[aliromman]

This enabling interface result audits the additional paid returns required after an explicit initial observation module. If the target closure adds Delta dimensions and the ordinary matrix algebra has dimension r, at least ceil(Delta/r) new single-row returns are necessary. The exact rational tool closes the supplied ordinary generators under products and handles retained observations, feedback and overwritten rows. The complete kappa calculation is unchanged.

Proof and scope, audit tool, exact checks, source pins, and registered replay. Base: canonical main 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision 7. Contribution head: 30ff28bfa7986b5534794f58de0a6359feb474cf.

The result covers finite rational matrices on named independent input versions, one field capacity and one new row per paid return. It is a necessary bound; an initial-closure membership result does not price storage, access, ownership, movement or erasure. No current physical caller, all-width native lift, supplier admission or stronger multiplication bound is supplied.

CPython 3.12.14 registered normal and optimized checks pass: 62 cached-subset cases, 96 literal chronological programs, all 1152 prefixes, and eight controls. These include a stale-version charge, its current-version zero-charge counterpart, and a generator-product closure control. Both modes print PASS_PAID_HISTORY_QUOTIENT; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE.

python3.12 -B scripts/research_package.py --package research/paid-history-quotient

The standard library suffices. Committed-byte checks pass. No selected cold replay, native compiler, Lean, Docker, independent human review or proof-kernel acceptance is claimed. The selected orientation helper stopped on sparse-omitted sources; no baseline pass is claimed. Hosted checks were not yet observed at publication.

Credit Chris R (@Clear1nsight) and the retained authors of PR #5425 and PR #5446 for the actual-return history principle and the retained-observation question. The exact pointwise predecessor is a proof precedent, not an imported executable. Classical module dimension and rational elimination retain their credit. PR #5427 tests frozen formal-word observations and PR #5441 checks physical transfer tables; neither is repeated here. The scoped claim keeps caller construction with its existing owners.

Next: extract one complete versioned physical boundary, validate its available observations, and apply this finite audit. No such physical ledger is assumed. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.