Scoped negative result: exact constant-row center shear optimum (no new kappa)

[RossRichardson]

The historical rank-h complex center admits infinitely many integral shears using its constant gather row. This note classifies every such shear that preserves the priced scalar alphabets, and proves that PR937's factor uniquely minimizes the complete gather/scatter bill for a fixed pivot and every even h≥10. At h=26 the minimum is 42,600. At h=8 the original size-two anchor wins instead, 702 versus 714.

Status: scoped negative result / research note; no new κ. PR937 supplies the construction and its savings. This contribution adds an exact boundary for an entire simultaneous-shear family, beyond the finite one-step and second-step searches in PR809/937.

The gather alphabet forces every integral offset to be binary. Permutation symmetry reduces all 2^(h−1) anchors to h cardinality classes; exactly sizes 0–4 preserve the signed scatter alphabet (0–3 for the original guard). Four exact polynomial identities have strictly positive constant coefficients and nonnegative binomial-basis coefficients at h−10, proving the all-h cost comparison. General nonconstant-row mixing, rational bases, rescaling, frame changes and cross-macro sharing are outside the result.

At h=26, the proof covers 33,554,432 binary anchors by symmetry, including 15,276 admissible ones. The complete bill includes every source read, both gathers, both scatters, dirty restoration and the unchanged side work. Helpers, frames and recursive profiles remain fixed, so this local boundary cannot produce a new exponent.

Reproduce offline, Python 3.11+ standard library:

python3 -I -B research/dyadic-center-anchor-bound/verify.py

Expected:

PASS_CENTER_ANCHOR_BOUND h=26 orbits=26 admissible=15276 optimum=42600 all_even_h_ge_10 controls=14 no_new_kappa

Executed validation on macOS/CPython 3.14.6:

• Complete 128-anchor h8 and 512-anchor h10 censuses; all 26 literal h26 representatives; rational polynomial identities checked coefficient by coefficient.
• Twenty exact central commutator replays, with 3,700 input-column/direction obligations, arbitrary dirty inputs, operation counts and 14 adverse controls.
• Nine package tests, including a separate direct integer matrix-product implementation, bounded integer-offset census, all h8 pivots, minimal optimized relocation, corrupted sources/certificates/typed receipts and output preservation.
• Four inherited complex-center tests, selected-record integrity and staged whitespace checks pass.

Base: 47481c89fd30a43f8f882eb4b391e89c29273fd0, selecting PR890 at conditional κ 0.001661957774647029. The selected certificate, T21 construction and maintainer review were read. Eight inherited inputs are SHA256-pinned. Predecessors: PR809 66cabb478385fbaa1a8060123cd57545d703d84e; PR937 b303fed6dfe2066dcd78c0d8ed74d2a3b153ba72. No pending code is imported.

OS assignment seed 33 chose the complex track and pivots for finite diagnostics. The local claim records three candidates, the exact class and stopping rule. No pinned sprint issue exists. The complete overlap refresh covered 717 open PRs through #946; predecessor heads were unchanged. Main advanced during publication to 007664306fff84fbb1c9a1b4053b69f439764034; a fresh fetch confirms all eight mathematical inputs are byte-identical and PR890 remains selected. The branch retains its exact starting base. The result corroborates PR937 without claiming its savings as new. PR863 changes a different side circuit; no composition is asserted.

Only research/dyadic-center-anchor-bound/ is added. upstream/, generators, selected frontier and existing artifacts are unchanged. Full make verify, selected cold replay, new global physical admission, Lean and remote CI were not run. Historical side/phase and all-size/compiler, realization/hosting, precision/tape and analytic obligations remain inherited. The all-h scoped result rests on the written proof; finite replay is not a proof of the multiplication theorem.

Credit Douglas R. Colkitt's original factor/invoice, Curtis Dransfield's PR809 basis-shear mechanism and signed pricing, and Ross Richardson's PR937 minimizing construction. Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance. Apache-2.0; no independent human review claimed. Exact receipts, source pins, proof, local claim and validation are included.