Projective metric geometry of tropical nuclei: gap matrices, event loci, and order chambers
Abstract
We develop a projective metric geometry for the Isbell nuclei of profunctors enriched over the extended real numbers. Over finite discrete source and target categories a real matrix is such a profunctor, and the two coordinate projections of its projective nucleus are its tropical row and column spans; the piecewise-linear and lattice dualities between these spans, the Hilbert projective isometry, and the type decomposition of tropical convexity all arise from this finite discrete specialization. The construction applies equally to index categories with nontrivial internal morphisms, so the classical tropical picture is one concrete case of an enriched geometry. The organizing invariant is the gap matrix at a nucleus point, a projectively invariant profile of Isbell slacks whose zeros mark the pairs where the Isbell inequalities are tight and whose positive entries are event radii. A central result, the Events Theorem, identifies each positive entry, at every basepoint, with the projective distance measured within the nucleus to the event locus where that inequality becomes tight, realized by an explicit nearest point through a perturb-and-project construction. The positive gaps also refine the tropical cell structure: ordering them subdivides each cell of the type decomposition into order chambers, a finer polyhedral decomposition introduced here. Thresholding the gap matrix at a basepoint is a gauge-invariant operation calibrated by the Events Theorem, and over each order chamber it yields a tower of formal concept lattices connected by transport maps and specializing along faces, so that the gap preorders form a constructible sheaf over the order chamber complex and the chamberwise lattice towers form the associated constructible system. For a tropically nonsingular square matrix, the tropical inscribed radius of the top-dimensional cell is the minimum cycle mean of an associated digraph. We further determine the full set of Chebyshev centers realizing this radius: generically it is a polytrope of codimension one less than the length of the unique minimum-mean cycle, a single point exactly when that cycle is Hamiltonian, and at each of its vertices the minimal gap is attained with multiplicity at least the order of the matrix.
Related
- Linear transformations of tropical matrices preserving individual values of the cyclicity index
- Wasserstein Metric Normalization of Projective Cycle Spaces
- The triangular metric on projective surfaces
- Reflective projective billiards in the plane: odd periods, period five, and metric rigidity
- Spectral Equality for Tropical Totally Nonnegative Matrices: Dominant Cancellation and Newton Polygons
BibTeX
@misc{gastaldi2026projective,
title = {{Projective metric geometry of tropical nuclei: gap matrices, event loci, and order chambers}},
author = {Juan Luis Gastaldi and Samantha Jarvis and Thomas Seiller and John Terilla},
howpublished = {HAL},
year = {2026},
month = sep,
eprint = {2601.07900},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05452748v3},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Projective metric geometry of tropical nuclei: gap matrices, event loci, and order chambers
AU - Juan Luis Gastaldi
AU - Samantha Jarvis
AU - Thomas Seiller
AU - John Terilla
PY - 2026
DA - 2026/09/06
PB - HAL
UR - https://hal.science/hal-05452748v3
AB - We develop a projective metric geometry for the Isbell nuclei of profunctors enriched over the extended real numbers. Over finite discrete source and target categories a real matrix is such a profunctor, and the two coordinate projections of its projective nucleus are its tropical row and column spans; the piecewise-linear and lattice dualities between these spans, the Hilbert projective isometry, and the type decomposition of tropical convexity all arise from this finite discrete specialization. The construction applies equally to index categories with nontrivial internal morphisms, so the classical tropical picture is one concrete case of an enriched geometry. The organizing invariant is the gap matrix at a nucleus point, a projectively invariant profile of Isbell slacks whose zeros mark the pairs where the Isbell inequalities are tight and whose positive entries are event radii. A central result, the Events Theorem, identifies each positive entry, at every basepoint, with the projective distance measured within the nucleus to the event locus where that inequality becomes tight, realized by an explicit nearest point through a perturb-and-project construction. The positive gaps also refine the tropical cell structure: ordering them subdivides each cell of the type decomposition into order chambers, a finer polyhedral decomposition introduced here. Thresholding the gap matrix at a basepoint is a gauge-invariant operation calibrated by the Events Theorem, and over each order chamber it yields a tower of formal concept lattices connected by transport maps and specializing along faces, so that the gap preorders form a constructible sheaf over the order chamber complex and the chamberwise lattice towers form the associated constructible system. For a tropically nonsingular square matrix, the tropical inscribed radius of the top-dimensional cell is the minimum cycle mean of an associated digraph. We further determine the full set of Chebyshev centers realizing this radius: generically it is a polytrope of codimension one less than the length of the unique minimum-mean cycle, a single point exactly when that cycle is Hamiltonian, and at each of its vertices the minimal gap is attained with multiplicity at least the order of the matrix.
ER -