| { |
| "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"} |
| ] |
| } |
|
|
