Research: shared-line rank-2 block fusion misses the 5% ceiling gate

research/complex-shared-line-rank2-seedceaca8/LICENSE

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+SPDX-License-Identifier: Apache-2.0
+
+Copyright 2026 Cinna Davis
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.

research/complex-shared-line-rank2-seedceaca8/NOTICE.md

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+# Notice
+
+This package contains an exact profile-level cost screen written for this
+research cycle. It imports the pinned repository ceiling calculator but does
+not alter or replace the selected source, certificate, or frontier.
+
+Adjacent work is acknowledged in `TASK.json`: PR #2130's rank-one-to-rank-two
+retiming, PR #2124's Phase-B mixed-rank block, Discussion #1987's Karatsuba
+block, and PR #1872's shared-product helper. This package studies only the
+favorable profile effect of pairing two rank-two children into one rank-three
+block when the target frames share a line. It does not claim a source-realized
+block or a new supplier.

research/complex-shared-line-rank2-seedceaca8/README.md

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+# Shared-line rank-two block: exact favorable ceiling screen
+
+**Scoped negative result. No source-realized block, supplier, or new κ.**
+
+The selected #1094 complex local profile is
+`{1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}` per call. I tested a new rank-shape
+mechanism: if two rank-two child frames have target planes that share exactly
+one line, replace the two rank-two children by one rank-three block child that
+materializes the shared direction once. This changes `2+2` paid rank units to
+`3`, frees one unit of coverage mass per pair, and would need a legal block
+interface to copy the shared output to both targets.
+
+## Exact upper-bound calculation
+
+Keep the selected model fixed: child size `m=73`, 896 families over 9 stages,
+centre charge 64, one unpaid coordinate, and idle cap `R=51`. Allow every one
+of the 418 rank-two events to pair, so `g` ranges from 0 to 209. For each `g`,
+the relaxed local profile is
+
+```text
+n2 = 418 - 2g
+n3 = 432 + g
+n1 = 2412, n4 = 142, n5 = 82, n8 = 8
+```
+
+All matching, source access, input/output adapters, copies, fanout, frame
+transitions, hosting, precision, and dirty-state restoration are charged zero.
+The freed rank mass is assigned to fractional rank-51 idle children by the
+pinned ceiling model. This is a favorable completed budget for the proposed
+profile substitution.
+
+More fusions cannot worsen this relaxation. For a fusion, four units of rank-2
+mass become three units of rank-3 mass plus one unit of idle mass. The idle
+unit is carried at rank 51. For `a >= 0`, the per-mass cost factor `r^(-a)` is
+non-increasing in `r`; the three rank-3 units and one rank-51 unit therefore
+cost no more than four rank-2 units. So `g=209` gives the best ceiling in this
+profile family.
+
+| Profile | Rank mass | Idle mass | Certified `a` ceiling |
+|---|---:|---:|---:|
+| `g=0` (selected) | 5,586 | 1,646 | `[0.001690054109088, 0.001690054109089]` |
+| `g=209` (every rank-2 event paired) | 5,377 | 1,855 | `[0.001768394682583, 0.001768394682584]` |
+
+Using the baseline upper endpoint, a 5% target is
+`0.00177455681454345`. Even the all-pairs upper endpoint is below it: the
+endpoint ratio is exactly `43131577624/41220831929` (about `1.0463`), less than
+`21/20`. The profile-only gain therefore tops out at about `4.6354%`, before charging
+any physical interface. Per the task gate, I stopped before searching or
+synthesizing selected-source pairs.
+
+## Scope and limits
+
+This rules out a 5% ceiling improvement from this profile substitution alone,
+under the unchanged coverage law, centre charge, and outer rank cap. It does
+not show that a shared-line block is impossible. Nor does it rule out a smaller
+profile gain, combinations with a changed coverage law, or other rank-shape
+mechanisms. The source event stream supplies frame IDs and ranks, but this
+screen does not prove that any particular two transitions admit the proposed
+joint interface. Physical compatibility and all adapter/copy/restoration
+costs remain unproved.
+
+## Open next questions
+
+- **Can a selected-source pair of rank-two frame transitions with a shared
+  target line be realized by one rank-three block with fully charged source
+  access, fanout, adapters, and dirty restoration?** A source-realized block
+  could still offer a sub-5% local gain or combine with a changed coverage law.
+  The first step is to identify one exact pair in the pinned local program and
+  compare its full Clifford transitions with a proposed joint rank-three block
+  interface. This was not needed after the ideal 5% gate failed.
+
+## Reproduce
+
+```sh
+python3 -I -B research/complex-shared-line-rank2-seedceaca8/verify.py
+python3 scripts/research_package.py --package research/complex-shared-line-rank2-seedceaca8
+```
+
+The verifier checks the selected replay archive and member hashes, independently
+recounts both directions of the local rank profile, recomputes both rational
+ceiling brackets using the pinned `complex-local-word-ceiling/ceiling.py`, and
+checks the maximal-fusion and 5% boundary controls.
+
+## Provenance
+
+Cycle X, seed `ceaca886bcd52d7e2a036a2b4d561117`, on exact base
+`0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2`; task `complex-rank-shape-decoder`,
+catalogue priority revision 4. Claim: [Discussion #1163 comment 18933950](https://github.com/CrocSwap/integer-mult-bounds/discussions/1163#discussioncomment-18933950).
+Focused review package: [PR #2151](https://github.com/CrocSwap/integer-mult-bounds/pull/2151).
+
+The initial claim comment contains transcription errors in its archive and
+canonical-word member SHA-256 strings. `SOURCES.json` and `verify.py` contain
+the correct hashes; the Discussion comment still needs a correction reply.
+
+Prepared by Cinna Davis with substantial OpenAI Codex assistance in selecting
+the profile-level mechanism, deriving the favorable bound, implementing exact
+checks, and writing the package.

