# Verification of the parameter hole theorem

The complete analytical proof has been accepted by originating-researcher
self-audit. It covers every nonzero L2 generator and arbitrary discrete
parameter set with a Bessel upper bound. No regularity, lower frame bound,
product structure or uniform separation is assumed. The raw zero-generator
exception is explicit and is not counted as another new result.

The main obligations checked in the argument are projective left covariance,
uniform local counting bounds, left Haar invariance, an explicit left Følner
sequence, vague compactness, nonzero limiting mass under hypothetical relative
denseness, compact cutoff passage of the Bessel bound, and the divergent
continuous coefficient integral by Plancherel and Tonelli. The density boxes
in the source are not Følner; their separate negative control prevents a
change in the target density definition.

The standard-library program checks 3,000 exact rational triples for group
identities and the cocycle, a symbolic Laurent-polynomial phase identity,
and the determinants of hand-derived Haar Jacobians. It also includes nine
numerical Gaussian coefficient checks, three slice integrals with a deliberately
missing-normalization mutation, and geometric diagnostics for 16 translated
Følner boxes. The source-box negative control has a symmetric-difference
ratio tending to one. Normal and optimized runs agree. Quadrature is floating
point and not interval-certified; these computations do not prove the
measure-theoretic theorem, novelty or source closure.

Eight primary source PDFs were retained in the research dossier, including
the recovered 2005 author preprint and current nearby density papers. The
mathematical claims are distinguished from the bounded novelty search:
continuous nonintegrability and invariant averaging are known. No inspected
source supplied the exact arbitrary-parameter conclusion, but exhaustive
priority is not asserted.

The desktop compiler succeeded. A separate local Tectonic build generated a
six-page PDF; all six pages were visually inspected for equations, references,
footnotes, clipping and layout. One underfull bibliography line is harmless;
there are no overfull boxes or unresolved references. See build_record.json
and rendered/visual_audit.json for hashes and the portable source build.

No independent review or proof-assistant verification occurred. No DOI,
remote draft, upload, website edit or Hugging Face post occurred. The author's
publication hold remains active.