# Verification report

The accepted statement is Theorem 1 and its initial-shift extension in the
six-page manuscript. Its proof is general for all positive real atoms,
including repetitions, every p,q>=0 and every admissible increasing index
tuple. The underlying qualitative classification is classical and is not
counted as a newly solved open problem.

Self-audit covered structural zeros, the two factor support conditions,
the constructive interlacing induction, p=0/q=0/p>=k cases, cap monotonicity,
uniqueness of the maximal feasible tuple, positivity of factorial minors,
existence of every coefficient in the stated interval, and initial shifts.

Exact computational diagnostics, normal and optimized modes agreeing:

- 143,103 scalar minors, including repeated atoms and initial shifts.
- 22,904 explicit positive Cauchy-Binet factor-chain witnesses.
- 900 full polynomial expansions, checking exact support and both endpoints.
- 80 independently summed full Cauchy-Binet polynomial identities.
- 900 dynamic-programming move counts versus factorial determinants.
- Four index/parameter negative controls.
- All 69 minors of a finite 4 by 4 TN scope-control block; the designated
  minor of its zero extension is exactly -1/4.

The general proof does not rely on finite testing. The package preserves
checker source, expected exact output, proof and source/novelty reviews.
The PDF was compiled successfully with the built-in LaTeX compiler and a
local TeX engine, and all six complete final pages were visually inspected.
Portable rebuild and checker replay results are recorded in build_record.json.

No independent reviewer or formal proof assistant has verified the theorem.
No exact earlier statement of the full quantitative formula was found in
the bounded search, but priority is not certified. This is an unrefereed
quantitative note, not a new closure of the already classified AIM record.