Pair-source connector calls: five-source universal path of exactly ten, 72-query support bound attained

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Summary

Contributor: Aaron Wang, with help from opencode.

This PR adds exactly one operation family to the connector alphabet of research/universal-connector-boundary: the pair-source connector call B_jk: z += B_j x_j + B_k x_k, one complete query per call, charged two coefficient/source accesses and one query record. Everything else of that interface is unchanged (dirty rows, paired readouts, restoration of z, no free copies).

Result on the five-source, all-31-nonzero-row matrix (package research/connector-pair-source-path-seed2cdb/):

Quantity	Fixed alphabet (predecessor)	With pair-source calls
Minimum universal source path	12	10
Complete queries, all-31 matrix	74	72
Support bound 2 x (5 + 31)	72, unattainable	72, attained
Source-side leaf cost, best labeling	120	120

The query optimum in the extended alphabet is exactly 72: 62 paired readouts (each nonzero row needs two A_i calls) plus a source path of length at least 10, attained by the certified word. In the fixed alphabet the same support bound of 72 was proved unattainable by the 945-pairing census, and parity ruled out 73.

The word

Closed walk on the five-bit parity space, one step per source query, step vectors of Hamming weight 1 (single B_j) or 2 (pair B_jk):

prefixes: 0  2  3  5  13  15  30  26  10  8  0
steps:    2  1  6  8   2  17   4  16   2  8
calls:   B_1 B_0 B_12 B_3 B_1 B_04 B_2 B_4 B_1 B_3

The prefix set {0,2,3,5,8,10,13,15,26,30} has full XOR-pair coverage of all 32 masks (verified two ways in verify.py). Each output's two A_i readouts go at the two boundaries whose prefix XOR is its row mask; characteristic two cancels the dirty connector. Source multiplicities are (2,4,2,2,2), all even, so z is restored.

Why ten is optimal

1. Each of the 31 nonzero masks needs its own unordered pair of prefixes, and C(8,2) = 28 < 31, so the prefix set has size at least 9.
2. A complete census of all C(31,8) = 7,888,725 nine-point sets finds zero with full XOR-pair coverage; maximum coverage is 29. Two independent implementations agree (recursive DFS in discovery, itertools.combinations in verify.py --full). So the prefix set has size at least 10.
3. The walk is closed (z restoration), so the prefix set size is at most the walk length; hence the length is at least 10, and the word attains 10.

This is a finite computational exclusion for the stated interface, not an impossibility theorem for other circuit models.

Pricing (predecessor fixture)

Fixture: B factor leaf prices 1, 4, 10, 16, 28; each A leaf price 1; every call doubled for the inverse in the complete cost; dispatch D paid per root query. Renaming sources preserves coverage, so free relabeling is granted on both sides.

Schedule	Queries	B leaves (min relabeling)	Total leaves
This word, best labeling	72	120	182
This word, fixture labeling	72	126	188
Predecessor ten-call path	74	118	182
Predecessor twelve-call path	74	120	182

At the best labeling this schedule ties the predecessor's leaf cost exactly and saves 2 dispatch records, so the completed cost is strictly lower for any positive per-query dispatch. At the fixed fixture labeling it pays 12 extra doubled leaf charges against 2 saved dispatch records: favorable iff one query's dispatch cost exceeds 6 leaf-price units. The tie at the best labeling is optimal among 72-query schedules: a length-10 source path needs at least one pair call (pure-single ten-call paths are excluded by the 945-pairing census), so its weight sum is at least 12.

Verification

python3 -I -O -B research/connector-pair-source-path-seed2cdb/verify.py

Expected last line: PASS_CONNECTOR_PAIR_SOURCE_PATH. Checks the word (two independent coverage recomputations), the dirty replay at L = 1,2,3 with seeded dense nonsymmetric noncommuting maps over F_2 and arbitrary dirty x, z, y state (exact boundary on all 31 outputs), three mutation controls (dropped call, mutated step, misplaced readouts) that must all be rejected, and the pricing numbers.

python3 -I -O -B research/connector-pair-source-path-seed2cdb/verify.py --full

Expected: CENSUS {"sets": 7888725, "covering": 0, "max_coverage": 29} then PASS_CONNECTOR_PAIR_SOURCE_PATH_CENSUS. This replays the exact nine-point census behind the lower bound.

python3 scripts/research_package.py --package research/connector-pair-source-path-seed2cdb --check-only

Expected: PASS_PACKAGE_METADATA_AND_PINS.

Scope and limits

• Exactly one added family (B_jk); a different extension is a different question. Pair calls are charged, never free.
• No instance of this interface is claimed inside the selected supplier; there is no kappa claim and the selected record is unchanged. The a < D/E diagnostic does not apply: no supplier moment or recursive profile changes here.
• The necessity side (62 readouts, two per nonzero row) inherits the predecessor's specialization argument unchanged; only the source-path side changes.
• The word was found by seeded random search over covering sets (seed 4242, trial 911); minimality rests on the census plus the counting bound, not on search exhaustion at length 10.

Credit and provenance

Predecessor: research/universal-connector-boundary (fixed-alphabet four/five-source boundary, 945-pairing census, 74-query optimum) and #934 (three-by-three result). Task card, scope pins and open next questions are in research/connector-pair-source-path-seed2cdb/TASK.json and README.md. Claim thread: #1166 (comment)

Substantial AI assistance: with help from opencode. New contributions are under the repository's Apache-2.0 license.