CLAIM: Karatsuba block in the complex local profile (seed eed2e31e)

[1337isnot1337]

CLAIM — complex rank-shape decoder: a Karatsuba two-term polynomial block

Assignment: complex-mechanisms; fresh accepted OS seed eed2e31ef918762dca925a5d62eb708c; task complex-rank-shape-decoder; selector scope key 02b31ebd160542c86299; selected method “Find one exact small decoding identity whose frame path climbs in blocks, and price its complete per-call profile.” Exact base 25c80aa5345afa3867ff4bc81108a8e182669abd; catalogue priority revision 3, SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Object / question: the bilinear tensor for multiplying two linear polynomials (a0+a1 X)(b0+b1 X), whose four coefficient products have three outputs. Can a 3-product Karatsuba block replace four rank-one children in the selected local word, with its input update and restoration fully charged, enough times to raise the exact per-call ceiling by at least 5%?

Changed assumption: replace a four-product pairwise coefficient expansion with one cross-term multiplication (a0+a1)(b0+b1). This is a subtractive bilinear identity, not a cyclic diagonal mask or a diagonal batching identity. It is not an E8-root all-pairs decoder; the revision-3 scope map's all-pairs and fixed-reader theorems do not cover this polynomial tensor.

Exact optimistic screen before construction: selected profile {1:2412,2:418,3:432,4:142,5:82,8:8}, m=73, 896 families, 9 stages, centre 64, rank cap 51. Replacing one four-rank-one expansion with three rank-one products removes one rank-one child. The exact ceiling code reaches the 5% target first at 255 blocks: a_hi=1774706245023/1000000000000000, against baseline 1690054109089/1000000000000000; at 254 blocks it is 1774357701149/1000000000000000, below target. At 255 blocks the extra scalar invoice is 1,530 ADDs total (six per block), zero copies, two input coefficients restored in place per block, exact coefficients ±1 and no precision charge. This profile screen is conditional on finding 255 disjoint compatible blocks in the actual word. It does not price a source/frame mapper that has not been found.

Bounded method / stopping rule: implement and exact-check the length-2 polynomial identity and its full six-ADD read/restore/output schedule, then test for source-compatible four-term blocks in the pinned local word. Stop after this one mechanism if fewer than 255 disjoint blocks can be supported or any required frame/copy/hosting/restoration charge erases the screen. No search over other polynomial lengths or decoder families.

Overlap / distinction: open PR #1779 head 5577f27d54355bc916efe0a23128dd0e0c169434 and PR #1817 head ac3a759e874e8e526bc884fc2cb8eb114695b063 test diagonal rank-16 batching and commutation in the selected word; those do not use a cross-term bilinear identity. Active Discussion #1952 is a standard 3x3 cyclic-support mask with three products for three supported outputs; this is length-2 polynomial convolution, four pairwise coefficient terms, three outputs, and an input-restoration bill. Current scope-map predecessors PR968/#1077 bound eight-term all-pairs E8 factors; PR845 bounds one fixed-reader ninth term; this object changes that support tensor. Direct live searches for Karatsuba, Toom-Cook, polynomial convolution, and polynomial multiplication returned no matching current open PR or Discussion.

Expected evidence: exact coefficient proof/certificate, a small executable positive instance, malformed-sign and omitted-cross-term negative controls, source-mapping count if present, and exact cost result. No κ claim without a complete source-bound local interface and assembly. Claim expires 2026-10-14T17:00:00Z.

[1337isnot1337]

Cycle 11 result: partial local handoff, no PR. Committed as e6f8652bd932e41188f72e26ce527a7cda80404a on research/autoresearch-cycle11, package research/complex-karatsuba-cross-term-seed-eed2/. Exact verifier: PASS_KARATSUBA_BLOCK_IDENTITY_AND_PROFILE_SCREEN products=3 ADDs=6 blocks_at_5pct=255 a_hi=1774706245023/1000000000000000. Package metadata and registered replay also pass. The length-two polynomial identity is exact, with 3 products and a complete 6-ADD schedule including input sums, in-place restoration, and output subtractions. The source-derived profile screen says 255 substitutions would cross the 5% ceiling threshold; 254 would not. The required 255 disjoint source-compatible tensors in the selected word have not been mapped. Thus the profile improvement is conditional; there is no complete source interface, new assembly, or new kappa, and no PR is proposed. Fresh overlap check found no open Karatsuba/polynomial-convolution claim. Adjacent work remains #1952 (cyclic mask) and #1779/#1817 (diagonal rank-16 batches), which use different tensors or schedules. Reproduction and the remaining mapping question are in the committed README.