Bound two-child serial affine COPY at 121 ranks and price the mixed-width gate

[Clear1nsight]

The mixed-width7+1 and affine-translation escapes left by #4014 cannot supply a cheap serial affine COPY. For the actual all-class rank-seven target, any two proper reversed children separated by forward affine address maps require total width at least121f. This package proves that bound and binds its favorable invoice to #4010's pooled profile.

The proof combines local-ring coordinate flags with reversible address information flow. Fixed outer57f coordinates force both child widths to reach that range; the middle14f quotient requires another7f of total width. The favorable hypothetical58+63 pair adds295488 rank mass across2592 COPY calls. Its zero-exponent moment is21915/18944>1, before source preparation, routing, restoration, disposal or reserve charges. An adaptive schedule would need theta<107/3078 at exponent zero under this invoice, plus a genuine frequency and producer-cost proof.

Scope: one serial source pullback with arbitrary affine lower-triangular maps and diagonal ancestor units. This does not exclude general multi-role masked Boolean circuits. An explicit zero-child masked translation identity is included as a constructive escape; it uses q^(7fw) branches in the literal full-scan realization. No new kappa or supplier admission is claimed.

Evidence includes complete729/32768 middle-factor enumerations,9120 local-ring cases, actual128/256-coordinate target cut profiles,81/6561-address masked translation checks, and an independently authored64^3 F2 product census. A proper-child toy attains the quotient bound, guarding against an overbroad impossibility claim. All four normal/optimized/export/independent commands pass; read-only Git-free execution rejects altered, missing, linked and corrupted inputs. Separate same-session AI review passed with unchanged scientific-file hashes.

Predecessors are pinned and credited: #4014 (99a6320), #3982's parabolic COPY, #4010 (92daee4) pooled requests/profile, #3967's caller catalogue, and the retained R437/OQ4 source/license lineage. This changes only the isolated research/r437-affine-two-child-bound package. Substantial OpenAI Codex assistance is disclosed; no shared/frozen source, selected frontier, merge or closure is changed.

Reproduce:
python3 -B scripts/research_package.py --package research/r437-affine-two-child-bound

Published head d4e989bb518f471291966cc771081c326ea45865 passes exact17-file package/input byte and canonical source-history preflight. Main20ec/priority6 refreshed unchanged at11:08UTC. All4 registered commands pass locally; hosted CI pending at creation. Canonical claim.