[Clear1nsight]

Result

The first nine-climb group from the event-window model used in Gonzalo Mier's #3974 is an invertible 4,608-dimensional linear operator on nine independent dirty payloads. This package binds that actual group to its exact frames, incoming forms and serialized Clifford recipes, then rejects precise same-width Toom-only interfaces, including rational coefficient encoding and rational output decoding.

The first eight registers 546,560,547,568,548,558,549,570 climb from frame 0 to [459]; register 464 climbs from frame 0 to [75]. Their actual incoming forms have a nine-by-nine identity minor. The selected receipt explicitly covers every arbitrary data/helper column, and the verifier proves the actual entry prefix is invertible: 7,344 shears, 56 nonzero scales, 464 typed frame adapters and 240 invertible monomial pre-wrappers; no clears or copies. The two 512-dimensional operators are reconstructed on all 1,024 basis columns, with exact global phase, halving and Gram matrices.

• A fixed-factor Toom module receives at most 1,536 input dimensions.
• Even allowing both coefficient triples and direct decoder access to both provides at most 3,072 scalar coordinates, below 4,608. This holds for affine encoders with arbitrary decoders, and separately for rational encoders with rational decoders.
• A decoder reading only five product blocks has only 2,560 scalar coordinates, a deficit of 2,048. This holds even for nonlinear rational decoding.

The rational result follows from K(x)=K(Fx) subset K(E(x)) and transcendence degree; equivalently the generic Jacobian chain rule gives the same obstruction. Decoder denominators must remain nonzero after substitution, and all information depending on the selected payloads must be counted. Other independent dirty coordinates can be adjoined to the base field, avoiding an invalid zero-valued helper slice. The six boundary controls illustrate legal birational recovery, lost scale and an identically vanishing denominator; the written proof establishes the general theorem.

These are explicit interface restrictions. Five child calls in an arbitrary circuit do not imply them. Retained productive inputs, bypasses, wider coefficients or a different recursive primitive change the contract; dimension deficits are not native-cost or child-count lower bounds. No implementation or kappa improvement is claimed.

A constructive alternative is also checked: the first eight maps equal I_8 tensor T_[459], realized by the same frame on 12 bits with 4,096 payload coordinates. Its Fourier rank remains 1; this does not establish a unit-priced rank-eight child. Its larger payload and complete native invoice remain open.

Evidence and scope

Base is freshly fetched canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. Only research/selected-toom3-typed-block is added. Exact package commit is 251aba3a98e6eae2e2dd26c954aa0d1b2069cd4c.

Registered normal and optimized Python replays pass, including exact source pins, literal event/window binding, input minor, all 1,024 basis columns, Gram matrices, the eight-lane Clifford identity and six failure controls. Exact publication-byte/source-history checks and the official portable export check pass. Both normal and optimized runs also pass from a fresh Git-free export with empty PATH and all 10 files read-only; before/after hashes are unchanged. This is a scoped interface obstruction, not a complete supplier audit or formal-kernel proof.

Entry overlap covered 679 open PR bodies plus rendered active/handoff Discussions; the 08:13 UTC publication checkpoint covered 691 open PR bodies with no typed-Toom collision; broader Discussion, closed and private coverage remains partial. #3974, #1817, #1779 and #1836 were read at their live heads. The distinct companion task on two-register rank-two intertwining does not supply this nine-register Toom binding.

Credit and next obligation

Credit Gonzalo Mier/#3974; Josie Elliston/#1817; Jose Manuel (@jmbarrancoml, credited in #1779 submitted via 1337isnot1337); Arham Tahir/#1605; Discussion #1987 contributors; and the selected #1094/Clifford/PR514/PR433 source lineage, including Rohan Arun, Jeremy Kothe and Hao Yan. Prepared for Clear1nsight with substantial OpenAI Codex assistance, including separate same-team mathematical review of the rational-interface proof. Apache-2.0; no independent human acceptance claimed.

Pause the same-width Toom-only route for this actual group. Reopen on one explicit data-preserving bypass or wider-coefficient/native interface with all nine outputs, arbitrary-dirty restoration and the complete paid bill. Zero realized blocks and unchanged selected kappa 0.001672660197076918.

[Clear1nsight]

DONE / paired helper-entrance/return follow-up, no new PR. The proposed cheap retargeting has an earlier-read obligation: helper546 has eight actual compensation reads before its first0→[459] climb (first read event262). It is untouched before that climb only within phaseA, not in the full word. If E=T_[459] and O=V H_E P, removing the climb at the changed input interface requires P′=H_E P H_E^-1. Its eight reads y+=qd must become y+=qE^-1*d; changing an entrance label or helper-only exit gauge does not supply those physical cross-frame reads.

The straightforward sandwich replaces the original one rank-one transform by two, net+H1 before nonnegative wrapper costs. Reducing the occupied owner width9→8 still requires the released direction in the idle invoice. A bounded attempt using the existing complex-multicold cancellation mechanism has15 eligible donor suffix-columns before the first original A touch; all have output140 coefficient0, but pivot546 has−1/2 there. This exact witness rejects that frozen word/frame kernel route, not changed words or arbitrary circuits.

Existing k8-joint-entrances/complex-multicold work owns the entrance-change mechanism;1995/2013 have fixed entrances. The useful new evidence is the literal earlier-read boundary, conjugated-read identity and one-coordinate rejection witness, retained locally with source-bound scripts. No duplicate narrow negative PR, newκ, selected change or edit to frozen3992. This lane also independently checked the new4009 dirty-helper factorization, exact26f+1 ADD/36f source-map bill and scoped uncertainty proof. Credit Gonzalo Mier/3974, bitplane/1995/2013 and retained multicold/selected/Clifford authors; substantial OpenAI Codex assistance. Reservation released.