{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"b19526a196b8f84aa4fcdf744391122f244df2f897ec6d6ae2e8ae6c9c0ccd19","created":"2026-10-03T07:17:57Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"3ae61010244dbccad8e0353a55aa42ce5cb5e9f746698906169ddfb715114501","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"bio.jamming-rigidity.omega4-modes","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"plain":"Computer glasses contain soft vibration patterns concentrated on a few dozen particles, and their number grows as the fourth power of frequency in every dimension tested. A finite-dimensional theory that derives this law is missing.","posed_since":"2016","precise":"Simulations of quenched glasses show non-phononic quasilocalized modes with density $D_{\\mathrm{loc}}(\\omega) = A_g\\,\\omega^4$ at low $\\omega$, with the same exponent in $d = 3\\ \\text{and}\\ 4$ and, with stronger finite-size effects, in $d = 2$, and with the prefactor $A_g$ decreasing for better-annealed glasses. Derive the $\\omega^4$ law and the dependence of $A_g$ on the preparation (parent) temperature from a controlled microscopic theory in finite d.","problem_ref":null,"references":"","settled_by":"A derivation of $D_{\\mathrm{loc}}(\\omega) \\sim \\omega^{4}$ and of $A_{g}(T_{\\mathrm{parent}})$ in finite d that matches simulation data quantitatively.","status_note":"The $\\omega^{4}$ law is established numerically (2016-2020) and several mean-field derivations exist, without consensus on the finite-d theory (2026).","title":"Why quasilocalized modes in glasses scale as $\\omega$ to the fourth","topic_ref":"e88cb77371bb7a86f0fbdb1f952ba3eab78bc8141039cb805db532ac7cced95b"},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"0d2afc9de390ac8507e02168918d96af763e31c1aeab1ae3dd1f518d15cd0fbb","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"8d082d3c9368279e4f53199484dd8fbc13629094940dce3f659c8f524bbdfc1e","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"dnH5A2p6434VTOAiMZ12oKpo-x3VMhi6WPlsnf5LugV2dcUTea-YD7IkzLD8Afz_COvDjEC4Q83lGKO9tkHxAw"},"schema":"pubphys.envelope/1"},"record_hash":"0d2afc9de390ac8507e02168918d96af763e31c1aeab1ae3dd1f518d15cd0fbb","leaf_index":789}