research/complex-shared-line-rank2-seedceaca8/RESULT.json

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+{
+  "schema_version": 1,
+  "status": "SCOPED_NEGATIVE_RESULT",
+  "objective": "5% increase in the fixed-word a ceiling from 2xrank-2 to rank-3 shared-line block substitutions",
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "source_profile_per_call": {
+    "1": 2412,
+    "2": 418,
+    "3": 432,
+    "4": 142,
+    "5": 82,
+    "8": 8
+  },
+  "model": {
+    "m": 73,
+    "families": 896,
+    "stages": 9,
+    "centre": 64,
+    "unpaid_coordinates": 1,
+    "idle_rank_cap": 51,
+    "interface_expenses_charged": "zero"
+  },
+  "selected_profile": {
+    "rank_mass": 5586,
+    "idle_mass": 1646,
+    "a_ceiling_interval": [
+      "52814190909/31250000000000",
+      "1690054109089/1000000000000000"
+    ]
+  },
+  "all_pairs_profile": {
+    "groups": 209,
+    "profile": {
+      "1": 2412,
+      "2": 0,
+      "3": 641,
+      "4": 142,
+      "5": 82,
+      "8": 8
+    },
+    "rank_mass": 5377,
+    "idle_mass": 1855,
+    "a_ceiling_interval": [
+      "1768394682583/1000000000000000",
+      "221049335323/125000000000000"
+    ]
+  },
+  "five_percent_target": "35491136290869/20000000000000000",
+  "all_pairs_upper_endpoint_ratio": "43131577624/41220831929",
+  "all_pairs_improvement_percent": "4.6353884809776",
+  "margin_below_target": "123242639189/20000000000000000",
+  "source_realized": false,
+  "new_supplier_or_kappa": false,
+  "remaining_obligation": "Find an exact selected-source pair and certify its rank-three shared-line block interface and complete costs if pursuing a smaller gain."
+}

research/complex-shared-line-rank2-seedceaca8/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "e8870e5917057d59b730eb14721a778639eca050",
+      "sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "provenance",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/selected-replay.tar.gz",
+      "usage": "input",
+      "git_blob": "c24daf067f9c543421d89ca2eeaa4eb78a57fb6f",
+      "sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6"
+    },
+    {
+      "path": "docs/research/community-batch1099-review.md",
+      "usage": "input",
+      "git_blob": "6cb6ec061a071a06a2d18cd4dc3e9a2b05d28099",
+      "sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    }
+  ]
+}

