Seven-source universal connector word in the singleton+DUO alphabet: 22 calls at 24 adds, batch 276 queries at 278 adds (no new kappa)

[navinor]

Seven-source universal connector word in the singleton+DUO alphabet (k = 22, W = 24)

Contributor: Aaron Wang, with help from opencode (substantial AI assistance; no independent human review). No kappa, supplier, certificate or selected-frontier change.

Scope and result

In the prefix-walk model of research/universal-connector-boundary (Ross Richardson, #943 lineage) extended by exactly the charged pair-source DUO family of research/duo-toggle-connector (DUO_{j,k}: z += B_j x_j + B_k x_k, one dispatch, two source adds, sources read-only, everything else unchanged), this package exhibits a universal seven-source word of 22 complete source calls at 24 toggle-adds covering all 128 masks of F2^7:

steps  64 32 8 20 64 32 1 16 64 2 4 64 1 4 34 8 16 1 4 32 1 16
weight-2 steps: 20 = B_2 + B_4, 34 = B_1 + B_5 (two charged DUO calls)
k = 22, d = 2, W = 24; label counts 4, 2, 4, 2, 4, 4, 4 (all even, word restored)

The closed walk's pairwise prefix differences realize every mask of F2^7; an independent dual test (mask m covered iff some prefix p has p xor m a prefix) agrees. This is a strict Pareto improvement on the merged singleton upper bound f(7) <= 24 in research/universal-connector-six-source (a 24-call Gray word at 24 adds).

For the frozen all-127-row batch, with two paired readouts per row and every call charged:

Schedule	Root queries	Toggle-adds	Source calls	Readouts
Singleton Gray word (research/universal-connector-six-source)	278	278	24	254
This package, DUO word	276	278	22	254

In the price model c_B = o + p, c_DUO = o + 2p, c_A = o + p the batch totals are 276o + 278p against 278o + 278p: strictly better for every o > 0, saving exactly two per-call overheads o, with total adds unchanged. Nothing is declared free: each DUO call pays two source adds and one dispatch, exactly as in research/duo-toggle-connector. With o = 0 there is no gain over the Gray word.

Evidence and reproduction

Package research/connector-seven-source-duo/ (12 files): literal words and bill in CERTIFICATE.json, deterministic verifier verify.py, seeded search tools search.py and walk_search.py (seed 20261015; the headline word was the first hit of the fixed-(k, d) = (22, 2) target), full metadata and pins.

python3 -I -B research/connector-seven-source-duo/verify.py
python3.12 scripts/research_package.py --package research/connector-seven-source-duo

Observed markers (Python 3.12, Windows):

PASS_SEVEN_SOURCE_DUO k=22 W=24 queries=276 adds=278 universal=128/128 controls=945+gray+side+mutations
PASS_REGISTERED_RESEARCH_PACKAGE research/connector-seven-source-duo outcome=completed

scripts/research_package.py --package research/connector-seven-source-duo --check-only prints PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check).

Controls in the deterministic replay (the search itself is not rerun): the five-source DUO witness of research/duo-toggle-connector (smallest falsifying-instance control: closed, recorded prefixes, universal); a complete independent re-enumeration of the 945 canonical five-source singleton words whose |D(w)| histogram {16: 42, 19: 120, 21: 180, 22: 56, 23: 45, 24: 220, 26: 110, 28: 150, 31: 22} reproduces the merged 945-pairing table of research/universal-connector-boundary exactly (and its example word covers 31 of 32 masks); Gray words for n = 4..7 and the merged six-source Gray literal; both certificate words checked for closure, per-label parity, step weights in {1, 2}, all 128 pairwise masks and the dual test; the complete batch bill and its tariff; three mutation controls (deleted call, DUO shrunk to a singleton, retargeted call) each detected; a SHA-256 digest pinning the two words.

A second word of 21 calls at 26 adds is recorded as search evidence only (not Pareto-optimal here). The search's best coverages at (20, 2) = 122/128, (20, 4) = 124/128, (20, 6) = 126/128, (21, 3) = 127/128 are bounded search failures, not exclusions.

Scope status and task card

Scope changed, recorded per the card's own trigger ("keeping the alphabet and only increasing the source count"): drawn card connector-alphabet-change (chooser OS seed 4e177efb421addcdb3ba4ff20efdaefd, track recursive-interfaces, priority revision 4). DUO is already added and priced by research/duo-toggle-connector and PRs #2128 and #2156, and six sources are entirely closed (construction and optimality certificate). This package keeps the alphabet and raises the source count to seven, the first boundary where universal words are superlinear; the card's acceptance question (does the one added operation family offer a favorable completed budget) is answered at that boundary: yes, 2o on the full batch. Task card: notes/agent-52-seven-source-duo-task-card.md (committed).

Pins: base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32. Branch research/agent-52-connector-seven-duo-4e177efb, head f43a8397 (single commit on base). Registered inputs: research/universal-connector-boundary/CERTIFICATE.json, research/universal-connector-six-source/EXPECTED.json, research/duo-toggle-connector/CERTIFICATE.json (pins in SOURCES.json, all matched at registration).

Overlap

Fresh complete overlap snapshot before submission (2026-10-15T07:23Z; 599 open PRs, 1126 discussion records; terms seven-source, seven source, f(7), universal word, DUO, additive basis, difference cover): no seven-source full-coverage (128-mask) claim in any alphabet. Distinct objects nearby, not superseded by them: discussion #2221 and PRs #2229 and #2236 cover the 63 weight-1/2/3 masks at seven sources with 14/18 singleton events; #2275 / PR #2282 concern root-row counts. No matches is not a novelty certificate.

Limitations and remaining obligations

No physical supplier instance of the connector interface exists (the predecessors' limitation, unchanged), so no compiler/tape integration, frames, owner admission, precision or outer analytic obligation is discharged and no recursive moment or kappa follows. Not claimed: minimality of k = 22 or W = 24 at n = 7 (open question in the package), any change to the singleton f(7) gap 18 <= f(7) <= 24, or any exclusion of shorter DUO words.

Credit

Model and five-source boundary: Ross Richardson (#943, #934). Six-source value f(6) = 16 and the n = 7 bounds: Douglas Colkitt (#1185) with Hanson Sun's independent verification (#1215). DUO alphabet, five-source Pareto profile and pricing: research/duo-toggle-connector (#1403 lineage). Six-source DUO word (14, 16): Aaron Wang/Navinor with opencode, #2128. Six-source DUO batch optimality and the basis value r[(C2)^6] = 14 (Volodymyr Gavrylkiv, JIS 20 (2017), Table 3): PR 2156 by Tairitsu-Aya. New work is Apache-2.0; no third-party code vendored.