Add five-source arity-four connector cost bound

research/five-source-arity-four-bound/OVERLAP.json

@@ -0,0 +1,85 @@
+{
+  "task_id": "connector-alphabet-change",
+  "checked_on_utc": "2026-10-19",
+  "canonical_main_refresh": {
+    "command": "git fetch --no-tags https://github.com/CrocSwap/integer-mult-bounds.git main",
+    "commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+    "priority_revision": 6,
+    "relevant_guidance_and_selected_record_diff_from_launch_base": "none"
+  },
+  "offline_search": {
+    "command": "python3 scripts/find_overlap.py --task connector-alphabet-change --offline --cache /home/user/int-bounds-pass/20261018-luna-xhigh/overlap.json --limit 8",
+    "cache_report": "/home/user/int-bounds-pass/20261018-luna-xhigh/overlap-intake.json",
+    "cache_searched_at_utc": "2026-10-19T02:10:06Z",
+    "status": "POSSIBLE_OVERLAP_NOT_A_NOVELTY_CERTIFICATE",
+    "coverage": {
+      "open_prs": 586,
+      "discussions_and_comments": 2536,
+      "complete_pagination": true,
+      "freshness_warning": "offline cache; refresh relevant live heads before submission"
+    },
+    "possible_matches": 46,
+    "relevant_hits": [
+      {
+        "number": 2249,
+        "title": "Research: high-arity connector toggle cost floor",
+        "state": "OPEN",
+        "head": "0c6ed7958433b2cbc0beaa331ff882aa08221a0d",
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2249",
+        "declared_scope": "Six sources; arity-four through arity-six source calls, charged one dispatch plus each constituent source addition."
+      },
+      {
+        "number": 2134,
+        "title": "Six-source one-TRIO path misses full paired coverage at 14 calls",
+        "state": "OPEN",
+        "head": "388a63c83cf24c3355be9798bde5a381188dadb3",
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2134",
+        "declared_scope": "Six sources; one atomic TRIO plus thirteen singleton source calls."
+      },
+      {
+        "number": 3586,
+        "title": "Charge one direct connector call on the one-by-one instance",
+        "state": "OPEN",
+        "head": "cde0d3c3a3be86de0c051dc1d2f270ff0f5ae2cd",
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/3586",
+        "declared_scope": "One source and one output; a direct product call is compared with the four-call B-A-B-A word."
+      },
+      {
+        "number": 1930,
+        "title": "Construction: three-stage persistent prefix shear chain (no new kappa)",
+        "state": "OPEN",
+        "head": "11cb2266f7e99279c440a301ea4b862587a9afa2",
+        "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1930",
+        "declared_scope": "Changes the boundary to a target-to-source prefix chain without a dirty connector; not the arity-four source call."
+      }
+    ]
+  },
+  "live_discussions": {
+    "start_here_1163": {
+      "updated_at_utc": "2026-10-16T02:54:20Z",
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1163",
+      "relevant_guidance": "Claims are advisory; state object, question and bounded domain; check overlap; distinguish provisional evidence from established results."
+    },
+    "handoffs_1166": {
+      "updated_at_utc": "2026-10-19T00:48:29Z",
+      "recent_comment_window": 100,
+      "recent_connector_match_count": 0,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166",
+      "bounded_snapshot": "/home/user/int-bounds-pass/20261018-luna-xhigh/handoffs.json",
+      "decision": "No direct connector match was identified in the recent bounded snapshot or live recent-comment window. This is not a novelty certificate; older DUO/TRIO and pair-source records remain scope evidence, not current reservations."
+    }
+  },
+  "live_open_pr_search": {
+    "query": "gh pr list --repo CrocSwap/integer-mult-bounds --state open --search connector --limit 100",
+    "results": [
+      {"number": 3779, "title": "Record the six-window connector gate bill", "head": "14aaaff18dbb8c2860108046e051921305c32224", "updated_at_utc": "2026-10-18T20:36:35Z"},
+      {"number": 2249, "title": "Research: high-arity connector toggle cost floor", "head": "0c6ed7958433b2cbc0beaa331ff882aa08221a0d", "updated_at_utc": "2026-10-15T18:21:03Z"},
+      {"number": 3586, "title": "Charge one direct connector call on the one-by-one instance", "head": "cde0d3c3a3be86de0c051dc1d2f270ff0f5ae2cd", "updated_at_utc": "2026-10-18T07:03:17Z"},
+      {"number": 3778, "title": "Research: exact dirty-connector cost with overlapping write windows (no kappa)", "head": "cc841651682e076fa8d4f2cec9b2055a8c91c5b1", "updated_at_utc": "2026-10-18T20:32:56Z"},
+      {"number": 1930, "title": "Construction: three-stage persistent prefix shear chain (no new kappa)", "head": "11cb2266f7e99279c440a301ea4b862587a9afa2", "updated_at_utc": "2026-10-14T17:28:27Z"},
+      {"number": 2134, "title": "Six-source one-TRIO path misses full paired coverage at 14 calls", "head": "388a63c83cf24c3355be9798bde5a381188dadb3", "updated_at_utc": "2026-10-15T17:59:07Z"},
+      {"number": 3728, "title": "Audit the 20-prefix confinement step of the length-22 exclusion (no new kappa)", "head": "0f000bb4aac86fa667906d8092eb36e49a76944a", "updated_at_utc": "2026-10-18T18:59:07Z"}
+    ],
+    "limits": "Search is by connector text in open PR title/body, not code diff; it does not certify novelty."
+  }
+}

