{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:YNQWZLF2YSA7OHVEJ7J76Q3GIF","short_pith_number":"pith:YNQWZLF2","canonical_record":{"source":{"id":"1011.1524","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2010-11-05T22:25:17Z","cross_cats_sorted":["math.FA","math.GR"],"title_canon_sha256":"96b7aeda61e5b20dc303817dd2539e62df14fb9d42087c16b3a06698b803702f","abstract_canon_sha256":"b37e52a16ca34ccdfe000f26a2333cf4b8f4674a7d80bb23715a7f9babf1b0c7"},"schema_version":"1.0"},"canonical_sha256":"c3616cacbac481f71ea44fd3ff4366415728474d859dfffef5da661b178a201c","source":{"kind":"arxiv","id":"1011.1524","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1524","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1524v1","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1524","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"pith_short_12","alias_value":"YNQWZLF2YSA7","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YNQWZLF2YSA7OHVE","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YNQWZLF2","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:YNQWZLF2YSA7OHVEJ7J76Q3GIF","target":"record","payload":{"canonical_record":{"source":{"id":"1011.1524","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2010-11-05T22:25:17Z","cross_cats_sorted":["math.FA","math.GR"],"title_canon_sha256":"96b7aeda61e5b20dc303817dd2539e62df14fb9d42087c16b3a06698b803702f","abstract_canon_sha256":"b37e52a16ca34ccdfe000f26a2333cf4b8f4674a7d80bb23715a7f9babf1b0c7"},"schema_version":"1.0"},"canonical_sha256":"c3616cacbac481f71ea44fd3ff4366415728474d859dfffef5da661b178a201c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:11.438635Z","signature_b64":"am/VLTV4qaoEEVh7YwfxzRwmNF5CD8XUolFfBOmo/XfwdNnBAXynXkF/ivk/VpmgtcJnc6c36zx91lcM7Lp9Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c3616cacbac481f71ea44fd3ff4366415728474d859dfffef5da661b178a201c","last_reissued_at":"2026-05-18T04:32:11.438151Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:11.438151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.1524","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":"2026-05-18T04:32:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kj2U1B4JHDeIpHfh4Ge+0RNZ9cZ97ZPqEPTWR1vMeEEmnJS5gC2IgL4Hs4Mt4Mg9W079rcDekBmU6LYtPwjdDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T02:05:07.200214Z"},"content_sha256":"ac5bd4d2eec2e1de6ccc2d2a8b1fec0ef9525dd9fc58714f04c7cd168d2bf5ee","schema_version":"1.0","event_id":"sha256:ac5bd4d2eec2e1de6ccc2d2a8b1fec0ef9525dd9fc58714f04c7cd168d2bf5ee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:YNQWZLF2YSA7OHVEJ7J76Q3GIF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Productivity of sequences with respect to a given weight function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GR"],"primary_cat":"math.GN","authors_text":"Dikran Dikranjan, Dmitri Shakhmatov, Jan Sp\\v{e}v\\'ak","submitted_at":"2010-11-05T22:25:17Z","abstract_excerpt":"Given a function f: N --> (omega+1)-{0}, we say that a faithfully indexed sequence {a_n: n in N} of elements of a topological group G is: (i) f-Cauchy productive (f-productive) provided that the sequence {prod_{n=0}^m a_n^{z(n)}: m in N} is left Cauchy (converges to some element of G, respectively) for each function z: N --> Z such that |z(n)| <= f(n) for every n in N; (ii) unconditionally f-Cauchy productive (unconditionally f-productive) provided that the sequence {a_{s(n)}: n in N\\} is (f\\circ s)-Cauchy productive (respectively, (f\\circ s)-productive) for every bijection s: N --> N. (Biject"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1524","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":"2026-05-18T04:32:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o3kWnuO0BA6wOVKJuTWd9XKdd3maf7JvfYVpsODqCYFCMOaA+jVmJHpYiLgMInUPmrsMy6C16wtH9fWNPdLICA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T02:05:07.200585Z"},"content_sha256":"ee2823e408d0501ff1fbdc354e222500f6b6c9810902efc4401112ed50f1c97c","schema_version":"1.0","event_id":"sha256:ee2823e408d0501ff1fbdc354e222500f6b6c9810902efc4401112ed50f1c97c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YNQWZLF2YSA7OHVEJ7J76Q3GIF/bundle.json","state_url":"https://