Exact exclusion for a six-source TRIO connector path

research/connector-trio-cycle25-seed1b82df/CERTIFICATE.json

@@ -0,0 +1,122 @@
+{
+  "schema_version": 1,
+  "claim": "No normalized simple 12-vertex F2^6 path with eleven unit edges and a closing weight-three edge has all 63 nonzero pairwise differences.",
+  "normalization": {
+    "dimension": 6,
+    "initial_prefixes": [
+      0,
+      1,
+      3
+    ],
+    "reason": "Translate q0 to 0; rotate the unique weight-three edge to closing; permute coordinates so the first two distinct unit edges are 1 and 2.",
+    "remaining_unit_edges": 9,
+    "raw_upper_bound": 1953125
+  },
+  "profile": {
+    "vertices": 12,
+    "unit_edges": 11,
+    "weight_three_closing_edges": 1,
+    "calls": 12,
+    "source_adds": 14,
+    "required_nonzero_differences": 63
+  },
+  "counts": {
+    "normalized_simple_walks": 1098204,
+    "closing_weight_three_candidates": 709812,
+    "coverage_histogram": {
+      "15": 4188,
+      "25": 3072,
+      "26": 22272,
+      "27": 40368,
+      "29": 49632,
+      "30": 70800,
+      "31": 50112,
+      "36": 24768,
+      "37": 8640,
+      "38": 70560,
+      "39": 4608,
+      "40": 50976,
+      "41": 86400,
+      "42": 44856,
+      "43": 70896,
+      "44": 4608,
+      "45": 52992,
+      "46": 13824,
+      "47": 31248,
+      "49": 4992
+    },
+    "maximum_coverage": 49,
+    "full_cover_witnesses": 0
+  },
+  "maximum_example_prefixes": [
+    0,
+    1,
+    3,
+    7,
+    6,
+    14,
+    15,
+    31,
+    29,
+    61,
+    57,
+    41
+  ],
+  "maximum_example_missing_masks": [
+    10,
+    11,
+    21,
+    22,
+    23,
+    33,
+    35,
+    37,
+    43,
+    44,
+    45,
+    48,
+    49,
+    53
+  ],
+  "endpoint_records_sha256": "e3035dbb174381d4ecfb35088dfdacd614b93cc14dde0f238bcbd7e509d201eb",
+  "positive_control_pr2128": {
+    "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
+    ],
+    "calls": 14,
+    "source_adds": 16,
+    "readouts": 126,
+    "root_queries": 140,
+    "source": "CrocSwap/integer-mult-bounds PR #2128, head 697369a971eed02c0fbcd75ebefec9f59945f08e"
+  },
+  "scope_limit": "This excludes only the stated normalized profile. It does not exclude other arities, other step-weight profiles, repeated boundary prefixes, deferred readout, or other interfaces."
+}

research/connector-trio-cycle25-seed1b82df/OVERLAP.json

@@ -0,0 +1,33 @@
+{
+  "schema_version": 1,
+  "refreshed_utc": "2026-10-14T20:54:54Z",
+  "command": "python3 scripts/find_overlap.py --refresh --all-terms 'six-source' 'TRIO' '12-call' 'weight-3' --limit 20",
+  "coverage": {
+    "open_pull_requests": {
+      "fetched_at_utc": "2026-10-14T20:52:59Z",
+      "complete_pagination": true,
+      "records": 399
+    },
+    "discussions": {
+      "fetched_at_utc": "2026-10-14T20:54:54Z",
+      "complete_pagination": true,
+      "records": 816
+    }
+  },
+  "exact_all_terms_matches": 0,
+  "warnings": [],
+  "nearest_distinct_open_work": [
+    {
+      "reference": "https://github.com/CrocSwap/integer-mult-bounds/pull/2128",
+      "head": "697369a971eed02c0fbcd75ebefec9f59945f08e",
+      "difference": "Uses two charged weight-two DUO calls at six sources, with (k,W)=(14,16). This census tests one charged weight-three TRIO call and the distinct (12,14) profile."
+    }
+  ],
+  "coordination": {
+    "claim": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2141",
+    "hub_pointer": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1167#discussioncomment-18933826",
+    "claim_expiry_utc": "2026-10-14T22:45:00Z",
+    "status": "active"
+  },
+  "limits": "The overlap utility searched open pull requests, discussion claims, and local task/readme records for the exact conjunction. It is not a novelty certificate and did not inspect closed code diffs or arbitrary package files."
+}

