[petalas]

Result

Claim type: finite certificate (explicit construction). Not a search exclusion, not an optimality proof, no new κ.

Aaron Wang / navinor's #2396 (research/connector-seven-source-duo) gives a seven-source universal word in the singleton+DUO alphabet at (k, W) = (22, 24) and asks whether that is the Pareto profile. It is not. This package gives two universal words:

Word	Calls k	DUO calls	Adds W	Batch queries	Batch adds
#2396 (cited)	22	2	24	276	278
new	21	3	24	275	278
new	20	6	26	274	280

(21,24) prefix states: 0 1 33 36 38 102 100 96 98 114 82 83 115 123 120 88 89 9 41 40 8
(20,26) prefix states: 0 2 16 112 120 88 92 60 61 29 21 85 81 115 123 59 58 26 14 12

The (21, 24) word strictly dominates (22, 24). In #2396's tariff (c_B = o + p, c_DUO = o + 2p, c_A = o + p, all-127-row batch with 254 paired readouts) it saves o for every o > 0, and 3o against the 24-call singleton Gray word. The (20, 26) word beats (21, 24) exactly when o > 2p. It also settles #2396's (20, d) row at d = 6, where #2396's search reached 126/128.

Evidence and reproduction

verify.py uses only the standard library and does not search. For each word it rebuilds the steps from the prefix states and checks step weights in {1, 2}, restoration, positive even label counts, all 128 masks of P + P plus an independent dual test, and the batch bill. It reads #2396's pinned certificate, confirms its (22, 24) word, and checks the dominance relations. Four mutations per word (deleted call, DUO shrunk to a singleton, retargeted call, dropped call) are rejected. Checks are explicit, so -O does not skip them.

From the repository root (local interpreter ~/git/integer-mult-bounds/.venv/bin/python3.12, CPython 3.12):

python3.12 -B research/seven-source-duo-21-calls/verify.py
python3.12 -B scripts/research_package.py --package research/seven-source-duo-21-calls
python3.12 -B scripts/check_research_commit.py --package research/seven-source-duo-21-calls --check-source-history

Local results: the verifier prints PASS_SEVEN_SOURCE_DUO_21_CALLS k21_W24_queries=275 k20_W26_queries=274 pred_queries=276 universal=128/128 mutations_rejected=8 in under 0.1 s, with and without -O. The runner prints PASS_REGISTERED_RESEARCH_PACKAGE research/seven-source-duo-21-calls outcome=completed. The commit check prints PASS_RESEARCH_COMMIT_BYTES.

The words were found by the included simulated-annealing program search/sa.c (./sa 21 24 1 5000000 20 and ./sa 20 26 7 5000000 20). The search is provenance only and is not part of the registered check.

Limitations

• No lower bound. Runs at (20, W <= 24), (19, W <= 34), (18, W <= 36) and W = 22 at k = 21, 22 found nothing. These are bounded search failures, recorded in CERTIFICATE.json, not exclusions. The counting bound gives only k >= 17.
• Inherited from Seven-source universal connector word in the singleton+DUO alphabet: 22 calls at 24 adds, batch 276 queries at 278 adds (no new kappa) #2396: no physical supplier instance of this connector exists, so no recursive moment, frame, compiler or κ consequence follows. With o = 0 the (21, 24) word gains nothing.
• Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Is 20 the minimum call count? First step: exactly enumerate 19-point sets P with P + P = F2^7, then test for a Hamiltonian cycle with edges of weight at most two.
2. Is 24 the minimum add count with DUO calls? f(7) = 24 (Research: exact seven-source singleton minimum f(7) = 24 #3058) covers only singleton words. First step: adapt Research: exact seven-source singleton minimum f(7) = 24 #3058's coordinate-deletion argument to weight-two steps.
3. Is (20, 24) attainable? It would dominate both new words.

Related open work

• Six-source one-TRIO path misses full paired coverage at 14 calls #2134 (1337isnot1337): six-source one-TRIO paths; different source count and alphabet. No shared number.
• Research: high-arity connector toggle cost floor #2249 (1337isnot1337): high-arity connector toggle cost floor; different alphabet. No shared number.

Credit

Ross Richardson (#943) for the connector model and paired-readout criterion; research/duo-toggle-connector for the DUO family and its pricing; Aaron Wang / navinor (#2396) for the (22, 24) word, the batch bill, the tariff and the open question; Douglas Colkitt (#1185) for the Gray baseline; conorholds (#3058) for f(7) = 24. Cited, not re-audited.

CI notes

At head 88dec5cb27a344c02cfcfab227df89777ef3feb5 both hosted checks passed, plan and replay (research/seven-source-duo-21-calls): https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40585061274

This account's other open PRs opened since the claim do not study connector words or the DUO alphabet, so Related open work is unchanged.