{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:2S4CUVMWY3H6WHEZ3IRXHCZHL5","short_pith_number":"pith:2S4CUVMW","canonical_record":{"source":{"id":"1312.3848","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-13T15:58:48Z","cross_cats_sorted":[],"title_canon_sha256":"5cfe05cc71713201f31c6b749c89f06e55eeb4efa95f6a3fe44cc110d2f6e9e6","abstract_canon_sha256":"c6ee7cbdc44f3ecb0f37fcfd6519e8ff5668769e902dbe16b5fa7e7820e769a7"},"schema_version":"1.0"},"canonical_sha256":"d4b82a5596c6cfeb1c99da23738b275f43fdc0d3d1ab52b1c3b01c43ba33a9a6","source":{"kind":"arxiv","id":"1312.3848","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.3848","created_at":"2026-05-18T03:04:47Z"},{"alias_kind":"arxiv_version","alias_value":"1312.3848v1","created_at":"2026-05-18T03:04:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3848","created_at":"2026-05-18T03:04:47Z"},{"alias_kind":"pith_short_12","alias_value":"2S4CUVMWY3H6","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2S4CUVMWY3H6WHEZ","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2S4CUVMW","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:2S4CUVMWY3H6WHEZ3IRXHCZHL5","target":"record","payload":{"canonical_record":{"source":{"id":"1312.3848","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-13T15:58:48Z","cross_cats_sorted":[],"title_canon_sha256":"5cfe05cc71713201f31c6b749c89f06e55eeb4efa95f6a3fe44cc110d2f6e9e6","abstract_canon_sha256":"c6ee7cbdc44f3ecb0f37fcfd6519e8ff5668769e902dbe16b5fa7e7820e769a7"},"schema_version":"1.0"},"canonical_sha256":"d4b82a5596c6cfeb1c99da23738b275f43fdc0d3d1ab52b1c3b01c43ba33a9a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:47.398257Z","signature_b64":"kCRaUFEj+VhErPJ4DJb4jl27zIqpkGA1ZzkzwUMgoL0IHnigkqxYSPgRgcUP/GubtaezTDzg7Fa5fCuNRDUhDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4b82a5596c6cfeb1c99da23738b275f43fdc0d3d1ab52b1c3b01c43ba33a9a6","last_reissued_at":"2026-05-18T03:04:47.397477Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:47.397477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.3848","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:04:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C9xfVPeigDdUkHpaTsaFE7aYeNlCpYGjXXQkWk8Xs0Mlad9RizUSoXgZ61rjOAASidvuiduhTQhswh4cKxcIAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T08:39:21.804605Z"},"content_sha256":"d39ad71186bc16ff6252286ee48503a79b284886f5e034abc2a3aa8dacfd4cf6","schema_version":"1.0","event_id":"sha256:d39ad71186bc16ff6252286ee48503a79b284886f5e034abc2a3aa8dacfd4cf6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:2S4CUVMWY3H6WHEZ3IRXHCZHL5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"M\\\"obius Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Will Murray","submitted_at":"2013-12-13T15:58:48Z","abstract_excerpt":"We introduce the M\\\"obius polynomial $ M_n(x) = \\sum_{d|n} \\mu\\left( \\frac nd \\right) x^d $, which gives the number of aperiodic bracelets of length $n$ with $x$ possible types of gems, and therefore satisfies $M_n(x) \\equiv 0$ (mod $n$) for all $x \\in \\mathbb Z$. We derive some key properties, analyze graphs in the complex plane, and then apply M\\\"obius polynomials combinatorially to juggling patterns, irreducible polynomials over finite fields, and Euler's totient theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3848","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:04:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nf1H44ILB0XHTW8jK71mxmwK25urhuNfHL3OxejbJHUaFtqhbw594aeZWK5mRovv1ORQ5Y6gY9D74H62cPdGDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T08:39:21.805215Z"},"content_sha256":"b6f6f87134c978991ca17174d5d14b128dc157cd53f961d4b68d7f754333fb90","schema_version":"1.0","event_id":"sha256:b6f6f87134c978991ca17174d5d14b128dc157cd53f961d4b68d7f754333fb90"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2S4CUVMWY3H6WHEZ3IRXHCZHL5/bundle.json","state_url":"https://pith.science