[lee-phys]

At block 3, #4628 leaves five arrays in fourteen calls open. Its published single-free-image obstruction gives a second-call gain at most 72, below the hook cap 80. This contribution constructs an explicit uniformly invertible two-call prefix whose second call instead uses two free images of the same earlier output and gains 80 simultaneously in eight specified coordinate cuts.

For addresses (x0,x1,x2), let U be CNOT from bit 0 to bit 2 and let V exchange bits 0 and 1. At every digit width f, the prefix implements

F_(2,f) = C_f (U_f + 2 V_f) C_f.

It uses one zero scratch register, clears that scratch, and has exactly two paid calls. All mixing coefficients are integers and no phase letters are used. The order-four permutation R = U^-1 V gives the explicit all-width inverse identity

(I + 2R_f)(I - 2R_f + 4R_f^2 - 8R_f^3) = -15 I.

The package recomputes exact ranks over Q(i), without external data or dependencies. For each of eight explicitly defined cuts in C^8 tensor exterior^4(C^8), the first call has potential 80; either summand alone gives total 152; the combined prefix has total 160, including after scratch cleanup. The target demand is 224. The caps/demands reproduce existing results and are credited as calibration, not new bounds.

A nearby control makes the limitation concrete: replacing coefficient 2 by 1 preserves all eight final potentials at 160 but gives nullity (8^f + 4^f - 2*2^f)/4. Thus tightness of these certificates alone does not certify injectivity.

Scope: a constructive equality-case lemma and search calibration. No full 5-in-14 unit, continuation, improved lower bound, or new kappa is claimed. It does not refute #4628's narrower single-image result. It shows why that result cannot be extended to arbitrary free linear combinations of one previous output.

Sources and task: branched from current main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51; only research/complex-two-image-hook-prefix/ is added. Card complex-rank-shape-decoder, with explicitly changed scope for the donor's small-unit assignment. #4628's README/PROOF are cited by exact source URLs and hashes at ee4fd87f064017c23e5a86fc1dcfe0a5237caed3; Sussman's single-term package is linked at 86e575e28ed94890872dcff1bb2621d7ff559cbd. Credit also to eumemic (#288), Swapnil Jain's P1/P2 as cited there, and OpenAI's engine. Coordination: #1166 (comment).

Validation: registered Python 3.12 normal and -O replays pass, with exact Gaussian-integer elimination, a separate determinant-formula check of all 313,600 first-call lift entries, the lift-square identity, seven controls, and independent cycle counts at widths 1–3. Metadata/source pins, committed-byte preflight, and package export checks pass. A fresh Git-free export replay with Python isolated mode also passes and leaves all exported files unchanged. No Lean, full-history ancestry check, or integer-multiplication replay was run. No independent human review is claimed.

Submission overlap refresh at 16:00 UTC covered all pages of 452 open PR bodies and 3,614 discussion records; it found #4628 and this investigation, with no competing result in the searched scope. Canonical main and #4628's head were unchanged. Local sparse-note limitations remain as recorded in the package.

GitHub CI passed on head 264c4b3cb13c82d171de352ecf60486dfb8d2bd4: https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39810593701 (onboarding checks and the registered package replay).

Prepared for lee-phys with substantial OpenAI Codex assistance. Apache-2.0. The package includes focused open next questions.