{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:L2KGMXSZEYHGQGDG6JL5PW3J2M","short_pith_number":"pith:L2KGMXSZ","canonical_record":{"source":{"id":"2010.08672","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-10-17T00:05:39Z","cross_cats_sorted":["cs.GT"],"title_canon_sha256":"bf396f87db73dbda2e24c31816ee0a52a87c6aa448772866fcbcf8a011427102","abstract_canon_sha256":"690a3c8435087dd8aca09eca404124af5a7aff06b7eb636dfd3176839755cc05"},"schema_version":"1.0"},"canonical_sha256":"5e94665e59260e681866f257d7db69d32fbd2d6a9e6064102eedc4a266cd9522","source":{"kind":"arxiv","id":"2010.08672","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2010.08672","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"arxiv_version","alias_value":"2010.08672v1","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.08672","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"pith_short_12","alias_value":"L2KGMXSZEYHG","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"pith_short_16","alias_value":"L2KGMXSZEYHGQGDG","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"pith_short_8","alias_value":"L2KGMXSZ","created_at":"2026-07-05T01:43:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:L2KGMXSZEYHGQGDG6JL5PW3J2M","target":"record","payload":{"canonical_record":{"source":{"id":"2010.08672","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-10-17T00:05:39Z","cross_cats_sorted":["cs.GT"],"title_canon_sha256":"bf396f87db73dbda2e24c31816ee0a52a87c6aa448772866fcbcf8a011427102","abstract_canon_sha256":"690a3c8435087dd8aca09eca404124af5a7aff06b7eb636dfd3176839755cc05"},"schema_version":"1.0"},"canonical_sha256":"5e94665e59260e681866f257d7db69d32fbd2d6a9e6064102eedc4a266cd9522","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:43:44.460132Z","signature_b64":"E3t8B9fBqSAvnYbUzUA8eSFLNBtf/JjNBDxxbdasGJ1eUjp8hqgxLYGODjGGKL62H+fBKSSy93vDF75BOuyFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e94665e59260e681866f257d7db69d32fbd2d6a9e6064102eedc4a266cd9522","last_reissued_at":"2026-07-05T01:43:44.459792Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:43:44.459792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2010.08672","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":"2026-07-05T01:43:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lf6aIE5K/KStGjzQ1N/0ZlDv8qB07gUZXzNstSZDsYKiP87hgq6vwU5Bg+AQMRHGlgsxvvjwzhi/CCNDf1EdAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T23:49:46.785617Z"},"content_sha256":"4a502ad54108fb88302acee39d8792458520db29edbef97c2d1455840f5c36b9","schema_version":"1.0","event_id":"sha256:4a502ad54108fb88302acee39d8792458520db29edbef97c2d1455840f5c36b9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:L2KGMXSZEYHGQGDG6JL5PW3J2M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Banzhaf and Shapley-Shubik Fixed Points and Divisor Voting Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.GT"],"primary_cat":"math.NT","authors_text":"Aaryan Prakash, Alex Arnell, Evelyn Choi, Miroslav Marinov, Nastia Polina, Richard Chen","submitted_at":"2020-10-17T00:05:39Z","abstract_excerpt":"The Banzhaf and Shapley-Shubik power indices were first introduced to measure the power of voters in a weighted voting system. Given a weighted voting system, the fixed point of such a system is found by continually reassigning each voter's weight with its power index until the system can no longer be changed by the operation. We characterize all fixed points under the Shapley-Shubik power index of the form $(a,b,\\ldots,b)$ and give an algebraic equation which can verify in principle whether a point of this form is fixed for Banzhaf; we also generate Shapley-Shubik fixed classes of the form $("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.08672","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2010.08672/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T01:43:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RjMXmnWVDw5nWcsT9kvds7eErRTUeOXlJKh10Bl9Pogx8sPoeQFAt/wSBKzcxpiDcftJ05QUUREgrN/iD3+BBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T23:49:46.786120Z"},"content_sha256":"c522029946e2f075d71992a77dc373425484ac3e3846fca40db44b9007e97f7b","schema_version":"1.0","event_id":"sha256:c522029946e2f075d71992a77dc373425484ac3e3846fca40db44b9007e97f7b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L2KGMXSZEYHGQGDG6JL5PW3J2M/bundle.json","state_url":"https://pith.science/pith