Power Concavity of Low-Dimensional Unimodular Fiber Counts
Scott Schubr — October 22, 2026

Contents
  Paper.pdf: complete mathematical manuscript.
  main.tex: complete editable source and bibliography.
  verify_counterexample.py: exact lattice-witness checker.
  verify_rounding.py: opposite-edge diagnostics and triangle witnesses.
  verify_A3_structure.py: exact full A3 chamber and polynomial certificates.
  verify_rank_three.py: exhaustive rank-three TU direction classification.
  README.txt: reader and build instructions.
  CITATION.bib: citation metadata.
  SHA256SUMS.txt: checksums of every other package member.

The main theorem proves E[N(A)^(1/6)] <= N(EA)^(1/6) for an integrable
integer-valued random demand A in a fixed unimodular system, with feasible
fibers almost surely, integral mean, and bounded nonempty central fiber
of affine dimension at most three. Logarithmic Jensen follows. Countable
support is allowed. The proof constructs a monotone concave extension
for all twelve raw A3 bounds, then supplies a common integral reduction
and a cap-free shortest-path argument. The exponent is not claimed
optimal; unrestricted higher-dimensional unimodular Jensen remains open.

The paper retains corrected planar count concavity, sharp leading
large-count exponents, continuous type-A3 power 4/13 concavity and block
logarithmic Jensen. It proves exact cube-root failures on integral K4
demands, including full-dimensional fibers. The best lattice-only K4 power
lies between 4/13 and the positive root of 2*13^alpha=1+51^alpha, below 1/3;
its exact value remains open.

The opposite-edge theorem allows arbitrary fixed cross-edge intervals
and all four independently varying endpoints on two opposite edges.
Directed midpoint rounding preserves count products. A full-dimensional
triangle example defeats every balanced parameter pair and every
mean-preserving law in the local midpoint unit cube. Its nonintegral mean
makes this a local-method obstruction, consistent with the global theorem.

Build with `tectonic main.tex`, or run `pdflatex main.tex` twice using a
standard LaTeX distribution with the packages named in the source.
No external bibliography, figures or datasets are required.

Optional checks (Python 3.9+, standard library):
  python -B verify_counterexample.py
  python -B verify_rounding.py
  python -B verify_A3_structure.py
  python -B verify_rank_three.py
All print JSON, write no files, and retain checks under `python -O`.
The first two check exact witnesses and finite directed-rounding tests.
The A3 checker reconstructs vertex cones, the required lower-degree
coefficients and cubic directional identities by exact Brion expansion
from six credited classical seeds, and verifies coverage and algebraic
certificates. The
rank-three checker tests all 1024 direction subsets and constructs the
unimodular normal-form maps. These finite exhaustive structural checks
support the specified algebraic identities; the manuscript supplies the
analytic concavity, wall, closure, lattice and countable-support proofs.
No checker proves a universal inequality by sampling.

Classical chamber enumeration, Brion theory, Lorentzian volume theory,
mixed Ehrhart polynomiality, directed rounding and L-convex marginal
theory are credited. AI-assisted internal checks are not external
refereeing or formal proof-assistant verification.
