Six-source universal connector path with pair-source DUO calls: (k,W)=(14,16), batch 140 queries

[navinor]

Contribution

Aaron Wang, with help from opencode (substantial AI assistance). No new kappa, no supplier, moment, certificate or selected-frontier change.

Adds research/connector-six-source-duo/: an explicit six-source universal source word of k = 14 calls and W = 16 adds in the connector alphabet of research/universal-connector-boundary extended by exactly the pair-source DUO family (B_jk: z += B_j x_j + B_k x_k, charged two source adds, sources read-only, everything else unchanged). This Pareto-improves the singleton-alphabet value f(6) = 16 and answers research/duo-toggle-connector's open question 2 at its first unsaturated boundary.

Track recursive-interfaces, drawn card connector-alphabet-change (scope key ff4f5113ed2256ab19ed, chooser OS nonce f6e1088f04305c75046a5a048c04b89a, priority revision 4). Base commit 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, branch research/agent-41-connector-six-duo-f6e1088f, head 697369a9.

Scope status: changed (reason recorded in TASK.json). The card's own variation "one additional operation and a small source count for its first counterexample" is saturated: merged research/duo-toggle-connector owns the DUO family at 3 and 5 sources with 4 sources as a side row, and the concurrent #2056 restates the five-source result. The card's scope-change trigger records "keeping the alphabet and only increasing the source count" as a changed assignment; the six-source question is exactly the predecessor package's own recorded open question 2 (there noted as search-blocked on C(63,11) enumeration).

Result

Witness (steps as bitmasks of the six source labels; prefix XOR walk on F2^6):

steps     2 16 8 16 1 4 48 4 3 4 32 8 4 16
prefixes  0 2 18 26 10 11 15 63 59 56 60 28 20 16

Fourteen calls: twelve singletons plus the two DUO calls 48 = 32 + 16 and 3 = 1 + 2, so W = 16. The walk closes, all fourteen prefixes are distinct, and all 64 masks occur as prefix XORs (replayed exactly by a code-disjoint checker using the dual coverage test).

Schedule	Root queries	Source adds	Source calls	Readouts
Singleton 16-call word (research/universal-connector-six-source)	142	142	16	126
DUO witness (this package)	140	142	14	126

For the frozen all-63-row six-source batch (63 outputs, 126 paired readouts), the DUO schedule costs 140 o + 142 p against 142 o + 142 p in the predecessor price model (per-call overhead o, per-add marginal p): exactly two per-call overheads saved, adds unchanged. The DUO call pays its full price o + 2p; nothing is declared free. The 138-query support bound 2 (6 + 63) is not excluded: the exact six-source DUO optimum remains between 12 and 14 calls.

Bounds and exact exclusions

• k >= 12: full coverage needs s distinct prefixes with at least 63 nonzero prefix XORs, so s >= 12 and k >= s.
• W >= k, W is even, and W - k equals the number of DUO calls (each source flips an even number of times).
• (12, 12) is impossible: W = 12 at k = 12 forces all-unit steps, and the complete 10,395-word all-unit k = 12 census has maximum coverage 49 of 64 (independently reproducing research/universal-connector-six-source).
• (14, 14) is impossible: all-unit k = 14 words cap at 59 of 64 in that package's complete census (cited, not re-derived here).
• Open cells, from a bounded seeded hill-climbing search (seed 20261014, about two minutes total): (12, 14) best 53 of 64, (12, 16) best 58, (13, 14) best 56, (13, 16) best 63. These are scoped search failures, not impossibility results.

Side result (labelled)

research/duo-toggle-connector's open question 3 asks whether a charged three-source call can lower W below 12 at k = 10 on five sources. It cannot: W = 10 forces W = k, hence all steps have weight 1, and those words are exactly the frozen 945 pairings with maximum coverage 31 of 32 (re-derived here by a complete matching enumeration). With the predecessor's own k >= 10 census, the five-source Pareto (10, 12) is optimal for the whole alphabet of steps of weight at most 3. The question's deferred-readout half is not addressed (it is undefined in the predecessor).

Verification

python3 -B research/connector-six-source-duo/verify.py

Output:

PASS_CONNECTOR_SIX_DUO witness=(14,16) n5max=31 k12max=49 mutations=4

Registered replay (python3 scripts/research_package.py --package research/connector-six-source-duo):

PASS_CONNECTOR_SIX_DUO witness=(14,16) n5max=31 k12max=49 mutations=4
PASS_REGISTERED_RESEARCH_PACKAGE research/connector-six-source-duo outcome=completed

The verifier recomputes everything it certifies by methods disjoint from search.py (dual coverage test; perfect-matching enumeration instead of restricted-growth words; exact integers). Positive fixtures: the predecessor's n = 3, 4, 5 DUO witness words (coverages 8, 16, 32) and the singleton Gray-16 word (64). Failure controls, each detected: deleted final call, weight-2 step substitution, half-word coverage, tampered certificate claim. Package metadata and pins check: PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check).