research/five-source-arity-four-bound/README.md

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+# Five-source arity-four connector bound
+
+**Scoped negative result; no supplier or kappa change.** Add one operation
+family to the five-source paired-readout connector: a call
+
+```text
+Q_r: z += sum_{j != r} B_j x_j       (r in {0,1,2,3,4})
+```
+
+Each `Q_r` is one dispatched connector event, and its four source-map additions
+are charged individually. The original singleton `B_j` source calls and
+`A_i: y_i += A_i z` readouts remain available. Sources and coefficients are
+read-only, `z` is arbitrary dirty state restored at the end, outputs are
+unobserved until completion, and each of the 31 nonzero output masks is
+required. No new frame, copy, extra connector, direct product call or free
+arithmetic is included.
+
+Under the explicit uniform ledger of one unit per event dispatch and one unit
+per source-map or output-map addition, **every schedule that uses at least one
+arity-four call costs at least 149 units**. The unchanged alphabet has exact
+minimum 74 unit-price complete queries from the pinned predecessor; under this
+ledger that is 148 units. Thus this charged extension cannot improve the
+five-source all-mask batch. This is a bound on the stated local ledger, not on
+every implementation cost model or a complete multiplication supplier.
+
+## Proof
+
+Project the source-side events of a schedule to vectors in `F_2^5`. A singleton
+event has weight one; `Q_r` has vector `11111 XOR e_r`, of weight four. Let
+`k=s+m` be the number of source events, where `m` is the number of `Q` events,
+and let `W=s+4m=k+3m` count their constituent source-map additions. Restoration
+for arbitrary dirty connector state and independent source terms forces each
+source's total multiplicity to be even, so `W` is even.
+
+Let `P` be the set of distinct source-prefix vectors at event boundaries and
+`D={p XOR q : p,q in P}`. A pair of output readouts realizes only a mask in
+`D`. Every nonzero output row needs an even positive number of readouts. If a
+mask is missing from `D`, its row needs at least four readouts. If `M` of the
+31 nonzero masks are missing, the total output-readout count is therefore at
+least `62+2M`. Since a closed `k`-event path has at most `k` distinct
+prefixes, `|D \\ {0}| <= choose(k,2)`, hence
+`M >= max(0,31-choose(k,2))`.
+
+The charged total is
+
+```text
+C = (k dispatches + W source additions) + 2 R output-readout units
+  >= 124 + 2k + 3m + 4M.
+```
+
+The only delicate case is `m=1, k=9`. Here `W=12`, and the exact finite
+census below gives `|D| <= 26`, so `M >= 6` and `C >= 169`. For `m=1` and
+`k>=11`, parity gives odd `k`, so `C >= 124+2*11+3 = 149`. For `m=2`, `W`
+even forces even `k`; the pair-count bound gives `k>=10`, so `C>=150`. For
+`m>=3, k>=9` gives `C>=151`. For `k<=8`, `M>=31-choose(k,2)` gives `C>=155`
+already at the weakest case `k=8,m=1`, with smaller `k` or larger `m` only
+increasing the bound. Therefore every schedule using the new family costs
+at least 149.
+
+For `m=1,k=9`, rotate the closed event cycle so the arity-four event is last;
+this translates all prefix vectors and preserves their pairwise differences.
+Relabel the five sources so that event toggles `01111`. The eight singleton
+events before it must then have XOR `01111`. The verifier exhausts all 23,184
+such ordered words, computes every pairwise prefix difference, and finds a
+maximum `D` size of 26 (including zero). The first maximizing word is
+`01020434`; its missing nonzero masks are `01010, 01011, 10010, 10011, 11010,
+11011`. Its histogram and ordered-census SHA-256 are pinned by the verifier.
+A positive control checks the predecessor's twelve-singleton path and obtains
