Recursive interface: order-four frames save 320 dirty-echo primitives (no new kappa)

research/z4-paired-frame-restoration-seed56/CERTIFICATE.json

@@ -0,0 +1,240 @@
+{
+  "assignment": {
+    "cycle_order": 4,
+    "os_seed": 56,
+    "track": "recursive interfaces"
+  },
+  "base_commit": "bce54fb3f895cbe5cd4c0eb477c3afeed100190f",
+  "boundary": {
+    "children": 5,
+    "dynamic_program_total": 140,
+    "eager_ADD": 270,
+    "eager_NEG": 30,
+    "eager_total": 300,
+    "eager_word_sha256": "90bf97554a5a1529e96f128ad93e54aeadbfeffbdd73f3c546ceb4f1e0d91273",
+    "exhaustive_path_count": 1024,
+    "exhaustive_path_minimum": 140,
+    "macro_alphabet": "canonical powers P^u and Q^v using the listed explicit words",
+    "optimized_ADD": 126,
+    "optimized_NEG": 14,
+    "optimized_word_sha256": "3459ba0aca8bd5e598696333a1a5b520d4a0e5353553d685732cf7d49acbd097",
+    "saved_per_batch": 160,
+    "states": [
+      0,
+      1,
+      2,
+      3,
+      0,
+      0
+    ],
+    "transitions": [
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+        "Q_power": 1,
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+      {
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+        "before": 2
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+      {
+        "P_power": 0,
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+        "after": 0,
+        "before": 3
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+      {
+        "P_power": 3,
+        "Q_power": 0,
+        "after": 0,
+        "before": 0
+      }
+    ]
+  },
+  "child": {
+    "checked_basis_columns": 96,
+    "distinct_block_checks": [
+      {
+        "ADD": 62,
+        "NEG": 6,
+        "basis_columns": 32,
+        "block": 0,
+        "inverse_word_sha256": "12a83c7b67cc26872a45569df66568dce8d6f47697ae23d08a7731d1c3e957d4",
+        "word_sha256": "24a9f259f8d8bcd826a8feec7f0dbbf22f10c5fa10e2ed0b5e101ea22bebfe95"
+      },
+      {
+        "ADD": 62,
+        "NEG": 6,
+        "basis_columns": 32,
+        "block": 1,
+        "inverse_word_sha256": "ed74b775d98f56e0b6a9b6153f5dce0e352d7597dd275e84bdfe7ef91db8282c",
+        "word_sha256": "33c9708615463d38ac54c283ff87d3755c2a02edfa6535a3fdce7555ed8091ea"
+      },
+      {
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+        "word_sha256": "2d9232ba4a81a06041bf4057a798aa538985524cbf97b40caca085fcefd309e4"
+      }
+    ],
+    "per_call": {
+      "ADD": 62,
+      "NEG": 6,
+      "total": 68
+    },
+    "selection_blocks": [
+      0,
+      1,
+      2,
+      0,
+      1
+    ]
+  },
+  "complete_echo": {
+    "all_arbitrary_dirty_basis_columns": 56,
+    "dispatches": {
+      "eager": 1328,
+      "optimized": 1008
+    },
+    "eager": {
+      "ADD": 1208,
+      "NEG": 120,
+      "total": 1328
+    },
+    "eager_word_sha256": "ee0e4afaf3cf2b1dbfc65e2eff076c92dba34fe8aa768fc8b506f9d22382c883",
+    "internal_C_v_example": {
+      "all_arbitrary_input_and_scratch_basis_columns": 56,
+      "eager": {
+        "ADD": 1262,
+        "NEG": 126,
+        "total": 1388
+      },
+      "eager_word_sha256": "b63466d5724bf7e60b6fcdb8bca73feaec48788326b43f70804dff1764cc07dc",
+      "optimized": {
+        "ADD": 974,
+        "NEG": 94,
+        "total": 1068
+      },
+      "optimized_word_sha256": "436ab81a02fc6013b6b167937c2294d4547b6f8ffd3888db5569f5c2cedc3131"
+    },
+    "layout": {
+      "output_D": 8,
+      "source_X": 24,
+      "total_data_coordinates": 56,
+      "working_W": 24
+    },
+    "map": "(D,X,W)->(D+sum_i R_i X,X,W)",
+    "optimized": {
+      "ADD": 920,
+      "NEG": 88,
+      "total": 1008
+    },
+    "optimized_word_sha256": "ecaf6008f7c81a6c495ce146b971b53d4c3a9b1fea6289645285045796235451",
+    "prefix_row_l1_peak": {
+      "eager": 11,
+      "optimized": 20
+    },
+    "reads": {
+      "eager": 2536,
+      "optimized": 1928
+    },
+    "saved_total_primitives": 320,
+    "writes": {
+      "eager": 1328,
