# Support augmentation verification report

The manuscript contains complete arguments for the explicitly stated finite
Cartesian augmentation problem. Its path formula follows from a proved
compatibility order and Dilworth; its fixed-graph hardness follows from a
polynomial reduction from partial Latin completion. The self-audit separates
the theorem scope from the general AIM marginal-cone question, which remains
unresolved here.

## Mathematical checks

Integer clique tables glue into a full-row multiset. Only at an optimum is
that multiset shown to have no old row or duplicate. Path compatibility is
transitive by running intersection. The flow network retains parallel arcs,
empty separators and the zero-demand boundary. The hardness construction
realizes its missing projections by an explicit observed support; it does
not impose structural zeros or hide an exponential unary expansion.

The 95-row branching example has three local optima five and global optimum
six. The general tree dynamic program retains separator multiplicities and
has a polynomial guarantee for fixed clique configuration count, not merely
fixed treewidth.

## Reproducibility and presentation

The exact checker covers 735 path instances and 85 partial-Latin instances,
with 51 completable and 34 noncompletable squares. The separate checker
compares matching, integer circulation and exact set cover on all 65,535
nonempty supports of a binary four-variable path, plus 81 boundary cases.
Normal and optimized outputs agree; the package build additionally requires
the extracted ZIP to reproduce all four reference outputs in isolated Python.
Raw outputs and hashes accompany the source. These bounded checks do not
replace the written general proofs.

The eight-page PDF compiled successfully with the built-in compiler and with
cached Tectonic resources. All eight rendered pages were visually inspected;
there are no clipped formulas, overlapping elements or missing glyphs. The
author name is alone at the top; affiliation and contact appear in a footnote.
AI assistance is disclosed in ordinary prose.

The prior-art audit credits the statistical existence criterion, integer bag
gluing, Dilworth, Latin-completion hardness, two-relation cover constructions
and earlier local-cover failures. This is originating-researcher self-audit,
not independent peer review, formal certification or proof of priority.