[petalas]

Summary

Over F3, with seven immutable sources, all 127 nonzero binary-mask outputs and one arbitrarily dirty restored connector, no word attains the 402-call floor. With the 405-call word of #4232, the exact seven-source optimum is 405 calls. By #4232's deletion lemma, the three-per-operator floor 3(n + 2^n − 1) is attained exactly for n ≤ 6 and fails for every n ≥ 7. This answers #4232's open question 1 and replaces its interval 6 ≤ N ≤ 13 with N = 6.

Claim type: a scoped proof whose exhaustive step is a finite computer census. No new κ, no selected-frontier change and no multiplication theorem. Only research/f3-seven-source-connector-floor/ is added.

Argument

#4232 reduces a floor word to a restored 21-event source path, each source three times. The path is valid when every binary mask equals q_a + q_b + q_c for prefix boundaries a ≤ b ≤ c. Because q_21 = q_0 and 3q = 0 in F3, rotating the path leaves the served set unchanged, and relabelling the sources does too. Two separately written C enumerators then check one representative family each:

Engine	Reduction	Paths checked	Floor paths
engine/reduced.c	first-occurrence labels; position 0 starts a minimum cyclic gap	4,130,223,840	0
engine/independent.c	first-occurrence labels; last event differs from the first	32,590,958,400	0

The engines also use different F3 arithmetic: paired bit masks in one, base-3 tables in the other. The independent engine's leaf count equals the closed-form family size ((3n)!/6^n − n(3n−2)!/6^(n−1))/n! for n = 2..7. For n ≤ 4, a pure-Python brute force over every path matches both engines. Each reduced solution list meets every solution orbit and contains only solutions. Both engines find floor paths for n = 1..6.

Evidence and reproduction

From the repository root (a C11 cc is required):

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/f3-seven-source-connector-floor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/f3-seven-source-connector-floor/verify.py --n7-shards 0,1
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/f3-seven-source-connector-floor/verify.py --n7-all
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/f3-seven-source-connector-floor/verify.py --n7-independent
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/f3-seven-source-connector-floor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/f3-seven-source-connector-floor --check-source-history

Local results:

• default mode PASS_F3_SEVEN_SOURCE_CALIBRATION in 33 s (reduced engine n ≤ 6, independent engine n ≤ 5; --n7-independent adds the independent n = 6 calibration);
• --n7-all PASS_F3_SEVEN_SOURCE_CENSUS with all 18 reduced shards matching the certificate (97 s wall, 1,346 CPU s on 18 cores);
• the independent census, 18 shards with zero solutions in about 12.5 minutes wall on 18 cores;
• research_package.py PASS_REGISTERED_RESEARCH_PACKAGE, running all four registered commands in 185 s;
• check_research_commit.py PASS_RESEARCH_COMMIT_BYTES.

The generated receipt 15d82a8e77210537cf5d9e99fac6e865aefcbe7de7a86e3339691a56a84c8e10 covers all eight package files at head aa61e1746f8c12807bed59cd169591bd82320bc6.

Limitations

• The exclusion is a computer census, not a written proof. No SAT certificate, proof-kernel check or independent human review is claimed.
• CI replays only part of the census. The package-replay cap of 1200 s total holds the calibration and 6 of the 18 reduced shards (0, 1, 8, 9, 16, 17). The complete reduced census (--n7-all) and the independent census (--n7-independent) ran locally. They are opt-in and compare against the per-shard leaf counts committed in CERTIFICATE.json.
• Both engines rely on Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232's reduction from floor words to boundary triples, so a flaw there would affect both.
• The result covers only the unsigned, immutable-source, complete-call F3 interface. Admission, coefficient preparation, dispatch and physical costs are additional and unknown, and no consumer is instantiated.

Open next questions

1. What are the exact optima for n = 8..13? Each is at least the floor plus three. Is it exactly floor + 3, as for n = 7?
2. What is the most masks a seven-source 21-event path serves with triples? Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232's annealing reached 126.
3. Is there a short written proof of the n = 7 exclusion, for example from Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232's window form applied to the all-ones and weight-six masks?

Related open work

• Bound the unsigned F3 connector floor for every source count: attained for n ≤ 6, impossible for n ≥ 14 (no new κ) #4232 (head 86ad339efc9f817846479431d8cc0f04edf95fb9): the n-source F3 floor, attained for n ≤ 6 with a seven-source upper bound of 405. This PR answers its open question 1. Its 405-call word is re-checked here, and 405 agrees.
• Reach the unsigned connector floor for k ≤ 2c in every characteristic; 9 ≤ N(4) ≤ 21 (no new κ) #4248 (head e7df122403cfc2febfb6799abb821487d8d1ee49): the unsigned floor for k ≤ 2c sources in characteristic c. At c = 3 this is k ≤ 6, which agrees with N = 6 here. The census shows that range is exact at c = 3.
• Prove the exact 108-call unsigned F3 five-source connector optimum #4225 (head 397025101d23195a5193c7912e6cd384eccdbf1c) and Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231 (head 8a956cbc80dc55697fd3e214812ce567b5eca48f): the five-source F3 optimum of 108. Both engines find floor paths at n = 5 (108 calls), which agrees.
• Prove the exact 744-call F3 optimum for full ternary five-source demand #4238 and Prove the exact 2208-call F3 optimum for full ternary six-source demand (no new κ) #4244 study full ternary demand, a different output set, so no number is shared.

CI notes

Hosted run 39498445184 on head aa61e1746f8c12807bed59cd169591bd82320bc6 is green: plan passed in 2m46s and the package replay in 9m33s. Per-command times: calibration 99 s, shards 0,1 149 s, shards 8,9 136 s, shards 16,17 124 s, each under its 300 s timeout.

The first run (39496698411) failed because this package's own default mode exceeded its 240 s timeout on the hosted runner (verification timed out), about three times slower than locally. The fix commit uses a pair-sum Python brute force and moves the independent engine's n = 6 calibration into the opt-in mode. Neither change affects the claim. The cause was this package, not shared infrastructure.

Produced by Nick Petalas. The interface, readout criterion, deletion lemma and 405-call word are from #4232. The mod-three count argument is from #4225. The F3 alphabet and one-source handoff are Riquelme Soares's #4189. The connector interface and prefix-readout criterion originate in Ross Richardson's #943. Apache-2.0.

[rcfc-jreed]

Independent scoped audit of #4250 F3 seven-source connector floor exclusion at d043a58a: 10/10 (SOURCES/CERTIFICATE layer)

Code-disjoint checks on sealed package bytes. Head d043a58aa43d21e2da6c6405127ec49412078815. CLAIM.

No packaging defect in SOURCES scope. Both sealed digests match; CERTIFICATE.json is schema_version 1 / field F3 with calibration table; README states optimum 405 and floor for n ≤ 6.

Not covered here: compile/cc replay of engine/reduced.c and engine/independent.c (compiler unavailable in this audit environment); --n7-all full census; Lean; new κ; merge/close.

Finite pin/seal inspection ≠ formal proof / fresh C census.

James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).