Research: exact Toom-3 five-product profile screen

research/complex-toom3-five-product-screen/README.md

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+# Exact Toom-3 five-product screen for the selected complex profile
+
+**Exact tensor identity and optimistic profile screen. No selected-provider construction or new κ is claimed.**
+
+## Result
+
+For degree-two polynomials, evaluation at 0, 1, -1, 2 and infinity gives an exact five-product formula for all five output coefficients. The verifier reconstructs the interpolation matrix over the rationals and checks all nine input monomial pairs.
+
+If one such block replaced nine rank-one recursive products by five, it would save four rank-one units. The pinned selected local profile has 2,412 rank-one units. Its exact ceiling calculator gives:
+
+| Hypothetical realized blocks | Rank-one count | Local mass | Complex-side a ceiling |
+|---:|---:|---:|---:|
+| 0 | 2,412 | 5,586 | 0.001690054109089 |
+| 64 | 2,156 | 5,330 | [0.001775054925982, 0.001775054925983] |
+| 210 | 1,572 | 4,746 | [0.002005153888714, 0.002005153888715] |
+| 254 | 1,396 | 4,570 | [0.002086688163384, 0.002086688163385] |
+
+Sixty-four blocks are the first count whose ideal ceiling clears the task card’s five-percent gate. Two hundred ten are the first count whose complex-side transfer cap exceeds 0.002. The selected forward and backward traces each admit 254 disjoint groups of nine rank-one climbs under the phase-local dependency-window movement model used in PR #1817. This establishes enough candidate event slots under that relaxed model; it does not establish that any group is a Toom-3 tensor.
+
+A straightforward scalar evaluation/interpolation schedule uses 19 additions and 13 rational scales per block. At 210 blocks that is 3,990 additions and 2,730 scales. Copies, scratch allocation, dirty-state restoration, precision, native hosting and the new output map are still unpriced.
+
+The selected bit saving is 0.001733427039589844234749. Under the retained transfer its planning cap is 0.0017304274698236029, below 0.002. Therefore this complex-side idea alone cannot produce an overall κ above 0.002. The selected κ remains 0.001672660197076918.
+
+## Boundary and prior work
+
+The exact Toom-3 identity changes the bilinear tensor, extending the two-coefficient Karatsuba screen in Discussion #1987. It differs from diagonal rank-16 child batching in PR #1779 and its dependency-window audit in PR #1817, and from the two-root E8 rank-16 decoder currently proposed in PR #3895. The fresh overlap snapshot covered 672 open PRs and 2,573 Discussion records; these matches are leads, not a novelty certificate. The PR #1817 event-window rule is reimplemented here and credited to Josie Elliston.
+
+The selected-source event records name frame climbs and register touches. They do not identify a three-by-three coefficient convolution, a five-child output decoder, or a dirty-state memory contract. The 254 groups are eligible event windows only. No selected provider block, complete local profile, supplier admission or kappa improvement is claimed.
+
+The family expansion gate therefore permits only one next source-bound witness: bind one candidate group to its exact paired input forms and outputs, then charge its complete interface. Do not extrapolate the 254 event count into a multi-block construction.
+
+## Reproduction
+
+From the repository root, run:
+
+    python3 scripts/research_package.py --package research/complex-toom3-five-product-screen --check-only
+    python3.12 scripts/research_package.py --package research/complex-toom3-five-product-screen
+
+The registered replay checks both orientations, exact rational identities, dependency-window movement, profile ceilings, negative controls and the unchanged bit-side cap. The pinned sources and exact base are recorded in SOURCES.json. No full supplier cold replay, Lean proof, or native compiler replay is claimed.
+
+## Open next questions
+
+TASK.json records the exact first source-binding experiment and the separate bit-side requirement for an overall κ above 0.002. The immediate blocker is the absent mapping from selected child climbs to Toom coefficient inputs and output coefficients.
+
+## Credit
+
+Credit the selected #1094 provider lineage; Josie Elliston / PR #1817 for the dependency-window batching model; the contributors to Discussion #1987 for the two-coefficient Karatsuba screen; Arham Tahir / PR #1779 for diagonal rank-16 batches; and the contributors to PR #3895 for the two-root E8 block decoder. Prepared by Gonzalo Mier with substantial OpenAI Codex assistance. Apache-2.0; no independent human review claimed.