+all 32 masks.
+
+The call's implementation charge is explicit: four constituent source-map
+additions plus its own dispatch. Any routing, frame, program-storage or
+additional scratch cost is outside this ledger and cannot be treated as free
+in a physical caller. A fused implementation that performs fewer than four
+charged additions would be a different premise.
+
+## Reproduce
+
+From the repository root:
+
+```sh
+python3 -I -O -B research/five-source-arity-four-bound/verify.py
+python3 scripts/research_package.py --package research/five-source-arity-four-bound --check-only
+```
+
+The first card control was also run from the pinned base:
+
+```sh
+python3 -B research/universal-connector-boundary/verify.py
+```
+
+## Scope and overlap
+
+The broad card is refined to the five-source arity-four family. Pair-source
+and trio extensions, connector frames, and the original all-singleton word
+have separate retained results. Open PR [#2249](https://github.com/CrocSwap/integer-mult-bounds/pull/2249)
+claims a charged arity-four-through-six floor for six sources; this result is
+the five-source complement case. Open PR [#3586](https://github.com/CrocSwap/integer-mult-bounds/pull/3586)
+claims a one-source/one-output direct product call, which is a different
+operation boundary. These open PR claims were not independently verified here.
+The contemporaneous PR and handoff overlap check is retained in
+[OVERLAP.json](OVERLAP.json); it is not a novelty certificate.
+
+The exact selected result at the base remains the maintainer-reviewed
+conditional finite construction `κ = 836330098538459/500000000000000000`
+(`0.001672660197076918`), reproduced by `make frontier1094-verify`. This work
+does not depend on that cold replay and did not rerun it. The selected record,
+review and source manifest were inspected; their exact paths and certificate
+binding are recorded in the session handoff.
+
+## Open next questions
+
+Does this family improve the complete bill for a concrete five-source caller
+with nonuniform source-map and dispatch costs? The answer could change which
+four-source subset is worth fusing. First bind a real caller's five source
+tariffs and word-storage/dispatch costs, then enumerate only the surviving
+prefix profiles. No such eligible caller or implementation contract is
+identified here; without one, further tariff optimization is not justified.
+
+## Credit
+
+This scope is based on Ross Richardson's five-source universal connector
+boundary (#943/#934 lineage) and Cinna Davis's six-source high-arity connector
+floor (#2249). Their source claims and limits are preserved above. Prepared by
+OpenAI Codex (macbook-09) with substantial AI assistance. No independent human
+review, proof-assistant check, hosted CI pass, supplier admission or new kappa
+is claimed. New package content is Apache-2.0.

research/five-source-arity-four-bound/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "input",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/universal-connector-boundary/CERTIFICATE.json",
+      "usage": "input",
+      "git_blob": "079e3e1441dc07c2f0536e9527acf290e51ea9c0",
+      "sha256": "e733746ba3e5151c829b4e8728bfff5ae3ed24d8d203eb77c735d20758e8266a"
+    },
+    {
+      "path": "research/universal-connector-boundary/README.md",
+      "usage": "input",
+      "git_blob": "d277f7c5830b1f2bc42e1ff41ab86e7d7564929f",
+      "sha256": "23415f7829ab8d80a783142f10f7d31abbce84579362769984680af3df954476"
+    },
+    {
+      "path": "research/universal-connector-boundary/verify.py",
+      "usage": "input",
+      "git_blob": "de6483bb7ff51940dbbb76c18f91af4c907e07da",
+      "sha256": "e4fa4700f9e1ae6ca09c3c76ebc569e8361b89b699425a67839d999d8812e42d"
+    }
+  ]
+}

