Scoped negative result: nonunit dyadic target pairs cannot improve X4 (no new kappa)

[ToikY]

Scoped negative result; no new kappa. Extending PR901's restored X4 target-pair method from unit signs to nonunit powers of two cannot improve its disjoint-pair family under the inherited binary-Horner lowering. Every nonunit pair costs at least 20 additional real primitives per orientation, including both basis shears and every coefficient operation.

The complete census covers all 112 × 111 ordered pairs, both signs, and exponents −3 through 3: 174,048 cases, including 149,184 losing nonunit cases. The 2,978 profitable unit cases corroborate PR901. A written binary-digit argument proves that every pair's cost strictly increases along both exponent tails, extending the exclusion to every nonzero integer exponent. Finite replay and the written universal argument are distinguished.

Only research/complex-x4-scaled-pairs/ is added. There is no installed circuit or change to the selected frontier, generators, or upstream/.

Reproduce

python3 -I -B research/complex-x4-scaled-pairs/verify.py
python3 -I -B research/complex-x4-scaled-pairs/crosscheck.py
python3 -I -B research/complex-x4-scaled-pairs/test_package.py

Expected main marker: PASS_X4_NONUNIT_PAIR_EXCLUSION cases=174048 nonunit=149184 tails=all_integer_exponents min_extra_real=20 no_new_kappa.

The independent Fraction checker prices literal primitive words and matches the main integer checker's complete record digest. It checks 137 coefficients, 548 dirty-work boundaries, and 56 wrapper basis cases in each orientation. Ten package tests pass on Python 3.9.6 and optimized Python 3.12.14, including missing-pin/source/typed-receipt corruption, incorrect cost/tail controls, a minimal offline relocation with spaces, and output preservation. Selected-record integrity and its five tests pass. All eight source files match their base Git blobs; the published tree equals the tested local tree.

Scope, provenance and credit

Exact base: f567884. OS seed 29, complex supplier track. The package consumes the committed maintainer replay and binds its canonical word and both programs to the accepted X4 pins. The original complete scalar invoices reproduce as 193,162 / 193,210. No full cold supplier replay, repository-wide suite, Linux/hosted CI, native frame/phase replay, global admission or Lean result is claimed.

Credit ToikY's #901 at dbe8dc01f8306bd7dab8d861a953b23e5cda6a70 and Sricharan Vagvala's #749 for the restored-pair method; the new evidence is the nonunit extension's exclusion. Credit Jeremy Kothe #799, Chafik Boukhalfa #741, icekylinx and the retained construction/emitter lineage. No pending implementation is imported. #883 changes the preceding precondition and #864 studies frame assignment; no combined result is asserted. The final pre-submission metadata refresh reached #905 and found no overlapping nonunit-pair result. No pinned sprint coordination issue was available; the local claim records the result and branch link.

The exclusion does not cover overlapping basis changes, other dyadic multipliers, helper bases, boundary-gate folding, coefficient sharing or alternative lowerings. It does not identify the unit-family optimum. Selected conditional κ remains 0.001661634653669567, with inherited all-size/compiler, realization, hosting, precision/tape and analytic obligations.

Prepared for ToikY with substantial OpenAI Codex assistance, including separate AI implementations and reviews. Apache-2.0; no independent human review or full multiplication-theorem proof is claimed.