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The interaction is of potential type, arising as the first variation of a quadratic energy, though the stationary system is not treated variationally.\n  Linearizing around the uniform equilibrium yields mode-wise $2\\times 2$ systems with dispersion $\\sigma_\\xi(\\gamma)=\\nu^2(2\\pi|\\xi|)^4+\\gamma(2\\pi|\\xi|)^2\\hat K(\\xi)$. If $\\hat K$ is negative for some mode, a finite threshold \\[ \\gamma_c=\\min_{\\hat K(\\xi)<0}\\frac{\\nu^2(2\\pi|\\xi|)^2}{|\\hat K(\\xi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.20213","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2026-04-24T23:10:34Z","cross_cats_sorted":[],"title_canon_sha256":"6c1633c633ee6b3c91179f16841f6a4e1627785c7d4c2996e1a1432ac4f7c13b","abstract_canon_sha256":"a5dd63c86020031d0fc38a79d544f5adad27b823665418c1bdfec1709f45baae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T00:04:21.331454Z","signature_b64":"3kU6wazYQpCvvx1npPgIfk10dpGcyqoAxpu1PyNhv7hPawQ+NiPGw9joxOISM1WkWEHikZde2iT633luxUjbDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f1feb29118f38588a2905d337243db9ff98faea8162e904abbde9658229431f","last_reissued_at":"2026-05-21T00:04:21.330743Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T00:04:21.330743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Phase Transitions in Turnpike Theory For Mean-Field Games","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Siddharth Karuturi","submitted_at":"2026-04-24T23:10:34Z","abstract_excerpt":"We study a translation-invariant mean-field game on the flat torus with interaction $F(x,m)=\\gamma (K*m)(x)$, where $K$ is smooth, even, and mean-zero. The interaction is of potential type, arising as the first variation of a quadratic energy, though the stationary system is not treated variationally.\n  Linearizing around the uniform equilibrium yields mode-wise $2\\times 2$ systems with dispersion $\\sigma_\\xi(\\gamma)=\\nu^2(2\\pi|\\xi|)^4+\\gamma(2\\pi|\\xi|)^2\\hat K(\\xi)$. 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