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In the ferromagnetic (FM) case, for $m=2$, the number of symmetric TISGMs varies with the temperature: here we identify a critical inverse temperature, $\\beta^1_{\\rm{cr}}$ ($=T_{\\rm{cr}}^{\\rm{STISG}}$) $\\in (0,\\infty)$ such that $\\forall$ $0\\leq \\beta\\leq\\beta^1_{\\rm{cr}}$, there ex"},"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":"math/0409047","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2004-09-03T10:39:43Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"8985bd45cdca58448c71dc6daa1e2c2c872d7d7bb00883ff1ca7419ae89b20e9","abstract_canon_sha256":"bca854b5a6ab0a6b36a4d5cd81575a0dc7a4fa4dd6a120a4d2bee14aa9600364"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:28.125970Z","signature_b64":"Ebq7ziTkmwgeqO2mCiwSfcBuXrA2u8UJPmBG9mF+T4AbZWc9USMtmhWU7A3Gg6Uxf6PIPvjvhcAQwGBJGVfjDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80cfa4583aea2ac3572c238cb7df871426e53027b62828f95e7b4ddc000f17e2","last_reissued_at":"2026-05-18T04:28:28.125522Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:28.125522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gibbs Measures For SOS Models On a Cayley Tree","license":"","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"U.A. 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