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The analogous problem in the one-dimensional lattice $\\mathbb{N}$ is also considered. %In this case we consider the potential $A: {\\cal B}=M^\\mathbb{N} \\to \\mathbb{R}.$ Let $A: \\mathcal{B}_i \\rar \\R$ be an {\\it observable} or {\\it potential} defined in the Bernoulli space"},"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":"1106.2845","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-06-14T23:46:40Z","cross_cats_sorted":["cond-mat.stat-mech","math-ph","math.MP","math.PR"],"title_canon_sha256":"419797c4173e2d6bce90ca56c8a227a6dd829d726b63c73ebfb6d929a41ec2c9","abstract_canon_sha256":"3e2261a9bbac61ebe27e13b1a44a1df0eebca090ea71c0040c0a0fc818f130f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:35.255365Z","signature_b64":"JGalOwpCKbPHWR1xLs/0eMLK5gzcHQBiqV9UsO/d33jkMknguRsrSbPz7NBQKFa+Zdkr8zVqbmNEe6wwX3pGBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5ce19d3843ceb2febe8b92fedd78dd8b69f6677788f6b4774cb1241acd85806","last_reissued_at":"2026-05-18T01:32:35.254840Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:35.254840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the general one-dimensional XY Model: positive and zero temperature, selection and non-selection","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math-ph","math.MP","math.PR"],"primary_cat":"math.DS","authors_text":"A. 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