# Verification of the recurrence criterion and counterexample

The accepted result is the fixed-multiplier scalar tail criterion
and its explicit separated-interval obstruction. The proof checks
both necessity and sufficiency for stabilization of S plus the
polynomial orthogonal complement. Sufficiency first removes the
orthogonal-complement component and then proves the degree bandwidth.
The ideal criterion uses Cayley-Hamilton only after the finite quotient
endomorphism is well defined.

The counterexample uses a bounded moment integral on [0,1] and an
exponentially growing test integral on [2,3]. It proves infinitely many
failures for every nonconstant multiplier. The mean-zero Legendre
control demonstrates that replacing S plus its complement by S alone
would make the necessity assertion false. The density argument limits
Hilbert projection onto the algebraic constrained space; it does not
invalidate finite-dimensional or oblique compression methods.

Exact regression runs use Fraction arithmetic. Across five families
and degrees through 18, two constructions agree on 90 basis polynomials.
There are 972 constraint and orthogonality checks, 628 positive-control
band-coefficient checks, 18 nonzero x-defects in the separated-interval
case, 18 density controls, a prescribed-gap test with gaps {1,4,9}, and
four rejected mutations. Normal and optimized runs produce identical
mathematical output. Finite testing is not an infinite or formal proof.

The five PDF pages were visually inspected. Both the built-in LaTeX
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missing glyphs, or unresolved references. The name-only author line,
affiliation footnote and prose AI disclosure follow the requested format.

The originating researcher's self-audit found no remaining mathematical
gap in the stated scope. The manuscript is not independently reviewed or
formally certified. The broad AIM source remains unresolved here, and
priority confidence is low.