+      "optimized": 1008
+    }
+  },
+  "controls": {
+    "count": 4,
+    "names": [
+      "rotation sign mutation",
+      "wrong output phase",
+      "missing terminal restoration",
+      "forward word substituted for inverse"
+    ]
+  },
+  "frame": {
+    "determinant": 1,
+    "intertwining_verified": true,
+    "order": 4,
+    "output_octets": 1,
+    "per_register": "two disjoint 4-cycles on one octet",
+    "phase_costs_per_octet": [
+      {
+        "ADD": 0,
+        "NEG": 0,
+        "phase": 0,
+        "total": 0,
+        "word_sha256": "4f53cda18c2baa0c0354bb5f9a3ecbe5ed12ab4d8e11ba873c2f11161202b945"
+      },
+      {
+        "ADD": 18,
+        "NEG": 2,
+        "phase": 1,
+        "total": 20,
+        "word_sha256": "652fe5774a3dc187f9df6fd2e578f0270a4a898e76edd1c5e57446ad5281d6fa"
+      },
+      {
+        "ADD": 12,
+        "NEG": 4,
+        "phase": 2,
+        "total": 16,
+        "word_sha256": "3db669a9548039bbfc034e92b0c3edf8a540d8b147cb95618eddd3e225e13bb4"
+      },
+      {
+        "ADD": 18,
+        "NEG": 2,
+        "phase": 3,
+        "total": 20,
+        "word_sha256": "278ea7a94adedef7155f32189da6002570d43d68701817674a62494c22d0ea22"
+      }
+    ],
+    "readers": [
+      0,
+      1,
+      2,
+      0,
+      1
+    ],
+    "workspace_octets": 3
+  },
+  "recursive_interface_cost": {
+    "complete_root_echo": "2*sum(child vectors) + (48,0) + 2*g(k_v)",
+    "depth_storage": "56 data coordinates for the root echo plus 24 per active nested internal C_v level",
+    "eager_comparator": "replace g(k_v) by k_v*(54,6)",
+    "g(k)": "minimum 4-state recurrence below; vector ADD/NEG invoice retained along chosen path",
+    "internal_C_v": "2*sum(child vectors) + (48,0) + 2*g(k_v) + (54,6)",
+    "leaf_forward": {
+      "ADD": 62,
+      "NEG": 6
+    }
+  },
+  "schema_version": 1,
+  "scope": "Exact finite dirty-state interface construction and macro-alphabet optimum for five children; no actual supplier admission, selected profile change, new kappa, or multiplication theorem.",
+  "selected_context": {
+    "decimal": "0.001672660197076918",
+    "kappa": "836330098538459/500000000000000000",
+    "review": "docs/research/community-batch1099-review.md",
+    "selected_source_pr": 1094
+  },
+  "source_pins": {
+    "certificates/selected-result.json": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+    "docs/research/agent-task-catalog.json": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce",
+    "docs/research/community-batch1099-review.md": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb",
+    "docs/research/current-status.md": "1f59d648763c77d574d18dc59ea8bf62d2ee74aaed3c202c9c8cfa4f901badbd",
+    "research/community-frontier1094-audit/ASSEMBLY.json": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045",
+    "research/community-frontier1094-audit/SELECTED-SOURCE.json": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+  },
+  "task": "z4-paired-frame-restoration-seed56"
+}

research/z4-paired-frame-restoration-seed56/CLAIM.md

@@ -0,0 +1,35 @@
+# Seed 56 task selection and overlap record
+
+**Track:** recursive interfaces. **OS seed:** 56. **Base:** `bce54fb3f895cbe5cd4c0eb477c3afeed100190f`. The seed maps to cycle order `2 + (56 mod 3) = 4`.
+
+## Three narrow candidates considered
+
+1. **Selected #1094 T16 same-owner call boundary.** Test whether one concrete repeated owner restore is immediately rebuilt by its next chronological call. The selected review and [heterogeneous owner assembly](../heterogeneous-owner-assembly/README.md) bound the frozen owner catalogue, while recent #1240/#1348 reports quantify selected-word packing headroom and changed coverage-law thresholds. PR #1252's `Z4` is a rank-pair phase polynomial in idle owner repartitioning, not a fourth-order output frame. The task still needed an exact identified two-call transition and a new legal return contract; no such concrete transition was established in this orientation pass.
