Log-concave polynomial prefix sums: algebraic certificates and a sharp zero sector
Scott Schubr, Luminary Oddities
Manuscript date: October 24, 2026.

Main result
Finite algebraic certificates also suffice: nonnegative log-concave factorial
coefficients with interval support and a nonnegative Q(UV,U+V) expansion
of h(U)h(V), with h(0)>0. This larger class is closed under products,
positive affine changes and nonzero derivatives. The explicit cubic
(t+1)(10t^2+20t+21) qualifies despite roots outside the sector below.

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. 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 concerns the stated concave logarithmic extension from integer prefix lengths.
The split-projective-bundle application holds in all dimensions. 

Public files
main.tex          Complete editable LaTeX manuscript, including bibliography.
Paper.pdf         Compiled manuscript, when supplied in the prepared package.
verify_algebraic.py  Exact cubic and derivative-certificate controls.
algebraic_results.json  Deterministic algebraic-control output.
verify_prefix.py  Self-contained exact certificate checker, Python standard library.
README.txt        This document.
CITATION.bib      Manuscript citation metadata.
CITATION.cff      Structured manuscript citation.
build_pdf.py     Portable Tectonic build helper.
verification_results.json  Actual deterministic checker output.
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
  python -B verify_algebraic.py
  python -O -B verify_algebraic.py
Each checker must succeed with status PASS; its normal and optimized
outputs must match each other. The two programs have different output schemas. 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 published companion Concavity of Lattice Counts on
Toric Threefolds. The companion paper supplies the cited Fano and threefold results. Classical tools and the substantial
Speyer lattice overlap are explicitly credited.  AI assistance is disclosed in the manuscript. 


Building the manuscript
-----------------------
The portable build helper accepts a local Tectonic executable:
  python build_pdf.py --tectonic /path/to/tectonic --output-dir Build
The output directory must not already exist. The helper uses a fixed
source-date epoch. Tectonic may fetch standard TeX dependencies on first use.
The bibliography is embedded in main.tex. CITATION.cff and CITATION.bib
identify this manuscript. The preceding published edition is available at
https://research.luminaryoddities.com/a-sharp-zero-sector-criterion-for-polynomial-prefix-sums/v/2
under its preceding title. verification_results.json records
the deterministic exact checker output. SHA256SUMS.txt lists every other
public package member and excludes its own checksum.