research/connector-trio-cycle25-seed1b82df/PR-BODY.md

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+## Contribution
+
+Adds an exact scoped census for a six-source connector alphabet with one
+charged three-source call. For a closed 12-call word with eleven singleton
+steps and one weight-three step, no path covers all 63 nonzero pairwise
+prefix differences. After symmetry normalization, the complete exact search
+contains 1,098,204 simple paths and 709,812 eligible closing-edge candidates;
+maximum coverage is 49, and no candidate covers 63 masks. This excludes only
+the stated call and source-add profile. No supplier, recursive bound, or
+`kappa` change is claimed.
+
+The fully charged hypothetical bill would be `138 o + 140 p`, including 126
+output readouts, versus PR #2128's `140 o + 142 p`. The TRIO call is charged
+`o + 3p`, and any additional implementation surcharge must be below
+`2o + 2p` for the modeled saving to survive. The complete census shows that
+this profile does not realize the saving.
+
+## Verification
+
+Run:
+
+```sh
+python3 -I -B research/connector-trio-cycle25-seed1b82df/verify.py
+python3 scripts/research_package.py --package research/connector-trio-cycle25-seed1b82df --check-only
+```
+
+The verifier independently scans all normalized paths by a product iterator,
+recomputes difference sets from scratch, checks the exact histogram and
+endpoint-record digest, confirms the PR #2128 full-cover fixture, and runs
+three negative controls. Direct local output under Python 3.14.7:
+
+```text
+PASS_TRIO_CYCLE_VERIFY walks=1098204 endpoints=709812 max=49 full=0 pr2128=63/63 controls=3
+```
+
+The metadata-and-source-pin check passes locally. The registered replay is
+configured for Python 3.12; it was not run locally because Python 3.12 is not
+installed.
+
+## Provenance and disclosure
+
+Seed `1b82dfd0673536e282d9d9e7adcb1a04`, track `recursive-interfaces`, task
+`connector-alphabet-change`, priority revision 4. Exact base:
+`0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`. Selected record SHA-256:
+`10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+All pins and the refined scope are in `TASK.json` and `SOURCES.json`.
+
+Nearest positive comparator and fixture: Aaron Wang, [PR #2128](https://github.com/CrocSwap/integer-mult-bounds/pull/2128),
+head `697369a971eed02c0fbcd75ebefec9f59945f08e`. The current result uses a
+different alphabet extension: exactly one charged TRIO call. Substantial
+OpenAI Codex assistance was used; no independent human review is claimed.

