Sharp all-times heat-content comparison and concentration on compact metric graphs
Scott Schubr, Luminary Oddities. October 20, 2026. Archive edition 3.

CONTENTS
main.pdf is the complete article. main.tex and eight
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 complete source is compiled during preparation; PDF byte identity
can depend on compiler and TeX resource versions. 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. The sharper comparison assumes every bridge has Dirichlet vertices on both
sides and uses the ND interval at four times the observation time, or the
DD interval with folded initial data. For datum one, single-time equality
is equivalent to the known L^3/12 torsion extremality. Balanced paired-edge
chains are sharp examples. The final section summarizes scope and consequences.
