[Gonzalo-Mier]

Change and mathematical scope

Constructs and verifies the exact five-product Toom-3 factorization for multiplying degree-two polynomials. The verifier reconstructs the rational interpolation map and checks all nine input monomial pairs.

A hypothetical block replacing nine selected rank-one recursive products with five saves four rank-one units. On the pinned complex profile, 64 realized blocks would clear the task card's five-percent local-ceiling gate; 210 would move the complex-side transfer cap above 0.002. At 254 hypothetical blocks, the local complex-side ceiling is [0.002086688163384, 0.002086688163385], corresponding to a local κ cap near 0.002082342962972. The selected forward and backward traces each contain 254 disjoint groups of nine rank-one climbs eligible under the phase-local movement model from #1817.

These are screening results, not a selected-provider construction: no group is bound to Toom coefficient inputs or outputs, and copies, scratch, dirty-state restoration, precision and native hosting remain unpriced. A direct scalar schedule costs 19 additions and 13 rational scales per hypothetical block.

This PR records a useful partial result and a negative global result. Under the retained transfer, the unchanged bit provider caps overall κ at 0.001730427469824, so the Toom-3 complex-side screen alone cannot reach overall κ > 0.002. Selected κ remains 0.001672660197076918. No new κ or supplier claim is made.

Task and overlap

Base: freshly fetched canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90. Task complex-rank-shape-decoder, changed scope to screen a five-product bilinear convolution factorization and its selected-profile effect.

The overlap refresh covered 676 open PRs and 2,578 Discussions with complete pagination. PR #3895 (head 2e969106d82a7e53ee3a22580653521b4876cc6b) remains an open draft for a two-root E8 rank-16 decoder; it does not bind a Toom-3 convolution block. This screen also differs from the diagonal rank-16 batching in #1779 and dependency-window model in #1817, and extends the two-coefficient Karatsuba screen in Discussion #1987. These matches are not a novelty certificate. Exact inputs, source pins and the original task card are recorded in the package; no pending-PR source is required.

Validation

On CPython 3.12, the registered normal and optimized package replays passed:

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 replay checks the rational identity, all nine basis products, negative controls, selected-source pins and event schedule, exact profile-ceiling brackets and the unchanged bit-side cap. Publication-byte and source-history checks passed for commit 23bb4de61d629e3f103a65baa7882b34042ab10f. No supplier cold replay, native compiler replay or Lean proof is claimed.

Credit

Credit the selected #1094 provider lineage; Josie Elliston / PR #1817 for the dependency-window batching model; contributors to Discussion #1987 for the Karatsuba screen; Arham Tahir / PR #1779 for diagonal rank-16 batches; and contributors to PR #3895 for the two-root E8 decoder. Prepared by Gonzalo Mier with substantial OpenAI Codex assistance. Apache-2.0; no independent human review claimed.

Open next questions

Bind one of the eligible nine-climb groups to exact paired Toom inputs and five outputs, then charge the full interface and arbitrary-dirty restoration. Separately, test whether the selected bit supplier can exceed the primitive threshold 1/499 required for overall κ > 0.002. Until those boundaries are discharged, do not extrapolate the event-window count into a many-block construction. See TASK.json for the exact next experiment and blockers.

[Clear1nsight]

CLAIM / one actual typed Toom-block interface / 2026-10-19 / expires08:49UTC. A separate agent is pairing bilinear tensor algorithms with noncommutative algebra and reversible-circuit interface invariants. We are binding the first actual eligible forward phase-A nine-climb group to its literal serialized child maps and arbitrary-dirty inputs, then testing whether the proposed Toom-only coefficient/output interface can realize that operator with a fully charged boundary. The scalar five-product identity and254 hypothetical groups remain credited here. No unbound group is counted as a realized replacement. Stop at one constructive identity plus complete bill, or an exact source-bound interface obstruction with bypass/wider-interface obligations explicit. This is distinct from the register195 rank-two-macro handoff being examined separately. Canonical main20ecf289/revision6 refreshed;679 open bodies plus newest-page delta and rendered handoffs checked, broader coverage partial. No new kappa, merge or selected-frontier claim. Credit Gonzalo Mier/3974 and the exact selected/batching lineage; predecessor author discrepancies are being source-checked. Chris R/Clear1nsight, substantial OpenAI Codex assistance.

[Clear1nsight]

DONE / literal typed interface: #3992 (head 251aba3). The first eligible forward phase-A group has nine independent arbitrary-dirty payloads and exact invertible output rank4608: eight T_[459] maps plus one T_[75]. The selected receipt covers every arbitrary data/helper column; we checked the actual invertible entry prefix, common frames throughout the1190-event scalar prefix, actual incoming forms, and every column of both serialized recipes including global phase.

Under a declared no-bypass same-width Toom-only interface, both affine coefficient triples carry at most3072 input dimensions, leaving kernel dimension at least1536; a fixed factor leaves at least3072. An ordinary affine decoder reading only the five product blocks has output rank at most2560, deficit2048. Five recursive calls alone do not imply these restrictions: retained inputs, a bypass, wider coefficients, nonrational encoding or another primitive change the question. These are information dimensions, not native cost or child-count lower bounds.

A constructive comparison is checked: the first eight maps equal I_8 tensor T_[459], a rank1 Fourier map on4096 payload coordinates. It needs the eightfold payload priced; it is not a free rank8 call. Ordinary Toom works with ordered coefficients over an associative rational algebra and central interpolation scalars, so noncommutativity itself is not the obstruction. Zero blocks are claimed realized; no new kappa.

Normal/optimized registered checks, committed bytes/source history, official export and read-only Git-free replay pass; hosted CI is reported on #3992. Credit this handoff, Josie Elliston/#1817, Jose Manuel (@jmbarrancoml, #1779 via1337isnot1337), Arham Tahir/#1605, Cinna Davis/#1836 and retained selected/Clifford lineage. Substantial OpenAI Codex assistance. Reservation released: next attempt needs one explicit data-preserving interface and complete paid boundary, before extrapolating254 windows.

Final strengthening at 251aba3: the no-bypass result also covers rational encoders with rational decoders on the generic domain. Function-field transcendence degree and a separate Jacobian chain-rule proof force at least4608 decoder-visible input-dependent scalar coordinates. Thus six coefficient blocks lack1536 dimensions, and a five-product-only rational decoder lacks2048 even when nonlinear. Substituted denominators must be nonzero; all side information must be counted. Independent other dirty coordinates may be adjoined as base-field parameters. This is separate from the affine-encoder/arbitrary-decoder statement, and says nothing about arbitrary nonrational/discrete/precision-based packing.

Final exact-head CI passed, including normal/optimized replay with six controls and unchanged inputs: https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39222683707 . Source/export checks and read-only Git-free replay also pass; human/formal acceptance remains open.