+2. **Signed-reader restoration coalescing.** A specific checkpoint-correct cut problem for fixed dirty readers could reduce restoration. This overlaps current #1199 (diagonal signed readers), #1203 (noncommuting reader restoration) and #1246/#1227 (paid reader-defect and output-sign repair), so it was not selected.
+3. **Order-four paired-cycle output-frame boundary.** Extend the already explicit cyclic output-frame mechanism to a genuinely different phase group and gate-cost profile. Current #1287 is order three; its recurrence does not cover the order-four paired cycles, whose phase powers require NEG accounting. Exact-term search found no matching order-four frame task.
+
+## Chosen scope and claim
+
+I chose candidate 3. The task is the five-child dirty echo with three workspace octets, one output octet, and selectors `(0,1,2,0,1)`. The checked macro alphabet is the explicit set of whole-register powers of the paired order-four frame. It compares a complete optimized echo with the eager per-child inverse frame.
+
+Claim posted on [coordination discussion #1163](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163#discussioncomment-18911404) at 2026-10-13 23:35 UTC; expected finish 2026-10-14 00:08 UTC, expiry 01:35 UTC.
+
+## Finding
+
+The certified path is `[0,1,2,3,0,0]`: the five-child boundary costs 140 primitives (`126 ADD, 14 NEG`) versus 300 eager (`270 ADD, 30 NEG`). The complete arbitrary-dirty echo costs 1,008 (`920 ADD, 88 NEG`) versus 1,328 (`1,208 ADD, 120 NEG`), saving 320. The corresponding internal `C_v` map and its literal inverse are also checked over all 56 input/scratch basis columns. The exact 9-ADD/1-NEG quarter-turn word was found during the final cost audit; an earlier three-pure-swap word was replaced before freezing the result. The final prefix row-L1 peak rises 11 to 20.
+
+## Overlap evidence
+
+At 2026-10-13 23:33 UTC, `python3 scripts/find_overlap.py 'order-four cycle output frame Z4 paired cycles' --refresh --limit 12` searched complete snapshots of **839 open PRs** and **89 discussions** and returned zero exact multiword matches. A later offline cached query at 23:44 returned PR #1287 for `cyclic output frames`, the posted seed-56 claim for `paired 4-cycles`, and PR #1252 for `Z4`. Direct inspection confirmed that #1252 uses `Z4` for a rank-pair phase polynomial in idle owner repartitioning, not an order-four output frame. The helper does not search arbitrary PR code diffs or nested package files; this is not a novelty certificate. Direct inspection established the nearest distinctions:
+
+- [#1287](https://github.com/CrocSwap/integer-mult-bounds/pull/1287), head `8606df05eca3c65032361c74ee3dbc2630b0cdc4`: order-three cyclic output frames with a three-state recurrence.
+- [#1199](https://github.com/CrocSwap/integer-mult-bounds/pull/1199), head `85ac2880ea5bda44f28632b71ffa2b4e35ec6b7d`: cancelling signed diagonal readers and checkpoint-correct partitions.
+- [#1203](https://github.com/CrocSwap/integer-mult-bounds/pull/1203), head `38459854fd5bb93099871a84324413bd0e338d8e`: noncommuting reader restoration under a paid dirty wrapper.
+- [#1227](https://github.com/CrocSwap/integer-mult-bounds/pull/1227), head `fea44a4d0d78570021d854881d15d9efe62e1713`: sharp output-sign restoration for a fixed-reader boundary.
+- [#1246](https://github.com/CrocSwap/integer-mult-bounds/pull/1246), head `5ed1ce17a9df4c853a93f9758ba903d2620c3443`: paid nonzero reader-defect repair.
+- [#1303](https://github.com/CrocSwap/integer-mult-bounds/pull/1303), head `bddf5ba853f1b798d021454362e53338c819f150`: exact ten-primitive lower bound for a restored local disjoint 2+3-cycle boundary, including finite dirty donors. It changes a local restoration word; this scope instead changes the child-facing frame contract.
+- [#805](https://github.com/CrocSwap/integer-mult-bounds/pull/805), head `ef34bd5b32fcf2d5640dae90b5b8f28488738d5a`: determinant-balanced cyclic recurrence leaves with a different coefficient and precision contract.
+- [#846](https://github.com/CrocSwap/integer-mult-bounds/pull/846), head `4bdb3e755ea7118170a7a34764df955880dec282`: stride-dependent cyclic inverse scheduling; different object and tape recurrence.