/pith/2S4CUVMWY3H6WHEZ3IRXHCZHL5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2S4CUVMWY3H6WHEZ3IRXHCZHL5/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T08:39:21Z","links":{"resolver":"https://pith.science/pith/2S4CUVMWY3H6WHEZ3IRXHCZHL5","bundle":"https://pith.science/pith/2S4CUVMWY3H6WHEZ3IRXHCZHL5/bundle.json","state":"https://pith.science/pith/2S4CUVMWY3H6WHEZ3IRXHCZHL5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2S4CUVMWY3H6WHEZ3IRXHCZHL5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:2S4CUVMWY3H6WHEZ3IRXHCZHL5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c6ee7cbdc44f3ecb0f37fcfd6519e8ff5668769e902dbe16b5fa7e7820e769a7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-13T15:58:48Z","title_canon_sha256":"5cfe05cc71713201f31c6b749c89f06e55eeb4efa95f6a3fe44cc110d2f6e9e6"},"schema_version":"1.0","source":{"id":"1312.3848","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.3848","created_at":"2026-05-18T03:04:47Z"},{"alias_kind":"arxiv_version","alias_value":"1312.3848v1","created_at":"2026-05-18T03:04:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.3848","created_at":"2026-05-18T03:04:47Z"},{"alias_kind":"pith_short_12","alias_value":"2S4CUVMWY3H6","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2S4CUVMWY3H6WHEZ","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2S4CUVMW","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:b6f6f87134c978991ca17174d5d14b128dc157cd53f961d4b68d7f754333fb90","target":"graph","created_at":"2026-05-18T03:04:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce the M\\\"obius polynomial $ M_n(x) = \\sum_{d|n} \\mu\\left( \\frac nd \\right) x^d $, which gives the number of aperiodic bracelets of length $n$ with $x$ possible types of gems, and therefore satisfies $M_n(x) \\equiv 0$ (mod $n$) for all $x \\in \\mathbb Z$. We derive some key properties, analyze graphs in the complex plane, and then apply M\\\"obius polynomials combinatorially to juggling patterns, irreducible polynomials over finite fields, and Euler's totient theorem.","authors_text":"Will Murray","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-13T15:58:48Z","title":"M\\\"obius Polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3848","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d39ad71186bc16ff6252286ee48503a79b284886f5e034abc2a3aa8dacfd4cf6","target":"record","created_at":"2026-05-18T03:04:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c6ee7cbdc44f3ecb0f37fcfd6519e8ff5668769e902dbe16b5fa7e7820e769a7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-13T15:58:48Z","title_canon_sha256":"5cfe05cc71713201f31c6b749c89f06e55eeb4efa95f6a3fe44cc110d2f6e9e6"},"schema_version":"1.0","source":{"id":"1312.3848","kind":"arxiv","version":1}},"canonical_sha256":"d4b82a5596c6cfeb1c99da23738b275f43fdc0d3d1ab52b1c3b01c43ba33a9a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d4b82a5596c6cfeb1c99da23738b275f43fdc0d3d1ab52b1c3b01c43ba33a9a6","first_computed_at":"2026-05-18T03:04:47.397477Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:04:47.397477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kCRaUFEj+VhErPJ4DJb4jl27zIqpkGA1ZzkzwUMgoL0IHnigkqxYSPgRgcUP/GubtaezTDzg7Fa5fCuNRDUhDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:04:47.398257Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.3848","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d39ad71186bc16ff6252286ee48503a79b284886f5e034abc2a3aa8dacfd4cf6","sha256:b6f6f87134c978991ca17174d5d14b128dc157cd53f961d4b68d7f754333fb90"],"state_sha256":"9998230f13d84aab411bfc4f33dc1075b513665cb251e1673b7ffe760c94d647"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wwVam66dGQ0fsqyjMKin2GmsQ/FTLILW4bLPr/aNXqSPWUv2CAZzwb5OfP7iuoHAwp0UTDEnNDjUQbI7Wmc7DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T08:39:21.808711Z","bundle_sha256":"f68f0e9ab161e36e0a2a6bb2aa334d2df1e53029bbfef8d5f36dae6ec501d45d"}}