Monotone center weights and the averaged odd-quintic theta gate
Scott Schubr

Read main.pdf for the theorem, proof, limitations and references. The main
theorem covers nonzero nonnegative nondecreasing center factors W with the
specified finite fifth-X moment, relative to s^2 Phi(s)^2 ds. It includes every
finite physical parameter y>=0, with the normalized derivative-limit at y=0.
The full degree-five moment matrix is positive definite. A separate corollary
gives local positivity on fixed shapes of at most six physical kernel points;
the scale depends on the shape and parameter. No arbitrary-spacing six-point,
all-degree or Fourier-positivity theorem is asserted.

Contents
  main.pdf                    complete manuscript
  main.tex                    editable source with bibliography included
  verify_prefix.py            global odd prefix and lower-order controls
  verify_tail.py              complete odd-quintic positive tail
  verify_even.py              complete even-block positive tail
  verify_separated.py         local mean and two-region variance certificate
  verify_counterexample.py    smooth log-concavity counterexample controls
  CITATION.cff                citation metadata
  MANIFEST.sha256             file hashes, excluding the manifest itself
  README.txt                  these reader instructions

Build with a current LaTeX installation:
  tectonic main.tex
or run pdflatex main.tex twice. Required standard packages: geometry, fontenc,
amsmath, amssymb, amsthm, booktabs, microtype and hyperref. No images, external
bibliography database or private files are needed. Tectonic can download its
normal TeX support bundle on first use.

The five checkers require only the Python standard library. They read no
numerical data files and reconstruct their own exact rational enclosures.
Run:
  python verify_prefix.py
  python verify_tail.py
  python verify_even.py
  python verify_separated.py 1024
  python verify_counterexample.py
Repeat with python -O in place of python. All mathematical acceptance gates
remain active. Normal and optimized mathematical outputs should agree exactly.
The separated verifier takes substantially longer than the small polynomial
checkers. Its default grid is finer; the explicit argument 1024 reproduces the
manuscript's recorded grid. It prints progress to stderr and its exact result
to stdout. Its status label EXACT_FINITE_INTERVAL_CHECKPOINT is deliberate:
the explicit numeric threshold gates do not replace the global analytic and
published-theorem dependencies. A failed gate raises an exception.

The tail verifiers reconstruct polynomial identities and full-series error
bounds. Analytic justification of the infinite tails and of the finite-to-global
lifting is in the paper. In verify_tail.py the preconditioned quantities are the
ones that certify the theorem; the unpreconditioned quantities are diagnostic.

One dependency is the published compact fixed-center classification cited in
the paper. Its complete historical classification computation is not reproduced
by these shorter scripts. The new proof uses its exact normalization, sign and
even-block statements, together with independently supplied unbounded tails.

The generic counterexample is a different smooth density. It does not refute
the actual theta theorem. Arbitrary positive center weights are not covered by
the odd theorem, and monotonicity is not asserted necessary or optimal.

No reuse license is specified in this package.