+- [#1252](https://github.com/CrocSwap/integer-mult-bounds/pull/1252), head `1b9db2faf82b8866b0bb0164501c9a627c7dca9b`: the literal `Z4` match is an exact owner rank-pair phase polynomial for `48+48 -> 45+51`, not a four-cycle frame.
+
+The targeted local-ledger search for these PR numbers returned no rows; the live heads and bodies above therefore control this overlap check. No selected construction or prior negative result is changed.

research/z4-paired-frame-restoration-seed56/PREDECESSORS.json

@@ -0,0 +1,79 @@
+{
+  "schema_version": 1,
+  "new_work_commit_base": "bce54fb3f895cbe5cd4c0eb477c3afeed100190f",
+  "predecessors": [
+    {
+      "pr": 1287,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1287",
+      "head": "8606df05eca3c65032361c74ee3dbc2630b0cdc4",
+      "author": "Ross Richardson",
+      "contribution": "Order-three cyclic output frames, equivariant-reader admission, and a three-state boundary recurrence with a full dirty echo.",
+      "relationship": "Closest method predecessor. This package uses a paired order-four frame, different ADD/NEG phase costs, and a separate five-child invoice; no implementation is copied."
+    },
+    {
+      "pr": 1227,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1227",
+      "head": "fea44a4d0d78570021d854881d15d9efe62e1713",
+      "author": "Sagar Thakrar",
+      "contribution": "Paid output-sign frames and exact restoration costs for fixed signed-permutation readers.",
+      "relationship": "Retained credit for output-frame restoration; the paired order-four permutation and five-child recurrence are distinct."
+    },
+    {
+      "pr": 1199,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1199",
+      "head": "85ac2880ea5bda44f28632b71ffa2b4e35ec6b7d",
+      "author": "Sagar Thakrar (PR body credits Navinor / Aaron Wang)",
+      "contribution": "Checkpoint-correct coalescing of restoration for fixed signed diagonal readers.",
+      "relationship": "Related dirty-restoration baseline; its cut planner does not contain the paired order-four phase model."
+    },
+    {
+      "pr": 1203,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1203",
+      "head": "38459854fd5bb93099871a84324413bd0e338d8e",
+      "author": "mightyOil",
+      "contribution": "Noncommuting reader-restoration identities with a paid dirty-state wrapper.",
+      "relationship": "Related reader-interface work; this package uses fixed block selectors and explicitly prices cyclic frame gates."
+    },
+    {
+      "pr": 1246,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1246",
+      "head": "5ed1ce17a9df4c853a93f9758ba903d2620c3443",
+      "author": "Hanson Sun",
+      "contribution": "Paid repair of nonzero reader defects at canonical cuts.",
+      "relationship": "Related correction mechanism; not used in the paired-cycle frame construction."
+    },
+    {
+      "pr": 1303,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1303",
+      "head": "bddf5ba853f1b798d021454362e53338c819f150",
+      "author": "mightyOil",
+      "contribution": "A scoped exact ten-primitive lower bound for a restored disjoint 2+3-cycle boundary, including arbitrary finite restored dirty donors.",
+      "relationship": "Relevant negative result for restored local cycles. It does not include this package's equivariant child maps or coupled P/Q frame phase schedule."
+    },
+    {
+      "pr": 805,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/805",
+      "head": "ef34bd5b32fcf2d5640dae90b5b8f28488738d5a",
+      "author": "Victor Liwentao",
+      "contribution": "A determinant-balanced cyclic recurrence leaf with coefficient, precision, and dirty restoration accounting.",
+      "relationship": "A different cyclic recurrence and data type; the present frame action is a finite signed-permutation interface."
+    },
+    {
+      "pr": 846,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/846",
+      "head": "4bdb3e755ea7118170a7a34764df955880dec282",
+      "author": "Rohan S. Guliani",
+      "contribution": "Stride-independent cyclic inverse cost on sequential tapes.",
+      "relationship": "Different object and tape recurrence; cited to distinguish the word cyclic in the overlap review."
+    }
+  ],
+  "selected_lineage": {
+    "selected_result": "certificates/selected-result.json",
+    "selected_supplier_pr": 1094,
+    "complex_word_predecessor_pr": 1087,
+    "bit_supplier_pr": 589,
+    "bit_lineage_pr": 555,
+    "relationship": "Source-pinned context only. The finite q4 interface is not integrated into the selected suppliers or profile."
+  },
+  "code_reuse": "No predecessor implementation or source archive is copied."
+}