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 under that hypothesis.
With the stronger finite sixth-X moment, the full degree-six moment matrix is
positive definite as well. Its even sextic Schur innovation has a strictly
positive mean and regression variance. At y=0 the mean is greater than
2,000,000,000,000. The degree-seven theorem assumes the corresponding seventh-X moment and adds
the odd-four block. Every finite upper-tail cutoff is positive. A separate
certificate proves positive upper-tail means of the two-column local residual.
The strict-variance proposition covers every fixed finite higher odd output list
and any nonzero nonnegative density weight with the required highest moment;
this does not make its mean residual positive for arbitrary weights.
Polynomial center weights give positive degree-seven jets for every associated
kernel index n>=1. The physical kernels have local positivity on fixed shapes
of at most eight points; the scale depends on shape and parameter.
No arbitrary-spacing, all-degree or global Fourier-positivity theorem is asserted.
The fifth- and sixth-X hypotheses of the earlier results are retained.

Contents
  main.pdf                    complete manuscript
  main.tex                    editable source with bibliography included
  verify_septic_tail.py        complete odd-septic unbounded positive tail
  verify_septic_compact.py     exact all-cutoff compact odd-four certificate
  verify_septic_residual.py    exact two-column residual upper-tail means
  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
  verify_sextic_initial.py    initial fixed-center even-sextic obstruction
  verify_sextic_tail.py       complete even-sextic positive matrix tail
  verify_sextic_mean.py       actual local sextic innovation integral
  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 eleven 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
  python verify_sextic_initial.py 256
  python verify_sextic_tail.py
  python verify_sextic_mean.py 1024
  python verify_septic_tail.py
  python verify_septic_compact.py 1024
  python verify_septic_residual.py 1024
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 sextic mean checker also takes longer than the small polynomial checks.
Its full inverse penalty and normalization are enclosed directly; its baseline
is not inferred by subtracting two averaged quantities. The initial sextic
checker proves a fixed-center negative square, not a counterexample to the
averaged theorem. Its negative sign and the compact published classification
give the needed sign transition; verify_sextic_tail.py covers the entire
remaining half-line, not merely a finite range of sampled centers.

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.

The two septic compact checkers enclose every real cutoff in [0,7/100]
using signed whole-cell loss intervals, not a sample grid of cutoff values.
They share one interval arithmetic engine. The residual checker splits the
cell containing 7/100. Both retain all theta atoms through derivative fourteen.
Their finite output is joined to the separate analytic tail and cited compact
classification in the paper. Normal/optimized agreement repeats one algorithm;
it is not an independent interval implementation or external human refereeing.