research/connector-trio-cycle25-seed1b82df/README.md

@@ -0,0 +1,104 @@
+# Six-source connector with one charged TRIO call: exact scoped exclusion
+
+**Scoped negative result; no supplier, assembled bound, or new `kappa`.** On
+the six-source connector over `F_2^6`, a closed source path with twelve calls,
+eleven singleton calls, and one weight-three call cannot produce all 63
+nonzero pairwise prefix differences. The complete normalized census contains
+1,098,204 simple paths; 709,812 have the required weight-three closing edge.
+Their maximum coverage is 49/63, attained by 4,992 candidates, with no full
+cover. This excludes only this exact call and source-add profile.
+
+The operation family is the universal connector's singleton source call
+extended by `TRIO_{jkl}: z += B_j x_j + B_k x_k + B_l x_l`. It reads three
+source rows and charges all three source additions; the rows stay read-only.
+For a closed source word, the prefix states are distinct elements of `F_2^6`,
+and each output mask must occur as a difference of two prefixes. The pairwise
+difference test is the connector endpoint condition for arbitrary dirty
+inputs.
+
+## Exact search and result
+
+There are twelve prefix vertices in a twelve-call closed word; the last call
+returns from the twelfth vertex to the first. Full coverage of 63 nonzero
+masks needs at least twelve distinct vertices because `binom(11,2) = 55`.
+Thus twelve calls meet the counting floor. In the tested profile, the eleven
+singleton edges form a simple path and the TRIO edge closes it.
+
+Normalize any candidate by translating its first vertex to zero, rotating the
+unique TRIO edge to the closing position, and permuting coordinates so the
+first two singleton edges are `1` and `2`. The vertices then start
+`(0, 1, 3)`. A self-avoiding unit walk adds nine more vertices, and its last
+vertex must have Hamming weight three so the closing TRIO returns to zero.
+This is a complete orbit representative: simplicity forces the first two
+unit edges to use distinct coordinates, and coordinate permutations act
+transitively on those ordered pairs.
+
+[`search.py`](search.py) uses a depth-first walk and incrementally packs the
+pairwise differences into an integer. [`verify.py`](verify.py) independently
+enumerates all `6^9` coordinate suffixes, filters repeated vertices, and
+recomputes each eligible difference set from scratch. Both exact methods
+agree on the census and a SHA-256 digest over all 709,812 endpoint records.
+The coverage histogram, maximal example, and exact census counts are in
+[`CERTIFICATE.json`](CERTIFICATE.json).
+
+The maximum example has prefixes
+`0, 1, 3, 7, 6, 14, 15, 31, 29, 61, 57, 41` and misses the fourteen masks
+listed in the certificate. A positive control replays the 14-call, 16-source-
+add witness in [PR #2128](https://github.com/CrocSwap/integer-mult-bounds/pull/2128),
+which covers all 63 nonzero masks under the same pair-difference test.
+
+## Fully charged comparison
+
+Both schedules use two readouts for each of 63 outputs, so each has 126
+readouts. Let `o` be the complete cost of each root call and `p` the cost of
+each source addition. The PR #2128 schedule has fourteen source calls and
+sixteen source additions: `140 o + 142 p` including readouts. A hypothetical
+schedule in this profile would have twelve source calls and fourteen source
+additions: `138 o + 140 p`. The TRIO is charged as `o + 3p`; it is not a free
+three-source operation. If an implementation adds a further TRIO surcharge
+`tau`, the modeled saving survives only when `tau < 2o + 2p`.
+
+This profile has no full-coverage path, so that favorable completed budget
+does not materialize here. Other arities, other weight profiles, mixed TRIO
+and DUO paths, implementation surcharges, and supplier availability remain
+open. No connector supplier is currently supplied by this experiment, so it
+does not establish a recursive saving or change the selected frontier.
+
+## Reproduction
+
+From the repository root:
+
+```sh
+python3 -I -B research/connector-trio-cycle25-seed1b82df/verify.py
+python3 scripts/research_package.py --package research/connector-trio-cycle25-seed1b82df --check-only
+```
+
+Expected verifier output:
+
+```text
+PASS_TRIO_CYCLE_VERIFY walks=1098204 endpoints=709812 max=49 full=0 pr2128=63/63 controls=3
+```
+
+The repository's registered replay runtime is Python 3.12. The local runtime
+used for this handoff is Python 3.14.7; Python 3.12 is not installed in the
+research environment.
+
+## Open next questions
+
+- Does a twelve-call six-source cycle with ten singleton edges and two
+  charged TRIO edges attain all 63 nonzero differences? Its source-add count
+  is sixteen, so a witness would match PR #2128's add count while saving two
+  complete calls before any TRIO-specific surcharge. First step: normalize
+  both weight-three edges under cyclic rotation and coordinate permutation,
+  then enumerate only simple paths whose total XOR is zero. Stop at a full
+  exact census of that fixed `10×weight-1 + 2×weight-3` profile. Blocked by:
+  no complete covering word has been identified.
+
+## Attribution
+
+The source-call operation and pair-difference endpoint follow the universal
+connector in `research/universal-connector-boundary/`. The cost comparator
+and positive witness are from Aaron Wang's [PR #2128](https://github.com/CrocSwap/integer-mult-bounds/pull/2128).
+Substantial OpenAI Codex assistance was used for the mathematical modeling,
+search implementation, and verification. No independent human review is
+claimed.

research/connector-trio-cycle25-seed1b82df/SOURCES.json

@@ -0,0 +1,43 @@
+{
+  "schema_version": 1,
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "e8870e5917057d59b730eb14721a778639eca050",
+      "sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "research/universal-connector-boundary/README.md",
+      "usage": "provenance",
+      "git_blob": "d277f7c5830b1f2bc42e1ff41ab86e7d7564929f",
+      "sha256": "23415f7829ab8d80a783142f10f7d31abbce84579362769984680af3df954476"
+    },
+    {
+      "path": "research/universal-connector-boundary/CERTIFICATE.json",
+      "usage": "provenance",
+      "git_blob": "079e3e1441dc07c2f0536e9527acf290e51ea9c0",
+      "sha256": "e733746ba3e5151c829b4e8728bfff5ae3ed24d8d203eb77c735d20758e8266a"
+    },
+    {
+      "path": "research/duo-toggle-connector/CERTIFICATE.json",
+      "usage": "provenance",
+      "git_blob": "accffc014ce13993ddce723abd4256dc0c884e5b",
+      "sha256": "a1ad3af8bbc872a8abcf8df543267eb82cfa0ede8d581a30b6c751307bb4f267"
+    }
+  ],
+  "external_references": [
+    {
+      "name": "PR #2128 positive comparator and fixture",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2128",
+      "head": "697369a971eed02c0fbcd75ebefec9f59945f08e"
+    }
+  ]
+}