| # Independent expert review guide |
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| The most valuable review is not a rerun of the examples. It is an attempt to break one of the four structural transitions below. |
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| ## 1. Direct nodal inertia |
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| 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. |
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| ## 2. Leaf-split evaluation functionals |
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| 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 2*r*d. |
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| ## 3. Exceptional openings and spectral index |
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| 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. |
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| ## 4. Quantitative inheritance |
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| 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. |
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| ## Priority questions |
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| 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. |
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| ## Reporting a counterexample |
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| 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. |
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