{"paper":{"title":"Sign-balance of random Laplace eigenfunctions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Random Laplace eigenfunctions are sign-balanced above a precisely determined scale with almost full probability.","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Igor Wigman, Stephen Muirhead","submitted_at":"2026-04-24T13:58:38Z","abstract_excerpt":"Motivated by the problem of the small-scale sign distribution of Laplace eigenfunctions, we introduce a strong notion of sign-balance for (eigen)functions, and prove that random eigenfunctions are sign-balanced above a precisely determined scale with almost full probability. The scale is proven to be optimal up to a logarithmic power of the energy. Our results include the important case of random spherical harmonics, as well as more general band-limited random waves on smooth Riemannian manifolds. Extending the notion of balance to arbitrary levels, we determine the precise optimum scale above"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"random eigenfunctions are sign-balanced above a precisely determined scale with almost full probability. The scale is proven to be optimal up to a logarithmic power of the energy.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The random eigenfunctions are modeled as centered Gaussian fields with covariance given by the spectral projector onto a narrow energy window; the strong notion of sign-balance is the appropriate one for capturing small-scale distribution.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Random eigenfunctions of the Laplace operator are sign-balanced above a precisely determined scale (optimal up to log factors of the energy) with almost full probability.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Random Laplace eigenfunctions are sign-balanced above a precisely determined scale with almost full probability.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b87a8f9f73cd7c3bdc4e3a75834cb7fd751e558e5ffa986a46861b140295cd3b"},"source":{"id":"2604.22567","kind":"arxiv","version":2},"verdict":{"id":"f79b6847-dc7c-4f75-969c-7852754e53a2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T10:07:43.862509Z","strongest_claim":"random eigenfunctions are sign-balanced above a precisely determined scale with almost full probability. The scale is proven to be optimal up to a logarithmic power of the energy.","one_line_summary":"Random eigenfunctions of the Laplace operator are sign-balanced above a precisely determined scale (optimal up to log factors of the energy) with almost full probability.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The random eigenfunctions are modeled as centered Gaussian fields with covariance given by the spectral projector onto a narrow energy window; the strong notion of sign-balance is the appropriate one for capturing small-scale distribution.","pith_extraction_headline":"Random Laplace eigenfunctions are sign-balanced above a precisely determined scale with almost full probability."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.22567/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T10:37:21.246229Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:53:22.460947Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"dbbc960733117bb963bfb547a01578bf15886edf1a0e8d03de98460b5133596c"},"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"}