Adversarial proof audit
Audit result
No unresolved internal gap was identified in the final stated arguments during this development. That is an internal audit conclusion, not an independent certification. The manuscript remains a proof-complete candidate. This audit does not use the numerical tests to replace any general argument.
The actual-metric tree step
The predecessor's mixed-tree necessity proof used a generic perturbation, a nodal-cell limiting process, and a collapsing-branch impedance step. This release does not assume that transfer. It first selects a fully supported eigenfunction at the actual extremal metric and proves an exact inertia identity there.
Each interior zero is one scalar trace condition. Splitting at all zeros gives r positive nodal ground states and s trace extensions. Their form matrix has blocks [0,B;B^T,H]. The cell-zero incidence graph is a tree. Every zero node has degree at least two, so a supported-kernel forest argument proves that B has full column rank s. The resulting inertia is s negative directions and r-s zero directions. This proves that the number of cells is the top spectral index even in the presence of branching zeros and multiplicities.
Review points: check the codimension s, the coercive positive complement, the nonzero derivative entries, and the full-column-rank forest argument. None requires a generic metric.
The nonnegative cell defect
Every mixed nodal cell must have a Dirichlet boundary point at positive eigenvalue. The all-Dirichlet and mixed length lower bounds are d and d/2. The number of cells containing natural leaves is at most the number of those leaves. The resulting exact identity contains only nonnegative terms. At equality every term vanishes, enforcing interval cells and one terminal half-cell per Neumann leaf.
Review points: Dirichlet leaves are excluded from the interior zero count; Neumann leaves cannot have zero value in a fully supported mode; a branching zero counts once but can create several cells.
Cycle opening
An arbitrary endpoint choice on cotree edges can create too many natural leaves. The proof cuts an incidence on a cycle only at a vertex of current degree at least three. That old vertex retains degree at least two; exactly one new natural leaf is created. New leaves are never recut. Repeating gives precisely beta independent leaf ports, a connected tree, and charge N+beta.
Review points: verify connectivity after cutting a cycle incidence; keep retained marking points even when suppressing degree-two vertices; do not identify a marking with a new leaf from another cut.
Entire-eigenspace inheritance
The largest sharp index is essential. The first k restricted directions have nonpositive shifted form. Their projection onto the unrestrained below-threshold subspace has a kernel at least as large as the entire threshold eigenspace. A strictly positive next gap forces that kernel into the threshold eigenspace. Thus every threshold mode survives.
Review point: preserving one chosen cosine does not suffice on a branched tree.
Leaf-flow rigidity and the two-leaf exception
On a saturated tree all branching values vanish, and sine amplitudes become conserved flows after a parity gauge. Evaluation on an edge separating leaf subset A is a scalar multiple of the sum of its leaf-flow coordinates. For at least three leaves, equality with one leaf coordinate forces A to be that singleton or its complement. This fixes the pendant edge. The remaining cosine equation forces a full even-cell period.
With two leaves, the coordinates are already proportional. That case is not passed through the branched-tree lemma: the opened interval is analyzed separately. Its reclosure orders give exactly barbell, figure-eight, and theta. This exception is a necessary part of the proof, not a numerical afterthought.
Correct spectral index
Mode reconstruction gives exact multiplicity, not just a lower bound on it. The published upper and lower estimates then coincide at the two ends of a block of that multiplicity and fix its top index. The wrong-parity theta (1,1,2) has a resonance at pi squared but ends at index four, not the target fifth index. Its exact certificate tests this distinction.
The finite-part claim is deliberately restricted
The implication R=0 => K=0 is proved only for admissible saturated-tree openings. General diagonal matrix examples show it is false without geometry. The proof derives maximal multiplicity first, then uses the reduced response nullity formula to show that the entire beta-by-beta finite part has zero kernel codimension.
Quantitative theorem
The spectral lift is obtained for every threshold vector by intersecting the first k restricted directions with the orthogonal complement of all lower modes. The estimate separates delta>=g from delta<g. The latter case yields an injective, hence onto, projection onto the threshold space. Constraint control is an energy estimate, not false boundedness of point evaluation in L2. The full path Gram matrix, including negative off-diagonal overlaps, is retained.
For zero ports, rho is defined as zero. The nontrivial matrix formulas require at least one port. The explicit graph gap applies to a saturated opened tree, not to arbitrary near-saturated trees.
Computational and presentation audit
The graph counts are exact rational inertia, not thresholded floating-point spectra. A second exact ODE matrix checks multiplicity. The classifier and the spectral counter are separate implementations. Numerical matrix tests and finite-element approximations are explicitly labeled. The final manuscript corrects empty-threshold counting notation, clarifies the zero-port convention, and removes LaTeX layout overflow. Render inspection is recorded separately in data/document_preflight.json.
Remaining external validation
The decisive independent checks are the nodal block inertia, leaf-split evaluation lemma, exceptional interval analysis, and quantitative lift. Historical review should specifically search for results on one-sided lower sharpness, not only two-sided sharpness. No numerical percentage is assigned to proof correctness or historical priority.