Power Concavity of Low-Dimensional Unimodular Fiber Counts
Scott Schubr, Luminary Oddities. October 22, 2026.

The main theorem proves corrected fourth-root Jensen with correction 1/12 for a fixed unimodular system, an integrable integer demand with integral mean, feasible fibers almost surely, and a bounded nonempty central fiber of affine dimension at most three. It implies uniform 2/11-power and logarithmic Jensen, including countable support. If endpoint counts are at least m, the power improves to (3 m-1)/(12 m-1).

A quartic Lorentzian certificate, shortest-path closure and one common integral rank-three reduction prove the theorem. For the specified continuous all-directed-bound type-A3 count extension,2/11 is the largest concavity power and 1/12 is the least corrected-fourth-root shift. Those optimality claims concern that real extension. The paper separately proves sharp corrected planar square-root Jensen, the leading 1/(8 m) exponent deficit, continuous K 4 power 4/13 and exact integral cube-root obstructions. Directed opposite-edge rounding and a triangle local obstruction illustrate the global method.

CONTENTS
Paper.pdf: full manuscript.
main.tex: full editable source and bibliography.
build.py: deterministic Tectonic helper.
verify_A3_structure.py: exact chamber, Brion polynomial and quartic identities.
verify_rank_three.py: exhaustive 1024-mask TU normal classification.
verify_counterexample.py: exact integer K 4 cube-root witnesses.
verify_rounding.py: finite rounding diagnostics and triangle witnesses.
CITATION.bib: citation metadata.
SHA 256 SUMS.txt: hashes of all other payload members.

BUILD
Python 3.10+ and Tectonic on PATH:
  python build.py --output-dir ../FiberBuild
The output directory must be new. Use --tectonic for an explicit compiler path or --only-cached for existing resources. The helper fixes SOURCE_DATE_EPOCH. With the same compiler/dependencies it reproduces the PDF bytes. A conventional LaTeX installation can also compile main.tex using its stated packages; no external bibliography, figures or datasets are needed.

EXACT AUXILIARY CHECKS
Python 3.9+, standard library:
  python -B verify_A3_structure.py
  python -B verify_rank_three.py
  python -B verify_counterexample.py
  python -B verify_rounding.py
All print JSON, write no files and keep checks active under -O. The structure checker reconstructs classical chamber polynomials from six credited seeds, checks all 62 sign patterns and 1092 actual quartic derivative identities using two Brion objectives. Its obstruction computation identifies the sharp real-extension constants. The rank-three checker constructs integral maps for every valid normal system. These exact finite certificates support the specified identities; the manuscript supplies the analytic Lorentzian, gluing, closure, lattice-transfer and countable-support arguments. The rounding script reports its finite diagnostic scope.

Classical chamber enumeration, Brion and mixed-Ehrhart theory, Lorentzian volume theory, L-convex marginals and discrete four-functions theorems are attributed in the paper.