/L2KGMXSZEYHGQGDG6JL5PW3J2M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L2KGMXSZEYHGQGDG6JL5PW3J2M/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":"2026-08-17T23:49:46Z","links":{"resolver":"https://pith.science/pith/L2KGMXSZEYHGQGDG6JL5PW3J2M","bundle":"https://pith.science/pith/L2KGMXSZEYHGQGDG6JL5PW3J2M/bundle.json","state":"https://pith.science/pith/L2KGMXSZEYHGQGDG6JL5PW3J2M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L2KGMXSZEYHGQGDG6JL5PW3J2M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:L2KGMXSZEYHGQGDG6JL5PW3J2M","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":"690a3c8435087dd8aca09eca404124af5a7aff06b7eb636dfd3176839755cc05","cross_cats_sorted":["cs.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-10-17T00:05:39Z","title_canon_sha256":"bf396f87db73dbda2e24c31816ee0a52a87c6aa448772866fcbcf8a011427102"},"schema_version":"1.0","source":{"id":"2010.08672","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2010.08672","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"arxiv_version","alias_value":"2010.08672v1","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.08672","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"pith_short_12","alias_value":"L2KGMXSZEYHG","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"pith_short_16","alias_value":"L2KGMXSZEYHGQGDG","created_at":"2026-07-05T01:43:44Z"},{"alias_kind":"pith_short_8","alias_value":"L2KGMXSZ","created_at":"2026-07-05T01:43:44Z"}],"graph_snapshots":[{"event_id":"sha256:c522029946e2f075d71992a77dc373425484ac3e3846fca40db44b9007e97f7b","target":"graph","created_at":"2026-07-05T01:43:44Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2010.08672/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Banzhaf and Shapley-Shubik power indices were first introduced to measure the power of voters in a weighted voting system. Given a weighted voting system, the fixed point of such a system is found by continually reassigning each voter's weight with its power index until the system can no longer be changed by the operation. We characterize all fixed points under the Shapley-Shubik power index of the form $(a,b,\\ldots,b)$ and give an algebraic equation which can verify in principle whether a point of this form is fixed for Banzhaf; we also generate Shapley-Shubik fixed classes of the form $(","authors_text":"Aaryan Prakash, Alex Arnell, Evelyn Choi, Miroslav Marinov, Nastia Polina, Richard Chen","cross_cats":["cs.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-10-17T00:05:39Z","title":"On Banzhaf and Shapley-Shubik Fixed Points and Divisor Voting Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.08672","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":"sha256:4a502ad54108fb88302acee39d8792458520db29edbef97c2d1455840f5c36b9","target":"record","created_at":"2026-07-05T01:43:44Z","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":"690a3c8435087dd8aca09eca404124af5a7aff06b7eb636dfd3176839755cc05","cross_cats_sorted":["cs.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-10-17T00:05:39Z","title_canon_sha256":"bf396f87db73dbda2e24c31816ee0a52a87c6aa448772866fcbcf8a011427102"},"schema_version":"1.0","source":{"id":"2010.08672","kind":"arxiv","version":1}},"canonical_sha256":"5e94665e59260e681866f257d7db69d32fbd2d6a9e6064102eedc4a266cd9522","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5e94665e59260e681866f257d7db69d32fbd2d6a9e6064102eedc4a266cd9522","first_computed_at":"2026-07-05T01:43:44.459792Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:43:44.459792Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E3t8B9fBqSAvnYbUzUA8eSFLNBtf/JjNBDxxbdasGJ1eUjp8hqgxLYGODjGGKL62H+fBKSSy93vDF75BOuyFDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:43:44.460132Z","signed_message":"canonical_sha256_bytes"},"source_id":"2010.08672","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a502ad54108fb88302acee39d8792458520db29edbef97c2d1455840f5c36b9","sha256:c522029946e2f075d71992a77dc373425484ac3e3846fca40db44b9007e97f7b"],"state_sha256":"d60b9bbaeb2d0206801b05f87ec14113ddb049753603dd8b776aede44503d7b9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gTfnWVqjUM0hS4u3a9/vfH0AnBNKt9Rx3r90XJxZXpU98ZxrumIiA9icrGnMfUFkUDMK/S6I0y64XwBf3VgkCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T23:49:46.790329Z","bundle_sha256":"8488eb533875383b9a07dadf15bec80c757ce0ac56ddcb5630ba58bdd79a49c8"}}