{"paper":{"title":"Multicriticality and Scaling: Mellin Spectral Theory, and the Decoupling of Geometric and Spectral Exponents","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.GM","authors_text":"Alejandro Frank, Laurence A. Jacobs","submitted_at":"2026-06-01T19:39:08Z","abstract_excerpt":"We develop a spectral theory of scale-invariant operators on the multiplicative half-line $(\\mathbb{R}_+, dx/x)$. A symmetric kernel $M(x, y)$ satisfying $M(kx, ky) = k^{-a}M(x, y)$ necessarily factorizes as $(xy)^{-a/2}F(x/y)$, where the shape function $F$ depends only on the ratio of its arguments. The Mellin transform diagonalizes such operators: the generalized eigenfunctions are $\\psi_\\omega(x) = x^{-a/2+i\\omega}$, and the eigenvalues are the Mellin multiplier $\\tilde{F}(\\omega)$. This structure reveals a fundamental decoupling of two exponents. The geometric exponent $a$, carried by the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07644","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.07644/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"}