{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:CDLPVDPMF47IEWU3YMJOFJVCG5","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":"70caaeb355115856e75242cccaf2e38979789973b6c699a3ac4823ebae24232b","cross_cats_sorted":["math.CO","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2022-04-28T15:07:34Z","title_canon_sha256":"70fcfd33517e56bef10f1333cddf010679f30236d730fc4272e3114fa5964753"},"schema_version":"1.0","source":{"id":"2205.00879","kind":"arxiv","version":7}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2205.00879","created_at":"2026-08-04T00:35:55Z"},{"alias_kind":"arxiv_version","alias_value":"2205.00879v7","created_at":"2026-08-04T00:35:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2205.00879","created_at":"2026-08-04T00:35:55Z"},{"alias_kind":"pith_short_12","alias_value":"CDLPVDPMF47I","created_at":"2026-08-04T00:35:55Z"},{"alias_kind":"pith_short_16","alias_value":"CDLPVDPMF47IEWU3","created_at":"2026-08-04T00:35:55Z"},{"alias_kind":"pith_short_8","alias_value":"CDLPVDPM","created_at":"2026-08-04T00:35:55Z"}],"graph_snapshots":[{"event_id":"sha256:1313f0640535a9e822587d11721d2d3e4dfd3447f6ae81ebbbc8ed9426d73855","target":"graph","created_at":"2026-08-04T00:35:55Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Combining ideas from various authors we are able to prove Newton's binomial theorem, Jacobi's triple product, the Rogers--Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan's partition congruences, generating functions of Stirling numbers and Jacobi's four-square theorem."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the ring operations and formal manipulations on power series (addition, multiplication, substitution, differentiation) suffice to establish the listed identities without any appeal to analytic properties or convergence."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"An expository lecture proving Newton's binomial theorem, Jacobi's triple product, Rogers-Ramanujan identities, Ramanujan partition congruences and related results using formal power series."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Formal power series ring operations prove Newton's binomial theorem, Jacobi's triple product, and Rogers-Ramanujan identities without analysis."}],"snapshot_sha256":"80b548040bdae8799f4add85a190088556e120b571a21c383f85214ae355826f"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"720d99be7e9d9ea13d56e1375edec543c697468794635278a1598d01b1333cb3"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2205.00879/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This is a lecture on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton's binomial theorem, Jacobi's triple product, the Rogers--Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan's partition congruences, generating functions of Stirling numbers and Jacobi's four-square theorem. We further discuss formal Laurent series and multivariate power series and end with a proof of MacMahon's master theorem.","authors_text":"Benjamin Sambale","cross_cats":["math.CO","math.NT"],"headline":"Formal power series ring operations prove Newton's binomial theorem, Jacobi's triple product, and Rogers-Ramanujan identities without analysis.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2022-04-28T15:07:34Z","title":"An invitation to formal power series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.00879","kind":"arxiv","version":7},"verdict":{"created_at":"2026-05-24T11:41:56.371768Z","id":"88b1b06c-14b3-4b53-acff-13d278e14bad","model_set":{"reader":"grok-4.3"},"one_line_summary":"An expository lecture proving Newton's binomial theorem, Jacobi's triple product, Rogers-Ramanujan identities, Ramanujan partition congruences and related results using formal power series.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Formal power series ring operations prove Newton's binomial theorem, Jacobi's triple product, and Rogers-Ramanujan identities without analysis.","strongest_claim":"Combining ideas from various authors we are able to prove Newton's binomial theorem, Jacobi's triple product, the Rogers--Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan's partition congruences, generating functions of Stirling numbers and Jacobi's four-square theorem.","weakest_assumption":"That the ring operations and formal manipulations on power series (addition, multiplication, substitution, differentiation) suffice to establish the listed identities without any appeal to analytic properties or convergence."}},"verdict_id":"88b1b06c-14b3-4b53-acff-13d278e14bad"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3921e9abf1d4e923393a6dab11f2652396efae94b4d3148d823b5b1a8cc70631","target":"record","created_at":"2026-08-04T00:35:55Z","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":"70caaeb355115856e75242cccaf2e38979789973b6c699a3ac4823ebae24232b","cross_cats_sorted":["math.CO","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.HO","submitted_at":"2022-04-28T15:07:34Z","title_canon_sha256":"70fcfd33517e56bef10f1333cddf010679f30236d730fc4272e3114fa5964753"},"schema_version":"1.0","source":{"id":"2205.00879","kind":"arxiv","version":7}},"canonical_sha256":"10d6fa8dec2f3e825a9bc312e2a6a2375eaa74552677c3bc29f5476caddfa114","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10d6fa8dec2f3e825a9bc312e2a6a2375eaa74552677c3bc29f5476caddfa114","first_computed_at":"2026-08-04T00:35:55.121069Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-04T00:35:55.121069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Kb2hjBSfXmfVEUGe3F4W77YqCB9YtPT7PY9m7gar70kPjYq7O9+h9P6mY4BaTF5VScQQQI4y2uPb/FV6TygkBg==","signature_status":"signed_v1","signed_at":"2026-08-04T00:35:55.122502Z","signed_message":"canonical_sha256_bytes"},"source_id":"2205.00879","source_kind":"arxiv","source_version":7}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3921e9abf1d4e923393a6dab11f2652396efae94b4d3148d823b5b1a8cc70631","sha256:1313f0640535a9e822587d11721d2d3e4dfd3447f6ae81ebbbc8ed9426d73855"],"state_sha256":"7465d955f3bdc402bee3ef1b27b70242540c516e150c68fef9b468142053f6dd"}