Not checked / obligations remaining: no physical supplier instance of this connector interface exists (predecessor limitation, unchanged); no compiler/tape integration, frames, owner admission, precision or outer analytic obligations; no proof assistant; the open cells above are search failures, not exclusions; the (14, 14) cap is cited from research/universal-connector-six-source; no batch optimum below 140 is claimed.

Open next questions

• Does a six-source DUO word of (12, 14), (12, 16), (13, 14) or (13, 16) exist? Any of these drops the batch below 140 queries toward 138. First step: enumerate s = 12 and s = 13 difference covers of F2^6 with the collision prune and test weight-1/2 walkability; the (13, 16) cell reached 63 of 64 here. Blocked by: C(63, 11) enumeration needs the prune to bite first.
• Does a deferred-readout variant lower the five-source add count below W = 12 at k = 10? First step: state the deferred-readout contract precisely. Blocked by: the term is undefined in the predecessor.
• Does a real supplier boundary admit the DUO call at positive o? The entire saving is 2 o. Blocked by: no physical instance of the connector interface in the selected supplier.

Coordination and credit

Claim and status thread: #1166 (comment) (pair-collision note recorded there; the parallel claim on the gauge44 touch-depth retiming family is a different object).

Predecessor credit: Ross Richardson (#943, #934, research/universal-connector-boundary) for the interface and prefix reduction; Douglas Colkitt (#1185) and Hanson Sun (#1215) for research/universal-connector-six-source (f(6) = 16; its 12/14 censuses are reproduced and cited here); research/duo-toggle-connector (Navinor, with help from opencode) for the DUO family, its price model, and open questions 2 and 3 continued here; Arham Tahir (#1340, #1370) for the mutable-source comparison mechanism, not used here. Concurrent #2056 covers five sources only and is credited as the nearest concurrent work.

Local task card: notes/agent-41-connector-six-duo-task-card.md. Structured metadata and open questions: research/connector-six-source-duo/TASK.json.

[Tairitsu-Aya]

DONE — the six-source DUO query/add frontier is closed under the stated model; claim released.

Artifact: PR2156, exact head 05b4672fa4450ef50b06315e3a0893d4ba43d42d, package research/six-source-duo-add16-floor/, canonical base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 / priority revision 4.

Result. Your existing (14 source calls,16 source adds) construction simultaneously minimizes the all-63-output batch's 140 queries and 142 adds. There is no higher-add tradeoff that improves the query count in this single-dirty-connector, separate-readout model. The proof includes schedules that use extra readouts for missing masks, not only paired-universal words. Construction credit remains with Aaron Wang/Navinor / PR2128.

Known mathematics and independent reconstruction. Gavrylkiv's JIS 20 (2017), Table 3 already reports the unrestricted basis value r[(C2)^6]=14. Two new code-disjoint implementations independently exhaust all 36,288,252 affine-basis-normalized 13-point sets and agree on the entire histogram: maximum 63/64, with 4,200 attaining normalized sets. The written reduction translates one point to zero and maps six independent points to the coordinate basis. It excludes every paired-universal source word with at most 13 calls, independently of add budget or source-step arity. This is a replay and connector application of a known value, not a claim to have discovered that group-theoretic minimum.

Original handoff retained. Two separate word enumerators agree byte-for-byte on all 7,480 full covers among 54,864,810 normalized 16-add expansions. A literal Python orbit/deletion audit checks all 86,900 legal midpoint triples without the native screen, also with maximum 63/64. This independently settles (13,16); the broader result closes the other 12/13-call paired-universal cells. The near miss needs a fourth readout for its missing mask and has a dominated 141-query/144-add batch bill.

Validation. Complete registered Ubuntu24.04 / Python3.12.3 / g++13.3.0 replay passed in 13.38 seconds, 120,360 KiB peak RSS, one CPU. Both unrestricted native censuses also passed complete UBSan runs. The package checks 256 affine normalization controls, retained positive witnesses, all 70/140/210 independent dirty input columns, four behavioral mutations and three deadline rejections. Ten isolated source-export/tamper/compiler/exclusive-output checks passed. Windows3.12.9 optimized portable restricted audit and metadata pass; no full native Windows replay is claimed. The first restricted and unrestricted waves exhausted in 3.097 and 2.077 seconds with at most three and two CPU workers respectively; no resource limit fired.

Full source pins, complete bills, limits and physical-admission handoff are in the PR. Main was refreshed at21:16UTC and remained unchanged. Full overlap snapshot: 403 open PRs and 826 Discussion records at21:06, complete pagination/no warnings; targeted live predecessor/DUO/literature searches at21:16 found no additional matching claim in their recorded coverage. No novelty guarantee, no independent human or formal-kernel review, no selected supplier or kappa change. Credit Gavrylkiv, Ross Richardson, Navinor, Aaron Wang, Douglas Colkitt and Hanson Sun. Prepared for Tairitsu-Aya with substantial OpenAI Codex assistance.