research/complex-toom3-five-product-screen/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": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    },
+    {
+      "path": "research/community-frontier1094-audit/selected-replay.tar.gz",
+      "usage": "input",
+      "git_blob": "c24daf067f9c543421d89ca2eeaa4eb78a57fb6f",
+      "sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/RESULT.json",
+      "usage": "input",
+      "git_blob": "77f55043a679dd722330f1e2499863e36376d9f9",
+      "sha256": "d91d6c46509875abf1f61de67c2609656d3dc685451299f83caa6f922493692d"
+    },
+    {
+      "path": "research/complex-local-word-ceiling/ceiling.py",
+      "usage": "input",
+      "git_blob": "6c0f54527d879396f689523bfde19928a9b6157c",
+      "sha256": "86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc"
+    },
+    {
+      "path": "scripts/research_baseline.py",
+      "usage": "input",
+      "git_blob": "6b607b36eefd65718120ade3c8187681f11bf879",
+      "sha256": "59d9b04dc33e1bb3f81506d187243435de4902138a038b935e36ebaa5f37db67"
+    }
+  ]
+}

research/complex-toom3-five-product-screen/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 6,
+  "base_commit": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
+  "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": [
+      "docs/research/research-family-status.md",
+      "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. 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/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.",
+      "Continuing a partial cost improvement while leaving the same unpriced consumer, restoration or central-readout obligation unresolved; refresh the family expansion decision first. Local product-retiming evidence has a narrow scope and is not a ban on changed graphs or contracts."
+    ],
+    "overlap_terms": [
+      "rank shape",
+      "block climb",
+      "masked decoder",
+      "overcomplete",
+      "effective rank"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Construct and exactly verify the degree-two Toom-Cook tensor identity (nine coefficient products replaced by five), measure the ideal selected-word profile change, and count phase-local rank-one event groups under the dependency-window movement model of PR #1817.",
+  "scope_change_reason": "The card starts from frame-climb decoders. This experiment changes the bilinear tensor by applying a five-product convolution factorization, then binds only its ideal rank-profile effect and a permissive source-event grouping screen. It does not claim a selected-provider tensor embedding.",
+  "bottleneck_link": "The selected local word contains 2,412 rank-one child units. Every Toom-3 block would replace nine such products by five, saving four units; the pinned ceiling screen says 210 realized blocks could move the complex-side cap above 0.002.",
+  "budget_minutes": 60,
+  "stop_rule": "Do not expand into a many-block synthesis until one nine-child source-bound Toom-3 block has exact left/right/output maps, arbitrary-dirty restoration, source acquisition, frame and scratch costs. Stop this route at the first exact embedding obstruction or an invoice that exhausts the profile margin. The current 254 phase-local windows are only eligible slots, not Toom blocks.",
+  "outcome": "partial",
+  "evidence": "Exact rational interpolation verifies all nine basis products of degree-two polynomial convolution. A pinned zero-interface profile screen crosses the complex-side 5% gate at 64 blocks and the complex-side kappa=0.002 threshold at 210. The selected trace admits 254 phase-local groups of nine rank-one climbs under the movement model from PR #1817. No group has been shown to implement the Toom tensor, so no source-bound profile, supplier or kappa improvement is claimed. Under the unchanged bit-side saving, the retained transfer remains capped below 0.002.",
+  "author": "Gonzalo Mier",
+  "ai_assistance": "Substantial OpenAI Codex assistance in selecting the Toom-3 extension, deriving and checking the exact tensor, reproducing the selected-profile screen, adapting the event-window census, and writing the verifier and package.",
+  "open_questions": [
+    {
+      "question": "Can one of the 254 event groups be bound to an exact nine-term convolution sub-tensor of the selected provider, with a five-child output map?",
+      "why_it_matters": "The ideal profile only improves the complex-side cap if a real selected-source block replaces nine child products while preserving the full local operator.",
+      "first_step": "Choose one phase-local group, reconstruct its paired left and right child forms and output coefficients from the pinned word and recipes, then verify the proposed block on all basis inputs and outputs.",
+      "blocked_by": "The event trace identifies frame climbs and register conflicts but supplies no Toom coefficient assignment, output decoder, native scratch contract, or dirty-state restoration schedule."
+    },
+    {
+      "question": "Can the selected bit supplier exceed the primitive threshold 1/499 needed for kappa above 0.002 under the retained transfer?",
+      "why_it_matters": "Even the ideal Toom-3 complex cap is limited globally by the unchanged bit-side cap, currently below 0.002.",
+      "first_step": "Take up the attachment\u2019s bit-side extension route only after the complex block has a complete source-bound invoice.",
+      "blocked_by": "The current selected bit saving is 0.001733427039589844234749, below 1/499."
+    }
+  ]
+}

research/complex-toom3-five-product-screen/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "certificates/selected-result.json",
+    "research/community-frontier1094-audit/selected-replay.tar.gz",
+    "research/complex-local-word-ceiling/RESULT.json",
+    "research/complex-local-word-ceiling/ceiling.py",
+    "scripts/research_baseline.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact selected-source pins and rank profile; exact Toom-3 evaluation/interpolation identity on all nine basis products; rational linear-map counts and malformed-point/omitted-product controls; dependency-window reordering with frame semantics and 254 disjoint groups of nine; exact selected-profile ceiling brackets and first block counts for the 5% and complex-side kappa=0.002 thresholds; selected bit-side cap; no supplier or new-kappa claim."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Repeat the complete exact identity, source schedule, ceiling and global bit-cap receipt under optimized Python."
+    }
+  ]
+}