# Verification report

The general all-tree proof factors the coefficient map through free
diagonal/edge-product coordinates, child polynomial gluing and a
leafwise rate-coordinate map. Two identical Sylvester coefficient
blocks produce the squared sibling resultants. Zero-rate
specializations prove the exact individual variable multiplicities.
The proof covers every finite rooted tree, not only tested sizes.

The final rational checker passes 1,973 explicit conditions in both
normal and optimized Python, with identical reports. It compares
Faddeev-LeVerrier coefficient derivatives to an independent tree
recurrence and Sylvester resultants, includes 154 increasing-parent
tuples through six vertices, paths/stars through ten, zero and signed
rates, and nested and quadratic-branch collisions. Coordinate signs
are fixed across nonsingular specializations. Separate classical SEM
checks are included and are not claimed as new.

All six final PDF pages were individually inspected. Equations,
footnotes, source qualifications, references and page transitions are
legible; no clipping or overlap was observed. The built-in compiler
reports success, and the exported PDF log has no errors, unresolved
references, missing glyphs, overfull or underfull boxes. The package
manifest binds the source, PDF, portable code and expected test report.

The theorem is accepted by originating-model detailed self-audit.
Neither finite testing nor PDF compilation is formal mathematical
certification. No independent review or absolute priority is claimed.
The complete tree and catenary conjectures are distinguished from the
unresolved general SEM covariance question in the selected corpus record.