# Proof Audit **Project:** Equality Rigidity in the Polya Bound for Compact Dirichlet Metric Trees **Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki **Release:** v1.0.0 - 24 September 2026 ## Audit verdict The proof is internally complete at the stated hypotheses. No unresolved implication was identified after adversarial checking. Independent expert verification remains necessary because the result addresses a problem explicitly listed as open in the current 2026 source paper. ## Dependency chain The main theorem depends on the following established inputs only: 1. The known lower bound `lambda_k >= pi^2 k^2/L^2` for compact Dirichlet trees. 2. The first-eigenvalue diameter bound `lambda_1(T) >= pi^2/diam(T)^2` for a Dirichlet tree. 3. Edge-length continuity of metric-graph eigenvalues. 4. Arbitrarily small generic edge-length perturbations for which the kth eigenfunction has exactly k nodal domains and no zero at original vertices, as used in Harrell-Kennedy-Ramos. 5. The standard nodal-domain fact that the restriction of an eigenfunction to one of its nodal domains is the positive first Dirichlet eigenfunction of that domain. 6. Standard interval Poincare inequality, sine-basis spectral gap, compact Sobolev embedding in one dimension, and min-max. No theorem from the author's other research programs is used as an external assumption. ## Critical transition A: defect squeeze For generic approximants `Gamma_n`, each nodal tree `T_nj` satisfies ```text L_nj >= D_nj >= d_n := pi/sqrt(lambda_k(Gamma_n)). ``` Summing gives ```text 0 <= sum_j(L_nj-d_n) = L_n-k d_n -> 0. ``` Since every term is nonnegative, each term tends to zero. Therefore ```text L_nj -> L/k, D_nj -> L/k, L_nj-D_nj -> 0. ``` **Audit:** exact; no cancellation is used. ## Critical transition B: diameter endpoints are Dirichlet A diameter endpoint of a finite tree is a leaf of that tree. In a generic nodal tree, an original interior vertex at which the eigenfunction is nonzero cannot become a nodal leaf: continuity gives the same sign on all incident edge germs near that vertex, so all such germs belong locally to the same nodal domain. Thus a diameter endpoint is either an original Dirichlet leaf or a nodal zero. **Audit:** valid. ## Critical transition C: mass concentration The normalized ground state `u` satisfies ```text |u(x)|^2 <= L_j * lambda_k ``` by integrating from any point to a Dirichlet leaf. Therefore ```text int_{T_j \ P_j} u^2 <= L_j lambda_k |T_j \ P_j| -> 0. ``` So the diameter contains asymptotically all L2 mass. **Audit:** valid; no assumption of uniform branch number is needed. ## Critical transition D: sine rigidity After rescaling the diameter to `[0,1]` and normalizing its L2 mass, the energy converges to `pi^2`, the first Dirichlet eigenvalue. Expanding in the sine basis gives an explicit gap of `3 pi^2` to the second mode, forcing all higher-mode coefficients to zero. Positivity selects the positive first sine. Strong H1 convergence implies uniform convergence. **Audit:** valid. ## Critical transition E: shrinking branch is not spectrally harmless For a rooted branch of total length `beta` with Dirichlet terminal leaves, every admissible `f` with root value `a` satisfies ```text |a|^2 <= beta int_B |f'|^2, int_B |f|^2 <= beta^2 int_B |f'|^2. ``` Therefore ```text int_B |f'|^2 - lambda int_B |f|^2 >= (1/beta - lambda beta)|a|^2. ``` Hence the one-port energy impedance diverges as `beta -> 0` for bounded lambda. In particular an eigenfunction with bounded total energy satisfies `u(root)->0`. **Audit:** valid. This closes the principal loophole of simply discarding a vanishing branch. ## Critical transition F: branch exclusion If an essential branch vertex remains at normalized coordinate `t in (0,1)` on a saturated diameter, the shrinking off-diameter branch forces the eigenfunction value to zero, while uniform sine convergence forces it to `sqrt(2) sin(pi t)>0` after normalization. **Audit:** contradiction is strict and stable. ## Critical transition G: limiting paths tile the graph There are finitely many combinatorial routes in a fixed finite tree. Pass to subsequences with fixed routes and convergent edge coordinates. A positive-length overlap of two limiting paths would force positive-length overlap of approximating paths for large n, impossible for different nodal domains. The k limits each have length L/k, so their union has length L. A finite union of compact arcs is closed. A nonempty complement would be open and contain a positive-length edge interval, contradiction. **Audit:** valid. ## Critical transition H: arithmetic edge conclusion No limiting cell may contain an essential branch vertex in its interior. With degree-two dummy vertices suppressed, each cell is contained in one essential edge. Since the cells tile the graph, every edge is a concatenation of an integer number of length-L/k cells. **Audit:** valid. ## Critical transition I: converse trial space Subdivide every commensurate edge into length-L/k cells. On each cell take the first Dirichlet sine, extended by zero. These functions belong to the quadratic-form domain even though their derivatives do not satisfy Kirchhoff matching at inserted subdivision points. Their supports are disjoint in measure, so every nonzero linear combination has the same Rayleigh quotient. Min-max gives the desired upper bound, which combines with the known lower bound. **Audit:** valid. ## Counterexamples checked against false shortcuts ### Equilateral stars An equilateral m-star saturates the first-eigenvalue diameter bound but is not an interval. This disproves the shortcut ```text diameter-bound equality => interval. ``` The actual proof uses the stronger condition `length-diameter -> 0`. ### Multiple target eigenvalues Equality metrics can be spectrally degenerate. The proof never chooses a preferred kth eigenfunction on the equality graph; it works on generic approximants and passes to the limit. ### Edge collapse The proof approximates a fixed proper equality metric by positive edge lengths and does not need to contract an essential edge of the target graph. Standard edge-length continuity is sufficient. ### Positive-measure overlap in the limit Endpoint-order convergence on each fixed edge excludes it. Only common endpoints can arise. ## Stronger results audited - Equality metrics on a labeled m-edge topology correspond exactly to positive integer compositions of k: count `C(k-1,m-1)`. - A fixed tree has an equality index iff all normalized edge lengths are rational. - If equality occurs, all equality indices are exactly multiples of the lcm of those rational denominators. - Equality at two coprime indices forces one essential edge, hence an interval. - The first possible equality index on a topology equals the number of essential edges; equality there is exactly the equilateral metric. All are elementary consequences of the main theorem. ## Claims intentionally not made - No universal optimal quantitative power law between spectral defect and distance to the commensurate set. - No extension to cyclic graphs or mixed Neumann/Dirichlet leaves. - No claim that the result has passed peer review. - No claim that no equivalent proof exists in unpublished or unindexed work.