Concavity of Lattice Counts on Toric Threefolds
Scott Schubr — October 24, 2026

Contents
  Paper.pdf: complete mathematical manuscript (in the final package).
  main.tex: editable source and bibliography.
  fan_data.json: eighteen explicit integer fan arrays and source-record links.
  verify_fano.py: exact fan, nef-cone, Brion and coefficient certificates.
  verify_bundle.py: exact toric and uniform weighted-prefix identities.
  README.txt: scope, provenance and reproduction instructions.
  CITATION.bib: citation metadata.
  SHA256SUMS.txt: checksums (in the final package).

Fano counts
The ordinary Riemann-Roch polynomial of every smooth toric Fano
threefold has concave fourth root on its entire closed nef cone. The
exponent 1/4 is uniformly optimal for these ordinary real polynomials.
Nine of the eighteen types have concave cube root; the other nine
actually fail cube-root concavity. Individual optimal exponents in those
nine cases and sharp lattice-only exponents are not determined.

Split bundles and sharp geometric bias
For P(O+O(a)) over P2, the ordinary real polynomial has optimal power
1/3 for integer twists a=0,1,2 and no positive concave power for a>=3.
Nevertheless the slice-weighted counts
  Z_(a,r)(k,y)=sum_(i=0)^k r^i binomial(y+ai+2,2)
have a nonlocal concave logarithmic extension for EVERY r>=1 and a>=0,
touching every integer k row at every real y>=0. Finite and countable
first-moment laws satisfy logarithmic Jensen at integral first-coordinate
means. For actual section multiplicities the twist and heights are integral.
For EACH fixed 0<r<1, integer-height midpoint Jensen fails at k=1 for
sufficiently large integer twists. Thus r>=1 is the exact universal
positive-bias class on the stated r>0 domain.

A broader quadratic theorem gives the same extension for prefixes of
Az^2+Bz+C when A,C>0, B>=0 and B^2>=2AC. This input criterion is sharp
for continuous-row contact, and allows some complex-root quadratics.
Affine factors Bz+C, B>=0,C>0, work for every positive bias. Exponential
times such quadratics work at every nonnegative exponential rate; affine
factors work at all real rates. The proof uses uniform upper-tail moment
identities, not a finite bound on Taylor or polynomial degree.

No positive power works uniformly over twists at bias r=1; every fixed
r>1 fails every positive power at sufficiently large integer prefixes.
This does not classify powers for a fixed unweighted twist. Ordinary
polynomial failure, integer midpoint failure and alternative logarithmic
extensions are distinguished explicitly.

Classical inputs and provenance
fan_data.json contains primitive normal arrays and maximal triples from
https://fanography.info/toric/ , preserving linked Graded Ring Database
URLs. These are secondary arrays; direct GRDB access was unavailable.
The verifier validates smooth Fano fans and all 153 pairwise lattice
inequivalences; the classical Watanabe-Watanabe/Batyrev count of eighteen
then proves coverage independently of secondary labels.
Classical Lorentzian theory, Brion, mixed Ehrhart polynomiality and toric
geometry are credited. Speyer's weighted marginal theorem already covers
unit-step integral-root/integer-height cases at every bias; the broader
real-step/real-row and all-twist bias results require the stated proof.
The Abel-transfer mechanism is classical and its scope is compared with
Johnson-Goldschmidt. Related earlier unimodular work is cited, without
counting its theorems as independent new results of this paper.

Build
Use `tectonic main.tex`, or run `pdflatex main.tex` twice in a standard
LaTeX installation. No external figures or bibliography files are needed.

Exact verification (Python 3.9+, standard library only)
  python -B verify_fano.py
  python -O -B verify_fano.py
  python -B verify_bundle.py
  python -O -B verify_bundle.py
Each prints one JSON object, writes no files and retains explicit checks
under optimization. Compare each normal/optimized stdout pair byte for
byte. No network is used. The Fano checker reads only adjacent fan_data.json;
the bundle checker reads no files.

The Fano checker reconstructs complete count polynomials with two generic
Brion objectives, validates all cones, positive spanning, complete nef
rays, exact inertia and coefficient bounds, and replays 152 direct counts
including boundary cases. The bundle checker retains exact toric witnesses
and verifies the full closed Bernstein identity for all lower tails, the
positive upper-tail expansion, the general quadratic product/row identities,
and the exact geometric-bias failure polynomial. The manuscript supplies
the all-parameter Abel, middle-strip, Legendre, countable-support and
asymptotic arguments. Bounded numerical tests are not theorem premises.
AI assistance is disclosed; internal checks are not external human
refereeing or formal proof-assistant verification.
