# Verification of the geometric sampling counterexample

The six-page manuscript proves the explicit counterexample by Gaussian
moment identities, residue calculus, finite harmonic synthesis and
triangular free-cumulant inversion. A separate positive-real-part argument
and Jensen's formula give the sufficient sampling uniqueness condition.
Neither theorem is inferred from a finite computation. The broad AIM
transform question is not declared newly solved.

The originating researcher's detailed proof and source audits are retained
in the reproducibility directory. They are not independent review or
proof-assistant verification. Classical lognormal Stieltjes perturbations
and R-diagonal analytic transforms are credited. Absolute priority remains
undetermined after a bounded literature search.

Exact rational and Laurent-polynomial tests cover:

- 78 telescoping synthesis coefficients and 12 cosine-power identities;
- 144 vanishing derivatives and 12 nonzero leading derivatives;
- 3,978 integer phase controls;
- 144 atomic-law/damping cases and 864 noncrossing moment reconstructions;
- Haar-unitary and circular cumulant controls through order 8.

Numerical checks use mpmath 1.3.0 at 70 decimal digits. Independent integrals
test nine off-grid contour values, 36 grid moment cancellations, nine first
separating moments and three off-grid negative controls. The largest observed
contour discrepancy is about 8.84e-74, an empirical floating-point value,
not an interval-certified error bound.

Both checkers previously passed under Python 3.13.5, optimized Python 3.13.5
and Python 3.12.14, with identical outputs per checker. The source archive
contains portable versions and their reference outputs. Its additional
normal and optimized replay receipts bind the exact extracted bytes; the
numerical checker requires the separately installed pinned mpmath dependency.

The built-in LaTeX compiler and exported PDF compilation succeeded. All six
complete page images were inspected: author footnote, mathematical formulas,
proofs and references are legible without clipping. The final log contains
no errors, missing glyphs, undefined references, overfull boxes or underfull
boxes. Package integrity and portable replay results are recorded separately
in the sealed verification receipts.