Sharp all-times heat-content comparison and concentration on compact metric graphs
Scott Schubr -- prepublication edition 1.0.0, 20 October 2026

CONTENTS
Heat_Content_Metric_Graphs.pdf: complete article, including supporting proofs and references.
main.tex and seven included TeX files: complete source; keep them in one directory.
CITATION.cff: citation metadata. SHA256SUMS.txt: integrity hashes for all other files.
No external figures, code, numerical datasets, or computational proof certificates are needed.

REBUILD
Use a current TeX distribution with geometry, amsmath, amssymb, amsthm, mathtools,
booktabs, array, fontenc, lmodern, microtype and hyperref.
Run pdflatex main.tex twice, or run tectonic -X compile main.tex.
The resulting main.pdf has the same article as Heat_Content_Metric_Graphs.pdf;
PDF metadata and byte hashes may differ between compilers and build times.
Tectonic may download its ordinary TeX support files on the first build.

MATHEMATICAL CONVENTIONS
Finite connected unweighted metric graph; nonempty Dirichlet vertex set;
standard Kirchhoff conditions elsewhere. Diffusion generator d2/dx2, no factor1/2.
Physical interval: Neumann at0, Dirichlet atL. Mass operator: Dirichlet at0, Neumann atL.
The comparison is integrated concentration, not pointwise rearranged temperature.
Single-positive-time equality is classified for initial datum one only.
Scope and remaining literature questions are stated in Section6 of the article.
