{"schema":"pubphys.bundle/1","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 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