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

Read main.pdf for definitions, complete proofs and scope. Full averaged moment
positivity through degrees five, six, seven and eight has the respective
fifth-, sixth-, seventh- and eighth-X integrability hypotheses on nonzero
nonnegative nondecreasing center weights W, relative to s^2 Phi(s)^2 ds.
Every finite physical y>=0 is included, with the normalized y=0 endpoint.
The degree-seven and degree-eight upper-tail matrices are positive at every
finite cutoff. Separate two-column local residual means are positive.

The general theorem proves, for every finite integer N>=1, local degree-N
positivity at X>=1000 N^2 with a99/100 reserve in center-dependent normalized
Hermite coordinates. An exact parity-block algorithm supplies a finite tail
threshold for any input degree and has a proved computable termination bound.
These results do not prove all orders at one fixed center or every averaged
order. General regression rigidity requires a genuinely positive pointwise
prefix, including s=0; its strict variance does not pay a negative mean.

Polynomial weights give degree-eight derivative jets for every associated
kernel index n>=1. Physical translation kernels have local positivity on fixed
shapes of at most nine points; their scale depends on shape and parameter.
The stronger all-order monotone-weight cone is FALSE: every sufficiently large
cutoff has a negative complete-theta Fourier value at an explicit frequency,
and thus some finite rational polynomial fails. The supplements certify a
negative point at A=23/2,x=154 and a closed negative frequency interval.
No explicit polynomial coefficients, numerical universal onset, arbitrary-spacing
positivity, global physical Fourier positivity or RH conclusion is asserted.
Classical generic mechanisms and source comparisons are credited.

Contents (24 flat members)
  main.pdf                      complete manuscript
  main.tex                      editable source, bibliography included
  cutoff_obstruction.tex        complete all-order obstruction proofs
  verify_theta_cutoff_fourier.py complete-theta negative point and interval
  verify_theta_cutoff_direct.py  independent original-integrand sign check
  verify_prefix.py              global odd prefix and lower-order controls
  verify_tail.py                complete odd-quintic positive tail
  verify_even.py                complete lower even-block positive tail
  verify_separated.py           quintic mean and regional variance certificate
  verify_counterexample.py      generic log-concavity counterexample controls
  verify_sextic_initial.py       initial even-sextic fixed-center obstruction
  verify_sextic_tail.py          complete even-sextic tail
  verify_sextic_mean.py          complete local sextic innovation mean
  verify_septic_tail.py          complete odd-septic tail
  verify_septic_compact.py       every compact cutoff, odd-four matrix
  verify_septic_residual.py      every compact cutoff, two-output odd residual
  verify_octic_tail.py           complete even-octic tail
  verify_octic_compact.py        every compact cutoff, even-five matrix
  verify_octic_residual.py       every compact cutoff, two-output even residual
  verify_hermite_tail.py         exact constants and generic Hermite controls
  verify_finite_order_tail.py    effective parity-block algorithm, input degree
  CITATION.cff                   citation metadata
  MANIFEST.sha256               complete hashes except manifest itself
  README.txt                    these instructions

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

The sixteen retained arithmetic scripts use only the standard library and reconstruct their
own arithmetic. 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
  python verify_octic_tail.py
  python verify_octic_compact.py 1024
  python verify_octic_residual.py 1024
  python verify_hermite_tail.py
  python verify_finite_order_tail.py 0
  python verify_finite_order_tail.py 8
  python verify_finite_order_tail.py 10
Repeat with python -O in place of python. Explicit exception acceptance gates
remain active. Each corresponding normal/optimized output should agree.
The complete-cell integrators are slower than the polynomial checks and print
progress on stderr. Specified grids reproduce the manuscript's finite checks.

Both septic and both octic compact checkers cover every real cutoff in
[0,7/100] by signed whole-cell loss intervals, not by sampling cutoff values.
Their shared outward rational arithmetic engine is not a collection of
independent arithmetic implementations. Residual integrators split the cell
containing7/100 and retain the full inverse penalty and output cross coupling.
Omitted theta atoms and all head-tail/tail-tail terms remain rigorously bounded.

The Hermite checker reconstructs121 generic formal coefficient pairs through
degree10 and exact rational constants. This bounded control is not an all-N
proof by testing; the analytic row/disc theorem in the paper supplies that proof.
The finite-order algorithm returns full halfline parity bounds. Its safe bound
is theoretical and can be very large; no efficient complexity is claimed for
arbitrarily high input degrees. Input degree0 is included; negative degrees
are rejected. Returned low-order thresholds are not new averaged theorems.

The inherited published compact fixed-center classification is a stated
mathematical dependency. Its historical complete computation is not rerun by
these shorter scripts. Complete-tail estimates and finite-to-global arguments
are proved in the manuscript. Normal/-O agreement repeats one algorithm and
is not external human refereeing or formal proof-assistant verification.

The fixed-center sextic obstruction and generic smooth-density quintic
counterexample do not refute the actual averaged theta conclusions. Arbitrary
nonmonotone weights are covered only in the explicitly stated supporting
variance or far-center statements. No reuse license is specified.

Complete-theta Fourier certificate reproduction
  python -m pip install python-flint==0.9.0
  python verify_theta_cutoff_fourier.py
  python verify_theta_cutoff_direct.py
Both require Python 3.10+ and write their certificate JSON to standard output.
The strict outward-enclosure gates, not requested quadrature tolerances,
establish negativity. Run with -O as an additional control: acceptance uses
explicit exceptions rather than removable assert statements. No finite atom
truncation or numerical plot is substituted for the paid complete tails.
