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Milnor conjectured that the affine algebraic curves $\\mathscr{S}_{k,n}$ are irreducible, for all $k \\geq 0, n>0$. In this article, we show the irreducibility of eventually $2$-periodic curves, i.e. $\\mathscr{S}_{k,2},\\; k\\geq 0$ curves. We also note that the curves, $\\mathscr{S}_{k,2},\\; k\\geq 0$, exhibit a possible splitting-merging phenomeno"},"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":"2305.19944","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2023-05-31T15:26:56Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"281e8457579d21a204f18d2e9621aac42255392c6958de40832b6a7c5b0595d8","abstract_canon_sha256":"c331211497ce4137cd1bacc99aa08f74f5173881935af295e8d5a6363606919b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:54:07.377010Z","signature_b64":"T3tPDLapMHIZ4738aZf9rKTgElWsDV4LuG9FUSVUT9dVKCnfw2Oy3zj89Ho+Ex7yZVEZB6mE62Yiiae6/I7mDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84777f842df48b49494453b618f6246cebd17fd942f1fb06d8850699f54af7c1","last_reissued_at":"2026-07-05T11:54:07.376359Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:54:07.376359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Irreducibility of eventually $2$-periodic curves in the moduli space of cubic polynomials","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Niladri Patra","submitted_at":"2023-05-31T15:26:56Z","abstract_excerpt":"Consider the moduli space, $\\mathcal{M}_{3},$ of cubic polynomials over $\\mathbb{C}$, with a marked critical point. Let $\\mathscr{S}_{k,n}$ be the set of all points in $\\mathcal{M}_{3}$ for which the marked critical point is strictly $(k,n)$-preperiodic. Milnor conjectured that the affine algebraic curves $\\mathscr{S}_{k,n}$ are irreducible, for all $k \\geq 0, n>0$. In this article, we show the irreducibility of eventually $2$-periodic curves, i.e. $\\mathscr{S}_{k,2},\\; k\\geq 0$ curves. 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