Independent expert review guide
The most valuable review is not a rerun of the examples. It is an attempt to break one of the four structural transitions below.
1. Direct nodal inertia
Read Section 3. Starting with a fully supported tree eigenfunction, verify that the finite block obtained by imposing Dirichlet conditions at all interior zeros has exactly s negative and r-s null directions. Pay particular attention to a zero at a vertex of degree greater than two. An arbitrary eigenfunction with a dead edge is outside this lemma.
2. Leaf-split evaluation functionals
Read Section 6. Confirm that parity gauges transform the saturated-tree Kirchhoff amplitudes into a leaf-flow space, and that evaluation on an essential edge is a scalar multiple of the associated leaf-split sum. Check the statement about a singleton split on an essential tree. Then check that equality of evaluations really forces distance 2rd.
3. Exceptional openings and spectral index
Read Sections 7–8. Verify the three orderings of the two interval reclosures, including common parity for the theta and even phases for the figure-eight. Check that reconstructed multiplicity is exact and that upper/lower bound comparison fixes the target index rather than merely locating an eigenfrequency.
4. Quantitative inheritance
Read Section 10. Check the dimension and injectivity argument for V intersect F_- orthogonal complement, the energy estimate for the high-mode correction, and normalization through the full tree-path Gram matrix. Replacing the largest residue eigenvalue by a trace or using diagonal path lengths alone would be a different and generally unjustified estimate.
Priority questions
Serio already establishes the two-sided-sharp/maximal-degeneracy equivalence and the known topology families. A useful literature report should ask specifically whether lower-sharp-only equality in the high branch has already been classified, whether the mixed-tree metric rules are elsewhere explicit, and whether the particular quantitative constrained-form inequality already appears under another notation.
Reporting a counterexample
Specify every edge, its positive length, D/N labels at leaves, the index k, and the eigenvalue-count convention. A matching resonance without the correct top index is not a counterexample. The JSON graph format in data/exact_cases.json is a convenient template. For a proof objection, identify the first unsupported implication and the exact hypothesis under which it fails.