Sharp all-times heat-content comparison and concentration on compact metric graphs
Scott Schubr, Luminary Oddities. October 20, 2026. Archive edition 2.

CONTENTS
Heat_Content_Metric_Graphs.pdf is the complete article. main.tex and seven
included TeX files are its editable source; retain them in one directory.
CITATION.cff and CITATION.bib identify the canonical Archive edition.
SHA256SUMS.txt lists every other distributed member, excluding itself.

REBUILD
Use a local Tectonic installation and the Python 3 standard library:
  python build_pdf.py --tectonic /path/to/tectonic --output-dir Build
The output directory must not already exist. The helper fixes the
source-date epoch and reports compiler failures. Tectonic may fetch
standard TeX dependencies on first use; the bibliography is embedded.
The output is Build/main.pdf. The supplied package was reproduced with
the documented build. The mathematical argument is analytic and includes
all proof dependencies explained in the article; no numerical verifier
or external data are needed.

MATHEMATICAL CONVENTIONS
Finite compact connected unweighted graph, nonempty Dirichlet vertex set,
standard Kirchhoff conditions elsewhere. The Dirichlet-cut interior may be
disconnected. Generator d2/dx2. Physical interval: Neumann at 0, Dirichlet at L.
Integrated mass operator: Dirichlet at 0, Neumann at L. The comparison orders
integrated rearranged mass for arbitrary nonnegative L^2 data. Single-time
equality is classified for initial datum one. Defect bounds use the actual
heat solution and its level slopes. Section 6 summarizes scope and consequences.
