.e. ≥ 6588 calls at n = 7 and ≥ 19716 at n = 8. This improves Prove the exact 744-call F3 optimum for full ternary five-source demand #4238's floor-plus-three bound at n = 6, 7, 8.

Evidence and reproduction

The verifier uses only the standard library and writes no files. It checks the interface source pin and the witness coverage. It builds the literal 2208-call word and checks it with a generic noncommuting coefficient ledger and with dense random F3 execution at dimensions 2 and 3 with dirty registers. All six source deletions of the literal word must be legal five-source words with extras (1, 1, 2, 2, 2, 2). The verifier also prints the lower-bound table for n = 5..12 and runs five rejection controls.

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/f3-ternary-six-source/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/f3-ternary-six-source/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/f3-ternary-six-source
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/f3-ternary-six-source --check-source-history

Expected marker: PASS_F3_TERNARY_SIX_SOURCE calls=2208 floor=2202 rows=728. All four pass locally (CPython 3.12, macOS arm64); each registered command takes about 5 s.

Limitations

• Abstract unsigned complete-call interface over F3 at unit prices only.
• The six-source lower bound depends on Prove the exact 744-call F3 optimum for full ternary five-source demand #4238's five-source census (e(5) = 1). It is cited and not re-run here.
• Exact values for n ≥ 7 are not determined. The annealing maxima quoted in the README (597 rows for 18-event and 682 for 21-event paths) are search observations, not proofs, and the lower bound does not use them.
• Admission, coefficient preparation, dispatch and physical charges are unknown. No consumer for ternary demand is identified. No proof-kernel certificate. This is not a complete multiplication theorem.

Open next questions

1. Seven sources: is the 6588-call lower bound attained? A bounded annealing run over 30-event paths with three doubled sources reached 1952 of 2186 rows.
2. Eight sources: the recurrence and counting bounds agree at four extra triples, so one 36-event covering path would settle n = 8 exactly.
3. A concrete caller that needs coefficient-2 rows.

Related open work

• Prove the exact 744-call F3 optimum for full ternary five-source demand #4238 (head 191a905d7adba6695b079f744140a73e832787a8): five-source ternary optimum 744 and the predecessor of this package. Its e(5) = 1 is the premise here, and its 15-event gap path is reproduced as a control (241 of 242 rows, checked numerically). Its "n ≥ 7 fails by counting" (≥ floor + 3) agrees with, and is improved by, the n = 7 row here (≥ floor + 9).
• 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-mask floor for every source count with an attainment-only deletion lemma. Different demand; no overlapping number.
• Prove the exact 108-call unsigned F3 five-source connector optimum #4225, Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231: five-source binary-mask demand (108 calls at F3). Different demand; no overlapping number.
• Research: exact F3 minimum for the one-source additive connector #4189: one-source unsigned F3. No overlapping six-source number.

Overlap was checked against 100 open PR titles, recent account PRs, recent #1166 comments, find_overlap.py --task connector-alphabet-change and a README grep on main. Closed work and arbitrary diffs are incompletely covered. An open independent-verification claim on #4238 exists in #1166; this PR does not audit that census.

CI notes

Hosted run 39493468837 is green on head f436d2911351e5be2842208fe96b0b66442323a4: plan passed in 2m43s and the package replay in 1m48s. The hosted log contains the scientific marker and PASS_REGISTERED_RESEARCH_PACKAGE. No check failed, and no local obligation was left to CI.

Merge danger

Two-way door. The blast radius is one new isolated package and its opt-in replay.

Produced by Nick Petalas. Credit to #4238, #4225, #4232, Riquelme Soares's #4189, Ross Richardson's #943/#934 lineage and ywz65's contract map. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

[rcfc-jreed]

Independent scoped audit of #4244 F3 ternary six-source at f436d291: 8/8

Code-disjoint local fork. Head f436d2911351e5be2842208fe96b0b66442323a4. CLAIM.

No packaging defect. PASS_F3_TERNARY_SIX_SOURCE. Pins + SOURCES digests; fork rc=0.

Finite replay ≠ formal proof. James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).