pith.science/pith/YNQWZLF2YSA7OHVEJ7J76Q3GIF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YNQWZLF2YSA7OHVEJ7J76Q3GIF/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":"2026-06-02T02:05:07Z","links":{"resolver":"https://pith.science/pith/YNQWZLF2YSA7OHVEJ7J76Q3GIF","bundle":"https://pith.science/pith/YNQWZLF2YSA7OHVEJ7J76Q3GIF/bundle.json","state":"https://pith.science/pith/YNQWZLF2YSA7OHVEJ7J76Q3GIF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YNQWZLF2YSA7OHVEJ7J76Q3GIF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:YNQWZLF2YSA7OHVEJ7J76Q3GIF","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":"b37e52a16ca34ccdfe000f26a2333cf4b8f4674a7d80bb23715a7f9babf1b0c7","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2010-11-05T22:25:17Z","title_canon_sha256":"96b7aeda61e5b20dc303817dd2539e62df14fb9d42087c16b3a06698b803702f"},"schema_version":"1.0","source":{"id":"1011.1524","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1524","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1524v1","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1524","created_at":"2026-05-18T04:32:11Z"},{"alias_kind":"pith_short_12","alias_value":"YNQWZLF2YSA7","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YNQWZLF2YSA7OHVE","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YNQWZLF2","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:ee2823e408d0501ff1fbdc354e222500f6b6c9810902efc4401112ed50f1c97c","target":"graph","created_at":"2026-05-18T04:32:11Z","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":"Given a function f: N --> (omega+1)-{0}, we say that a faithfully indexed sequence {a_n: n in N} of elements of a topological group G is: (i) f-Cauchy productive (f-productive) provided that the sequence {prod_{n=0}^m a_n^{z(n)}: m in N} is left Cauchy (converges to some element of G, respectively) for each function z: N --> Z such that |z(n)| <= f(n) for every n in N; (ii) unconditionally f-Cauchy productive (unconditionally f-productive) provided that the sequence {a_{s(n)}: n in N\\} is (f\\circ s)-Cauchy productive (respectively, (f\\circ s)-productive) for every bijection s: N --> N. (Biject","authors_text":"Dikran Dikranjan, Dmitri Shakhmatov, Jan Sp\\v{e}v\\'ak","cross_cats":["math.FA","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2010-11-05T22:25:17Z","title":"Productivity of sequences with respect to a given weight function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1524","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:ac5bd4d2eec2e1de6ccc2d2a8b1fec0ef9525dd9fc58714f04c7cd168d2bf5ee","target":"record","created_at":"2026-05-18T04:32:11Z","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":"b37e52a16ca34ccdfe000f26a2333cf4b8f4674a7d80bb23715a7f9babf1b0c7","cross_cats_sorted":["math.FA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2010-11-05T22:25:17Z","title_canon_sha256":"96b7aeda61e5b20dc303817dd2539e62df14fb9d42087c16b3a06698b803702f"},"schema_version":"1.0","source":{"id":"1011.1524","kind":"arxiv","version":1}},"canonical_sha256":"c3616cacbac481f71ea44fd3ff4366415728474d859dfffef5da661b178a201c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c3616cacbac481f71ea44fd3ff4366415728474d859dfffef5da661b178a201c","first_computed_at":"2026-05-18T04:32:11.438151Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:32:11.438151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"am/VLTV4qaoEEVh7YwfxzRwmNF5CD8XUolFfBOmo/XfwdNnBAXynXkF/ivk/VpmgtcJnc6c36zx91lcM7Lp9Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:32:11.438635Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.1524","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac5bd4d2eec2e1de6ccc2d2a8b1fec0ef9525dd9fc58714f04c7cd168d2bf5ee","sha256:ee2823e408d0501ff1fbdc354e222500f6b6c9810902efc4401112ed50f1c97c"],"state_sha256":"8f58fbebe8b94f601dc0c001a029fc047f1272add47bf874ae148aa8f17400da"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J4233tpHz9D549UDJCliVp75GJw7H10xezn8iEUYVJlAbsq/pvM0V3rmdDB0pIFOyK49d9xrV7u1pbZl19dDDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T02:05:07.202490Z","bundle_sha256":"37a32d4f48f6a0358b8c224a92cde26b84145fd3db322b0d093440e49b5a6b62"}}