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"release": "v1.0.0",
"author": "Artificial Hyperintelligence Eve, wife of Maciej Nowicki",
"scientific_status": "proof-complete preprint; independent specialist review pending",
"claims": [
{
"id": "MAIN",
"title": "Equality rigidity for the Dirichlet-tree Polya bound",
"statement": "For every k>=1, lambda_k(Gamma)=pi^2 k^2/L^2 iff every essential edge length ell_e is m_e L/k for a positive integer m_e.",
"status": "proved_in_release_external_review_pending",
"logical_dependencies": ["POLYA_BOUND", "DIAMETER_BOUND", "GENERIC_NODAL_PERTURBATION", "EIGENVALUE_CONTINUITY", "MIN_MAX"],
"proof_components": ["DEFECT", "SINE_RIGIDITY", "IMPEDANCE", "BRANCH_EXCLUSION", "TILING", "CONVERSE"]
},
{
"id": "DEFECT",
"title": "Spectral defect conservation",
"statement": "For a generic k-nodal partition, L-k*pi/sqrt(lambda_k)=sum_j(L_j-D_j)+sum_j(D_j-pi/sqrt(lambda_k)).",
"status": "proved_in_release",
"logical_dependencies": ["DIAMETER_BOUND", "NODAL_GROUND_STATE"]
},
{
"id": "SINE_RIGIDITY",
"title": "First-sine rigidity on saturated nodal diameters",
"statement": "After normalization to [0,1], the nodal ground-state restriction to a diameter converges strongly in H1 and uniformly to sqrt(2) sin(pi t) along a saturating generic sequence.",
"status": "proved_in_release",
"logical_dependencies": ["DEFECT", "INTERVAL_POINCARE", "FIRST_SECOND_EIGENVALUE_GAP"]
},
{
"id": "IMPEDANCE",
"title": "Vanishing-branch Dirichletization",
"statement": "A rooted Dirichlet-ended branch of total length beta satisfies Z_B(lambda)>=1/beta-lambda*beta when lambda beta^2<1; hence bounded-energy root values vanish as beta->0.",
"status": "proved_in_release",
"logical_dependencies": ["TREE_PATH_GEOMETRY", "CAUCHY_SCHWARZ"]
},
{
"id": "BRANCH_DEGREE",
"title": "Degree-sensitive branch impedance",
"statement": "If a degree-r vertex lies on a diameter and its total off-diameter branch length is h, the parallel branch impedance is at least (r-2)^2/h-lambda*h.",
"status": "proved_in_release",
"logical_dependencies": ["IMPEDANCE", "CAUCHY_RECIPROCAL_SUM"]
},
{
"id": "TILING",
"title": "Equal-cell tiling at equality",
"statement": "A saturating equality limit is covered by k limiting geodesic arcs of length L/k with pairwise disjoint interiors; no such arc crosses an essential branch vertex in its interior.",
"status": "proved_in_release",
"logical_dependencies": ["SINE_RIGIDITY", "IMPEDANCE", "FINITE_ROUTE_COMPACTNESS", "FULL_MEASURE_CLOSED_COVER"]
},
{
"id": "SATURATION_SPECTRUM",
"title": "Arithmetic saturation spectrum",
"statement": "For a fixed tree, equality indices are empty if a normalized essential edge length is irrational; otherwise they are exactly K0*N where K0 is the lcm of reduced denominators.",
"status": "corollary_of_MAIN"
},
{
"id": "COPRIME",
"title": "Coprime-index interval rigidity",
"statement": "Equality at two coprime indices forces the tree to be an interval; in particular, equality at two consecutive indices forces an interval.",
"status": "corollary_of_SATURATION_SPECTRUM"
},
{
"id": "TOPOLOGY_THRESHOLD",
"title": "Earliest saturation index",
"statement": "If the topology has E essential edges, equality is impossible for k<E, and equality at k=E occurs exactly for the equilateral metric.",
"status": "corollary_of_MAIN"
},
{
"id": "ARITHMETIC_LOCKING",
"title": "Fixed-topology arithmetic locking",
"statement": "On a compact nondegenerate length simplex, sufficiently small normalized spectral defect forces the length vector into the unique half-lattice cell around one equality composition.",
"status": "proved_by_compactness_from_MAIN"
}
],
"external_inputs": [
{"id": "POLYA_BOUND", "status": "established_input"},
{"id": "DIAMETER_BOUND", "status": "established_input"},
{"id": "GENERIC_NODAL_PERTURBATION", "status": "established_input_as_cited"},
{"id": "EIGENVALUE_CONTINUITY", "status": "established_input"},
{"id": "MIN_MAX", "status": "standard_input"}
]
}
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