Scoped negative result: volume-aware selector batching of complex children

[g-guthrie]

An attempted replacement of two equal-rank complex children by one selector-batched call gives a false moment saving if the joined array's volume is omitted. This scoped negative result supplies the exact circuit, restoration of arbitrary dirty padding, and the complete volume-weighted budget on a source-pinned pair of rank-2 calls from the accepted #626 jp8 supplier. No new κ or supplier is claimed.

At the accepted supplier's complex exponent, the unweighted 2 C_2 -> C_3 + C_1 comparison gives ratio 0.999786029497691095. The actual joined array has twice the volume, giving 1.999572058995382190. Even granting the inverse selector and all routing/frame work for free leaves ratio 1.499003977340941911. Exact rational enclosures are included.

The note distinguishes the two uniform scaling regimes: a fixed pair uses one selector bit and has the same leading asymptotic moment; a selector slot scaled by f requires 2^f lanes and cannot retain the two-stream volume. A support argument excludes replacing selector correction by monomial routing alone. These are comparisons of the inherited cost bounds, not lower bounds on arbitrary algorithms or a defect claim against production admission.

All new research material is in research/selector-batching/, plus one focused CI workflow. The selected record, supplier code, and upstream/ are unchanged.

python3 -B research/selector-batching/verify.py

Expected: PASS_SELECTOR_BATCHING_IDENTITIES_AND_VOLUME_OBSTRUCTION.

Validation completed locally on Python 3.14.6 / macOS:

• 34 full forward/inverse operator cases, 2,776 exact basis columns / 753,568 entries, including 60 arbitrary dirty-padding columns. Integer Gaussian butterflies agree with a separate closed-form Kronecker reference.
• Six operator controls and three incomplete-invoice controls rejected; concrete operator counterexamples retained.
• Normal and isolated optimized replays pass. A relocated minimal checkout with a recompressed input gives the same certificate. Changed native input and corrupted expected certificate both fail under -I -O.
• make selected-record-check, local documentation links and staged whitespace checks pass.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Assignment seed 22, complex supplier. The initial review covered 487 open PRs; the pre-submission refresh covered 495 through #675, followed by an immediate check of #676–#678. No pinned sprint issue existed at either check; the local claim records the completed scope. #646/#651 cover the other candidate tasks. This experiment differs from within-array tensor batching, #451's split-residual routing, #665's spectral dilation, and #673's nonlinear dirty echo.

The all-size conclusion relies on the written tensor identity and monotonicity argument, not extrapolation from finite replay. Full supplier/repository-wide replay and Lean were not run locally for this isolated research addition. Focused Linux CI passed on Python 3.11 and 3.14 at the submitted head dce137e540acd2b689281afe222cd32c0ec1f1ae. The broader inherited CI is separate and is not claimed complete.

Prepared by Greer Guthrie with substantial OpenAI Codex assistance. The package credits Kothe, Boukhalfa, eumemic, icekylinx, DreamingOfClouds, Arun and the retained construction lineage; all new material is Apache-2.0.

[Mister-Meeseeks]

Integrated at ee055dc44815497927ef618fc985b93c2dcbda2a in the 20-PR pilot, retaining your original commit, source notices and contributor credit.

Exact selector circuit and dirty-padding repair; volume-weighted budget defeats this specific batching proposal. Tensor identity and support/volume argument reviewed; no universal algorithmic lower bound.

The scoped package check passed at dce137e540acd2b689281afe222cd32c0ec1f1ae. This is research integration; the selected κ and inherited theorem assumptions are unchanged. Review, evidence and attribution. Thank you for the contribution.