[Lyy-iiis]

The invertible two-call prefix in #4688 attains the hook rank bound, but on three cuts its quotient rows are misaligned with the desired target. For its literal use as the first two calls on one fresh input, the remaining target deficit is 1016, greater than twelve further calls can remove (960). This is a scoped obstruction to that beginning, not to five-in-fourteen generally.

This follow-up constructs an explicit phase repair, P=SUS, Q=SVS, S e_x=i^popcount(x)e_x, using the same two calls and one zero scratch. All eight prefix ranks remain 160, and all joint target-prefix ranks become 224. The package proves these equalities for every natural lifted hook, gives a uniform inverse, and supplies an exact target-plus-one-call residual identity for continuation work.

This is a constructive local interface and a necessary-condition repair. No completed five-in-fourteen unit, improved kappa, or selected-frontier change is claimed. Other invariant tests and the actual multi-input continuation remain open.

Validation: exact Gaussian-integer ranks and four-point pencils; predecessor calibration and seven controls; independent Gaussian-rational 8-by-8 residual identities; normal and optimized Python replay; registered Python 3.12 package replay; committed-byte and export integrity checks. A separate agent reviewed the proof. No human, Lean, or full multiplication-theorem review is claimed. The optional full-history source preflight declined this shallow checkout; source bytes and original blob equality were verified separately, and retained sources are in this branch's history.

Reproduce:

python3.12 scripts/research_package.py --package research/phase-aligned-two-call-prefix

Credit: lee-phys/#4688 (264c4b3cb13c82d171de352ecf60486dfb8d2bd4) for the unchanged rank engine, two-image prefix and controls; Jacob Sussman/#4628 (ee4fd87f064017c23e5a86fc1dcfe0a5237caed3) for the model and hook bound. Prepared by Yiyang (@Lyy-iiis) with substantial OpenAI Codex and parallel-agent assistance; Apache-2.0.

Based on canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Overlap coverage and its limitations are recorded in OVERLAP.json (473 open PR bodies plus 210 recent Discussion records, not complete Discussion history).

Canonical finding and continuation handoff: #1166 (reply in thread)

Hosted Ubuntu/Python 3.12 replay now passes at b0d5192390af493d626054b060921895df3c94d2: https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39824365361 . The initial 240-second limit expired after six successful cuts on the slower hosted CPU; the same unchanged exact checker passed with a 600-second bound.

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