Finite complex construction: two center shears save 5,562 primitives per invocation (no new kappa)

[RossRichardson]

Two commuting basis shears reduce the historical h=26 complex center's complete gather/scatter bill from 45,381 to 42,600. Compared with Curtis Dransfield's PR #809 single shear, this saves a further 1,309 operations in G+H. Source access, both gathers/scatters, dirty restoration and the unchanged side circuit are charged.

Status: finite complex-supplier construction / research note; no new κ.

The explicit replacement is V[0,T]=1, V[i,T]=t_i, U[S,0]=3*s_0-1, U[S,i]=s_i-s_0 for i≥1. Its exact product is |S∩T|-1. It is the original factor conjugated by A=I+E10+E20; A changes the tables and is not executed on dirty center registers. Coefficients stay in the original alphabet.

The full historical three-stage scalar bound decreases 15,959,102,640,000 → 15,846,305,280,000, saving 112,797,360,000. Independent instrumentation of the actual signed invocation counts 786,938 → 781,376 primitives in either direction. These are operation counts, not runtime measurements. Helpers, grouped incidence frames (including zeros), recursive children, W/m/s and rank profile stay unchanged. No selected-bound improvement follows.

Reproduce with Python 3.11+ standard library:

python3 -I -B research/dyadic-center-two-shear/verify.py

Expected: PASS_DYADIC_CENTER_TWO_SHEAR h=26 G+H=45381->42600 saved=112797360000 no_new_kappa.

Validation includes all 6,760,000 h26 central coefficients; a separate implementation checking 6,777,536 entries across h8/10/26 and all 5,226 independent h26 central input columns in both directions; full signed symbolic invocations at h8/h10; dense h26 execution; reused-bank three-stage replay; complete side/frame/recurrence accounting; 14 mathematical adverse controls; five package tests including minimal optimized relocation, changed selection metadata, source corruption and an underpriced receipt; nine relevant inherited tests; and selected-record integrity before/after rebase. See VALIDATION.md and deterministic EXPECTED.json.

Seed 39 selected the complex track. The declared census completed all 1,300 second shears after PR809's witness, with 602 admissible and a unique 43,128 minimum. One recorded algebraic simplification then produced the 42,600 factor. No unrestricted optimality claim is made. The factor is valid but more expensive at h8; the written cost proof shows improvement for even h≥10.

Starting base: 91a2094f2e022930069e8cfdc810a3ea1b81cd09. Rebased onto refreshed main 47481c89fd30a43f8f882eb4b391e89c29273fd0; all mathematical inputs remain byte-identical. SOURCES.json pins 21 inputs and the frozen original selection separately. The newly accepted PR890 conditional κ 0.001661957774647029 is acknowledged, not attributed to this work. PR809 remains pinned at 66cabb478385fbaa1a8060123cd57545d703d84e.

The final overlap check includes new PRs through #936. PR863 changes the distinct side DAG; PR922/924/931 concern the eight-register factory; PR932 changes selected X4 forests. None is composed here. No pinned coordination issue exists; the local claim records the contribution. Only research/dyadic-center-two-shear/ is added. upstream/ and the selected frontier are unchanged.

All-size compilation, realization/hosting, precision/tape and analytic transfer remain inherited. Selected-supplier transport, global invoice/owner admission and outer assembly are separate obligations. No full repository suite, selected cold replay, Lean proof or remote CI pass is claimed.

Credit Douglas R. Colkitt's original factor/circuit/phase/recurrence work, Curtis Dransfield's PR809 basis-shear mechanism, and retained contributor notices. Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance, including scoped AI implementation and adversarial review. Apache-2.0; no independent human review claimed.