A Sharp Zero-Sector Criterion for Polynomial Prefix Sums
Scott Schubr
October 24, 2026

Main result
For a real polynomial h with h(0)>0 and all zeros in the closed sector
|arg(-z)|<=pi/4, every real step a>=0 and geometric bias rho>=1 give
H_k(y)=sum_(i=0)^k rho^i h(y+ai) a finite concave logarithmic extension
from integer k>=0 and real y>=0 to the nonnegative quadrant. This proves
neighboring-row inequalities and Jensen for arbitrary finite-first-moment laws
with integral mean prefix length. The angle is sharp as a universal
sufficient sector, not a necessary condition for every individual
higher-degree polynomial. Every fixed bias 0<rho<1 has an integral-height
quadratic counterexample for a sufficiently large integral step.

The truncated kernel also has the sharp universal real exponent range
p,q>=2, proved by monotone fractional marginalization; every positive
exponent below two has a radius-one midpoint obstruction.

The proof includes a computer-assisted finite exact algebraic certificate
for the real truncated-square kernel. Complete geometry cases, formulas,
normalizations, gluing, continuous convolution, Abel induction and the
concave-contact argument are stated in the paper. The conclusion is not
concavity of an ordinary polynomial continuation in the prefix variable.
The split-projective-bundle application holds in all dimensions. No
arbitrary toric-variety or unrestricted unimodular-fiber claim is made.

Public files
main.tex          Complete editable LaTeX manuscript, including bibliography.
Paper.pdf         Compiled manuscript, when supplied in the prepared package.
verify_prefix.py  Self-contained exact certificate checker, Python standard library.
README.txt        This document.
CITATION.bib      Citation metadata for this manuscript.
SHA256SUMS.txt    Exact public-file identities, when supplied in the package.

Verification
Use Python 3 from this directory:
  python -B verify_prefix.py
  python -O -B verify_prefix.py
Both runs must succeed and print identical JSON with status PASS. No
external files, downloads, packages, floating tolerances, random searches
or input data are required. The checker writes no files. It reconstructs
the chamber and determinant identities, power sums, elementary moment
sectors, input-factor identity and sharpness counterexample identities.
These are exact polynomial checks over unbounded parameters, not sampled
parameter tests. The analytic portions of the proof must be read as well.

Compile main.tex with an ordinary LaTeX installation supporting the stated
standard packages. Run enough passes to resolve citations and references.
The bibliography is embedded; no bibliography download is required.

Provenance and scope
This paper develops the arbitrary-degree real-parameter prefix transform
from the earlier unpublished companion Concavity of Lattice Counts on
Toric Threefolds. Its separate Fano and threefold conclusions are not
reproduced as new results here. Classical tools and the substantial
Speyer lattice overlap are explicitly credited. No worldwide-priority
or external-refereeing claim is made. AI-assisted development and internal
verification are disclosed in the manuscript. Preparation of these files
is distinct from public publication or acceptance.
