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dirichlet-tree-polya-equality-rigidity

ID	Result	Status	Dependencies
K1	`lambda_k >= pi^2 k^2/L^2` for compact Dirichlet trees	Established literature	Berkolaiko-Kennedy-Kurasov-Mugnolo; noted by Harrell-Kennedy-Ramos
K2	`lambda_1(T) >= pi^2/diam(T)^2` for a Dirichlet tree	Established literature	Standard metric-tree spectral geometry
N1	Exact defect law `L-kd = sum(L_j-D_j)+sum(D_j-d)`	Proved here	K2 + nodal domains
N2	Mass on each saturated nodal tree concentrates on a diameter	Proved here	N1 + root-to-leaf Cauchy-Schwarz
N3	Rescaled diameter state converges to `sqrt(2) sin(pi t)`	Proved here	N2 + interval spectral gap
N4	Rooted branch impedance `Z_B(lambda) >= 1/beta-lambda beta`	Proved here	Elementary 1D path estimates
N5	Essential branch vertex cannot remain inside a saturated limiting cell	Proved here	N3 + N4
N6	k limiting cells of length L/k tile the whole tree	Proved here	Finite-route compactness + N5
MAIN	Equality iff every essential edge length is in `(L/k) N`	Proved here; external review pending	K1-K2, generic perturbation, N1-N6, min-max
C1	Labeled m-edge equality metrics are positive compositions of k; number `binom(k-1,m-1)`	Consequence	MAIN
C2	Equality indices of a fixed tree are empty or `K0 N`	Consequence	MAIN + elementary arithmetic
C3	Equality at two coprime indices forces an interval	Consequence	C2
C4	Earliest possible equality index is number of essential edges; equality there iff equilateral	Consequence	MAIN
S1	Generic near-equality controls total branching and axial defects explicitly	Proved here	N1
S2	On a fixed nondegenerate topology, spectral defect -> 0 iff distance to finite equality set -> 0	Proved here by compactness	MAIN + eigenvalue continuity
S3	Below a positive topology-dependent threshold, nearest arithmetic composition is unique and recovered by rounding	Proved here by compactness/lattice separation	S2

Status vocabulary

Established literature: source exists before this release.
Proved here: a complete proof is included in the manuscript.
Consequence: follows immediately from the main theorem and elementary arithmetic/combinatorics.
External review pending: internal proof audit completed, but independent specialist verification has not yet occurred.