research/five-source-arity-four-bound/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "connector-alphabet-change",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "original_card": {
+    "id": "connector-alphabet-change",
+    "question": "Does one explicit extension of the connector alphabet offer a favorable completed budget?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/universal-connector-boundary/README.md",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Construct and charge a legal extra operation on a minimal instance.",
+      "Prove a target-specific bound after adding one specified operation family."
+    ],
+    "acceptance_gate": "State which fixed-alphabet premise changes and charge its implementation; do not declare it free. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/universal-connector-boundary/verify.py"
+      ],
+      "expected": "PASS_UNIVERSAL_CONNECTOR_BOUNDARY",
+      "inspect": "Reproduce the 945-pairing control, then read Exact interface to list the allowed calls.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Add exactly one legally priced operation and test a tiny dirty-input instance before enumerating larger cases.",
+    "bottleneck_link": "A new legal connector operation could change recursive interface cost; a larger census in the same alphabet is a separate side result.",
+    "allowed_variations": [
+      "Choose one additional operation and a small source count for its first counterexample."
+    ],
+    "scope_change_triggers": [
+      "Keeping the alphabet and only increasing the source count; record this as a changed assignment.",
+      "Extending a source-count census without naming a supplier/interface decision it could change; audit #2973 pruning concerns before relying on the affected predecessor exclusions."
+    ],
+    "overlap_terms": [
+      "universal connector",
+      "source-toggle",
+      "connector alphabet"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "For five sources and all 31 nonzero paired-readout masks, does adding the arity-four source call Q_r: z += sum_{j != r} B_j x_j lower the complete ledger when each Q_r pays one dispatch and four source-map additions?",
+  "scope_change_reason": "Refines the broad card to the five-source complement arity-four call, disjoint from the retained pair/trio/frame results and complementary to open PR #2249, whose high-arity floor is for six sources.",
+  "bottleneck_link": "docs/research/research-family-status.md: New coding, geometry and recursive boundaries / retain exploration; the result tests whether a paid connector alphabet change removes any cost in the all-mask caller.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop this family at the five-source uniform charged ledger unless an actual caller supplies nonuniform source tariffs or a cheaper legal fused implementation. Do not infer a supplier or global obstruction from this local result.",
+  "outcome": "negative-result",
+  "evidence": "The original 945-pairing verifier passes. A dirty two-dimensional four-source call control passes. The exact normalized one-Q/eight-singleton branch contains 23,184 words with maximum prefix-difference set size 26/32. Pair-count and parity bounds cover all remaining event-count cases and prove every schedule using Q costs at least 149 ledger units, versus the predecessor's exact 148-unit baseline.",
+  "author": "OpenAI Codex (macbook-09)",
+  "ai_assistance": "Substantial OpenAI Codex assistance in proof design, verifier implementation, source inspection and writing. No independent human or proof-assistant review is claimed.",
+  "open_questions": [
+    {
+      "question": "For an actual five-source caller with nonuniform B-map prices, can the complement arity-four family beat the complete caller bill?",
+      "why_it_matters": "The present bound uses one uniform unit per source-map addition and dispatch. A real caller might price source subsets differently, which could change whether this operation saves anything.",
+      "first_step": "Bind a concrete caller's five complete B-map prices, dispatch cost and program-storage cost, then enumerate only event profiles that survive those prices.",
+      "blocked_by": "No eligible five-source caller or legal implementation boundary for Q_r is identified in this checkout; further enumeration has no supplier decision to change until that contract exists."
+    }
+  ]
+}

research/five-source-arity-four-bound/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "docs/research/agent-task-catalog.json",
+    "research/universal-connector-boundary/CERTIFICATE.json",
+    "research/universal-connector-boundary/README.md",
+    "research/universal-connector-boundary/verify.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Dirty connector semantics and restoration; exact normalized one-Q/eight-singleton coverage census and histogram; cyclic/source-label symmetry controls; twelve-singleton positive coverage control"
+    }
+  ]
+}