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

notes/agent-41-connector-six-duo-task-card.md

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+# Task card: connector-alphabet-change, six-source DUO Pareto (agent-41)
+
+Task ID / proposed scope key: `connector-alphabet-change` (drawn), scope key
+`ff4f5113ed2256ab19ed`; chooser OS nonce `f6e1088f04305c75046a5a048c04b89a`;
+track `recursive-interfaces`; priority revision 4.
+
+Base commit; exact predecessor and dependency heads: base
+`0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2` (origin/main at session start).
+Predecessors (read-only, in-tree): `research/universal-connector-boundary`
+(PR943/PR934, Ross Richardson), `research/duo-toggle-connector` (PR1403
+claim lineage, Navinor), `research/universal-connector-six-source`
+(PR1185/PR1215). Concurrent external: https://github.com/CrocSwap/integer-mult-bounds/pull/2056
+(head `c713ff9e`, pair-source five-source restatement, claimed 17:42Z,
+released 18:22Z).
+
+Concrete object and question: in the connector model extended by exactly the
+pair-source DUO family (charged weight-2 steps), what is the six-source
+universal-path Pareto profile (k calls, W adds)? Any witness with k < 16 and
+W <= 16 Pareto-improves singleton f(6) = 16 and drops the all-63-row batch
+below 142 root queries.
+
+Scope status: **changed**; reason: the card's own first-counterexample domain
+(3/4/5 sources) is saturated by `research/duo-toggle-connector` and the
+concurrent PR above; 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 the predecessor's own recorded open question 2. Side
+result (labelled): counting closure of that predecessor's open question 3
+(three-source calls cannot lower W below 12 at k = 10 on five sources).
+
+Existing restriction being changed (or exact claim being audited): the
+singleton-alphabet value f(6) = 16 and the frozen support bound 2(n + rows)
+for the all-63-row batch; the changed premise is the added DUO operation
+family of `research/duo-toggle-connector`, already defined and charged there.
+
+Method; explicit parameter ranges/candidate alphabet: exact prefix-difference
+reduction; seeded hill-climbing over closed walks on F2^6 with weight-1/2
+steps, k in {12, 13, 14, 15}, number of DUO calls in {1..6}, total budget
+about two minutes (seed 20261014); exact lower bounds and complete all-unit
+censuses (945 and 10,395 canonical words) for exclusions; code-disjoint
+replay of every certified fact.
+
+Nearest overlapping PRs and why this adds something:
+https://github.com/CrocSwap/integer-mult-bounds/pull/2056 (five-source, same
+DUO family; different source count and open question);
+https://github.com/CrocSwap/integer-mult-bounds/pull/1185 (singleton f(6)=16,
+improved here by the DUO witness); `research/duo-toggle-connector` (five
+sources; its open questions 2 and 3 are exactly this package's main question
+and side result).
+
+Optimistic full cost budget; required observable improvement: batch 140
+queries at 142 adds (saving 2o in the predecessor price model); if only
+singleton-length words were found, no improvement and the route dies.
+
+Success evidence; falsification test; time/compute cap: explicit witness with
+all 64 masks covered (exact replay); falsification by the four mutation
+controls and the two complete censuses; compute cap two minutes of search.
+
+Predecessor authors; intended artifact and reproduction command: Ross
+Richardson, Douglas Colkitt, Hanson Sun, Navinor, Arham Tahir (comparison).
+Artifact: `research/connector-six-source-duo/` with
+`python3 -B research/connector-six-source-duo/verify.py` ->
+`PASS_CONNECTOR_SIX_DUO witness=(14,16) n5max=31 k12max=49 mutations=4`.

research/connector-six-source-duo/CERTIFICATE.json

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+{
+ "model": "closed walks on F2^6, steps weight 1 (B_j) or 2 (DUO B_jk)",
+ "seed": 20261014,
+ "controls": {
+  "duo_n3": {
+   "k": 5,
+   "W": 6,
+   "closed": true,
+   "distinct_prefixes": 5,
+   "coverage": 8
+  },
+  "duo_n4": {
+   "k": 6,
+   "W": 8,
+   "closed": true,
+   "distinct_prefixes": 6,
+   "coverage": 16
+  },
+  "duo_n5": {
+   "k": 10,
+   "W": 12,
+   "closed": true,
+   "distinct_prefixes": 10,
+   "coverage": 32
+  },
+  "singleton_n6_gray16": {
+   "k": 16,
+   "W": 16,
+   "closed": true,
+   "distinct_prefixes": 15,
+   "coverage": 64
+  },
+  "singleton_n6_nonsubspace16": {
+   "k": 16,
+   "W": 16,
+   "closed": true,
+   "distinct_prefixes": 16,
+   "coverage": 64
+  }
+ },
+ "counting": {
+  "singleton_n5_k10": {
+   "words": 945,
+   "max_coverage": 31
+  }
+ },
+ "singleton_k12_census": {
+  "words": 10395,
+  "max_coverage": 49,
+  "histogram": {
+   "22": 132,
+   "26": 495,
+   "29": 880,
+   "31": 660,
+   "32": 79,
+   "33": 1452,
+   "35": 528,
+   "36": 840,
+   "37": 22,
+   "38": 1962,
+   "40": 414,
+   "41": 264,
+   "42": 807,
+   "44": 1146,
+   "47": 528,
+   "49": 186
+  }
+ },
+ "search": {
+  "time_budget_s": 110.0,
+  "results": [
+   {
+    "k": 12,
+    "n_duos": 2,
+    "W": 14,
+    "best_coverage": 53
+   },
+   {
+    "k": 12,
+    "n_duos": 4,
+    "W": 16,
+    "best_coverage": 58
+   },
+   {
+    "k": 13,
+    "n_duos": 1,
+    "W": 14,
+    "best_coverage": 56
+   },
+   {
+    "k": 13,
+    "n_duos": 3,
+    "W": 16,
+    "best_coverage": 63
+   },
+   {
+    "k": 14,
+    "n_duos": 2,
+    "W": 16,
+    "best_coverage": 64,
+    "witness_steps": [
+     2,
+     16,
+     8,
+     16,
+     1,
+     4,
+     48,
+     4,
+     3,
+     4,
+     32,
+     8,
+     4,
+     16
+    ],
+    "witness_prefixes": [
+     0,
+     2,
+     18,
+     26,
+     10,
+     11,
+     15,
+     63,
+     59,
+     56,
+     60,
+     28,
+     20,
+     16
+    ]
+   },
+   {
+    "k": 15,
+    "n_duos": 1,
+    "W": 16,
+    "best_coverage": -1
+   },
+   {
+    "k": 14,
+    "n_duos": 4,
+    "W": 18,
+    "best_coverage": -1
+   },
+   {
+    "k": 15,
+    "n_duos": 3,
+    "W": 18,
+    "best_coverage": -1
+   }
+  ],
+  "note": "heuristic search; a missing word is a scoped search failure, not an exclusion"
+ }
+}