research/complex-shared-line-rank2-seedceaca8/TASK.json

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+{
+  "schema_version": 1,
+  "track": "complex-mechanisms",
+  "seed": "ceaca886bcd52d7e2a036a2b4d561117",
+  "scope_key": "cad3ba853d7f324f16fa",
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 4,
+  "catalogue_sha256": "d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7",
+  "base_commit": "0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2",
+  "selected_record_sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32",
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1163#discussioncomment-18933950",
+  "pull_request_url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2151",
+  "claim_update_status": "Pending correction reply because GitHub GraphQL returned an API rate-limit error; correct hashes are recorded in SOURCES.json and verify.py.",
+  "overlap_snapshot": {
+    "searched_at": "2026-10-14T20:59:11Z",
+    "open_prs": 401,
+    "discussions": 821,
+    "complete_pagination": true,
+    "warnings": [
+      "Exact terms were queried offline against the complete live snapshot fetched at 20:57:09Z and 20:59:04Z; refresh again if submitting after cache expiry."
+    ],
+    "exact_grid_matches": 0
+  },
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a decoder deliver the local paid mass in fewer, larger-rank children (block frame climbs instead of rank-1 to rank-3 steps)?",
+    "starting_points": [
+      "research/complex-local-word-ceiling/README.md",
+      "research/complex-decoder-field-extension-seed44/README.md",
+      "research/complex-frame-pair-decoders/README.md",
+      "research/selected-k8-unrestricted-ends/README.md",
+      "research/eisenstein-reduction-copy-floor/README.md"
+    ],
+    "methods": [
+      "Find one exact small decoding identity whose frame path climbs in blocks, and price its complete per-call profile.",
+      "Bound the best effective child rank achievable for one fixed support alphabet, including copies, frames and restoration."
+    ],
+    "acceptance_gate": "A per-call local profile, with all copied information, frames and restoration counted, whose ceiling exceeds the selected word's by at least 5%; masked or overcomplete supports must change an assumption of the eight-term all-pairs bound.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "research/complex-local-word-ceiling/verify.py"
+      ],
+      "expected": "PASS_COMPLEX_LOCAL_WORD_CEILING",
+      "inspect": "Read the selected word's per-call local profile, its packing ceiling and the lever table; run levers.py for the ideal budget of the lever you target.",
+      "scope": "Positive control and planning budget only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Before any synthesis, write the candidate per-call local profile (and centre charge / unpaid coordinates if changed) and screen it with ceiling.ceiling(); reject it if even the ideal ceiling does not beat the selected word's by your stated margin. Ideal budget: rank-16 delivery is 2.31x (levers.py).",
+    "bottleneck_link": "Rank-1 children are about half of recursive cost; the selected word's effective child rank is about 3.9. Larger-rank delivery lowers recursive cost per unit of paid mass.",
+    "allowed_variations": [
+      "One masked, overcomplete or block decoder family with exact labels and supports.",
+      "A scoped lower bound on effective rank for one alphabet.",
+      "If the coverage law, template or guard changes, derive an applicable fully charged replacement bound instead of applying the incumbent ceiling outside its scope."
+    ],
+    "scope_change_triggers": [
+      "Repacking idle remainders of an unchanged word (covered by the packing ceiling).",
+      "Repeating the eight-term all-pairs search inside its proved scope.",
+      "Only changing helper terminal subspaces and monotone frames within the fixed scalar word, entrances and cleanup contract already covered by #1995/#2013.",
+      "Searching below C=40 in the #2017 Eisenstein binary scalar-rectangle or fixed-support block-linear additive pairing model; specify a changed premise instead. This bound does not cover every complex decoder."
+    ],
+    "overlap_terms": [
+      "rank shape",
+      "block climb",
+      "masked decoder",
+      "overcomplete",
+      "effective rank"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Upper-bound the best possible fixed-word ceiling from replacing pairs of rank-two children by rank-three blocks under a shared target-frame line, with all physical interface expenses set to zero.",
+  "bottleneck_link": "The selected complex side binds; reducing rank-2 paid mass could improve the fixed-word ceiling, but this ideal block family alone does not reach the 5% screen.",
+  "scope_change_reason": "This is a distinct geometric block hypothesis within rank-shape decoder research. The profile gate gives a strict stopping result before source synthesis.",
+  "budget_minutes": 60,
+  "stop_rule": "Allow all 209 pairings of the selected 418 rank-two events at zero interface cost. Stop before source matching if the resulting exact ceiling is below 1.05 times the pinned baseline ceiling.",
+  "outcome": "negative-result",
+  "evidence": "verify.py recomputes the selected profile and exact ceiling intervals. Even all 209 rank-two pair fusions give a ceiling ratio 43131577624/41220831929 < 21/20.",
+  "author": "Cinna Davis",
+  "ai_assistance": "Substantial OpenAI Codex assistance in mechanism selection, exact cost derivation, verifier implementation, and package writing.",
+  "open_questions": [
+    {
+      "question": "Can a selected-source pair of rank-two frame transitions with a shared target line be realized by one rank-three block with fully charged source access, fanout, adapters, and dirty restoration?",
+      "why_it_matters": "A source-realized block could still offer a sub-5% local gain or combine with a changed coverage law, although this package's profile gate rejects a 5% gain from this substitution alone.",
+      "first_step": "Identify one exact pair in the pinned local program and compare its full Clifford transitions with a proposed joint rank-three block interface.",
+      "blocked_by": "No physical paired transition was needed after the ideal 5% cost gate failed."
+    }
+  ]
+}

research/complex-shared-line-rank2-seedceaca8/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/community-frontier1094-audit/selected-replay.tar.gz",
+    "research/complex-local-word-ceiling/ceiling.py",
+    "research/community-frontier1094-audit/ASSEMBLY.json",
+    "docs/research/community-batch1099-review.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Replay archive and member hashes; independent forward/backward profile counts; exact baseline/all-pairs ceiling brackets; maximal pair count; exact 5% ratio; impossible-pair, overfull-mass, and rank-cap negative controls"
+    }
+  ]
+}