{"paper":{"title":"Distinguishing finite metric spaces via similarity spectra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The q-spectrum from similarity matrices distinguishes all finite metric spaces on at most four points and a large class of larger ones.","cross_cats":[],"primary_cat":"math.MG","authors_text":"Jun O'Hara","submitted_at":"2025-02-13T05:33:41Z","abstract_excerpt":"We study spectra and characteristic polynomials of similarity matrices associated with finite metric spaces, where the similarity matrix of a finite metric space $X=\\{x_1,\\dots,x_n\\}$ is given by $\\displaystyle Z(q)=(q^{d(x_i,x_j)})_{i,j},$ where $d(x_i,x_j)$ denotes the distance between $x_i$ and $x_j$. %\nWe introduce two spectral invariants of finite metric spaces, the $q$-spectrum and the normalized $q$-spectrum, defined respectively from $Z(q)$ and its normalized transition matrix. In the case of graphs, these invariants recover the adjacency spectrum and the Laplacian spectrum in the limi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The q-spectrum completely distinguishes a large class of finite metric spaces and all metric spaces on at most 4 points. The transition q-spectrum distinguishes spaces for which the multiset of pairwise distances is independent over the rational numbers, along with all spaces on at most 3 points.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The similarity matrices are constructed so that their spectra encode enough metric information to separate the claimed classes; the paper invokes this construction without an independent verification that the matrix definition is the minimal or canonical choice that preserves distinction power.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Introduces q-spectrum and transition q-spectrum invariants for finite metric spaces that recover graph spectra in a limit and distinguish all spaces with at most 4 and 3 points respectively under stated conditions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The q-spectrum from similarity matrices distinguishes all finite metric spaces on at most four points and a large class of larger ones.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c2a42ebe87aa437b6ec22439d561cda6eaad7705973f11843387626d9f5ba846"},"source":{"id":"2502.08980","kind":"arxiv","version":6},"verdict":{"id":"6ebc7451-65bc-4f26-91d0-d3cd70b94fab","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-23T03:55:00.584443Z","strongest_claim":"The q-spectrum completely distinguishes a large class of finite metric spaces and all metric spaces on at most 4 points. The transition q-spectrum distinguishes spaces for which the multiset of pairwise distances is independent over the rational numbers, along with all spaces on at most 3 points.","one_line_summary":"Introduces q-spectrum and transition q-spectrum invariants for finite metric spaces that recover graph spectra in a limit and distinguish all spaces with at most 4 and 3 points respectively under stated conditions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The similarity matrices are constructed so that their spectra encode enough metric information to separate the claimed classes; the paper invokes this construction without an independent verification that the matrix definition is the minimal or canonical choice that preserves distinction power.","pith_extraction_headline":"The q-spectrum from similarity matrices distinguishes all finite metric spaces on at most four points and a large class of larger ones."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.08980/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}