{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:2MSZS5PRAMEXOACADCOE2I3A7G","short_pith_number":"pith:2MSZS5PR","canonical_record":{"source":{"id":"2605.12717","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-05-12T20:26:37Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"2033f25ac2c4fe99f6388d8642007658c5c673bb3d92d64cc7094117ca7b4ffc","abstract_canon_sha256":"d75dc86c1844e7a8dd84391413281390a657c75edf6c0027e3995c1dc7b68d8d"},"schema_version":"1.0"},"canonical_sha256":"d3259975f10309770040189c4d2360f990301ce34659fd7fb2556c9e10e25da3","source":{"kind":"arxiv","id":"2605.12717","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12717","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12717v1","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12717","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"pith_short_12","alias_value":"2MSZS5PRAMEX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"2MSZS5PRAMEXOACA","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"2MSZS5PR","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:2MSZS5PRAMEXOACADCOE2I3A7G","target":"record","payload":{"canonical_record":{"source":{"id":"2605.12717","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-05-12T20:26:37Z","cross_cats_sorted":["cs.AI"],"title_canon_sha256":"2033f25ac2c4fe99f6388d8642007658c5c673bb3d92d64cc7094117ca7b4ffc","abstract_canon_sha256":"d75dc86c1844e7a8dd84391413281390a657c75edf6c0027e3995c1dc7b68d8d"},"schema_version":"1.0"},"canonical_sha256":"d3259975f10309770040189c4d2360f990301ce34659fd7fb2556c9e10e25da3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:49.458984Z","signature_b64":"T+VNz+7mND02Rmy8R2WDH71RuKw//dV3ilTfvnwNXMSFOK27j4GZ9LzHSs2ShrKEbwgWYM4s06sRFCNxBgd6Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d3259975f10309770040189c4d2360f990301ce34659fd7fb2556c9e10e25da3","last_reissued_at":"2026-05-18T03:09:49.458299Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:49.458299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.12717","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":"2026-05-18T03:09:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EhZ1CwmZoGMdTuk7zJVJqX1KDtVCe3xIIWRgJf8k6WIzbTFG6ArHc+PPZVt2NmEngR5QvZUfsjJKD9wcOtTNDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T20:03:44.816458Z"},"content_sha256":"a682ed7ef8527e584fca6fc616402eaf71034a842666c76e8c5da529b6e1491a","schema_version":"1.0","event_id":"sha256:a682ed7ef8527e584fca6fc616402eaf71034a842666c76e8c5da529b6e1491a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:2MSZS5PRAMEXOACADCOE2I3A7G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings.","cross_cats":["cs.AI"],"primary_cat":"cs.GT","authors_text":"Bailey Flanigan, Carmel Baharav, Maximilian T. Wittmann, Niclas Boehmer","submitted_at":"2026-05-12T20:26:37Z","abstract_excerpt":"AI alignment and participatory design motivate a new democratic design problem: how to collectively choose a decision rule to use repeatedly. We study this problem for linear ranking rules, which repeatedly rank items $x_j$ within batches $X=(x_1,\\dots,x_m)\\in(\\mathbb{R}^d)^m$, where each item's ranking is dictated by its score $\\langle \\theta^*,x_j\\rangle$ according to a fixed scoring vector $\\theta^*$. Given voters' preferred scoring vectors $\\theta^{(1)},\\dots,\\theta^{(n)}$ and their population fractions $\\alpha^{(1)},\\dots,\\alpha^{(n)}$, we ask how to choose a collective vector $\\theta^*$ "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Our main result is that, surprisingly, there is a simple rule that does satisfy long-run IP: the angular mean, the spherical analog of the arithmetic mean.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That long-run average proportionality is acceptable even when per-batch outcomes can deviate substantially, and that voter preferences remain fixed across batches.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The angular mean of voter scoring vectors satisfies long-run individual proportionality for sequential linear ranking decisions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c8a0a0cf58106f9c53006213f855fd3b6aca16c14a34bcadcccf662a3b1d7e15"},"source":{"id":"2605.12717","kind":"arxiv","version":1},"verdict":{"id":"ce602db4-2870-4808-94e8-ad1673f33dcb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:52:24.096293Z","strongest_claim":"Our main result is that, surprisingly, there is a simple rule that does satisfy long-run IP: the angular mean, the spherical analog of the arithmetic mean.","one_line_summary":"The angular mean of voter scoring vectors satisfies long-run individual proportionality for sequential linear ranking decisions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That long-run average proportionality is acceptable even when per-batch outcomes can deviate substantially, and that voter preferences remain fixed across batches.","pith_extraction_headline":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings."},"references":{"count":50,"sample":[{"doi":"","year":2013,"title":"On the convergence of gradient descent for finding the Riemannian center of mass","work_id":"63fa1ffd-f1c8-440a-ad4c-688565a2a94d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"The moral machine experiment","work_id":"f3db6ff3-741e-4e73-9145-61daa7f87012","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"On the stability of moral preferences: A problem with computational elicitation methods","work_id":"2aa99587-1a15-4b27-ace5-d9eff8d192c7","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Optimal budget aggregation with single-peaked preferences","work_id":"6fe369b5-00d3-4d1b-9767-8e6df37f8ff6","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Justified representation for perpetual voting","work_id":"3a0ce6bc-8d7b-46eb-aa02-c68e86208775","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":50,"snapshot_sha256":"bf3a5ee429ad6fde8e84e430e1590463b2bd60f14745e807fca9c078b49f6696","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"57ef7755063a80e809e2cef5b1a1a2f6623d8549d7389426ed99fa04dc846fa6"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"ce602db4-2870-4808-94e8-ad1673f33dcb"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:09:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YRd1RPdUcEqhV9Kyqo6uTOnfqHSDM9E1EDp2bVNSxV/b+lSvpZLHScqCNjlOcIGY/AeiVyQtd3rT0/aEFNgoBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T20:03:44.817403Z"},"content_sha256":"96fc55cfd96647aaabb9e9bd9c005d4d21a9b7e7c886e82ad746d190de804f98","schema_version":"1.0","event_id":"sha256:96fc55cfd96647aaabb9e9bd9c005d4d21a9b7e7c886e82ad746d190de804f98"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2MSZS5PRAMEXOACADCOE2I3A7G/bundle.json","state_url":"https://pith.science/pith/2MSZS5PRAMEXOACADCOE2I3A7G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2MSZS5PRAMEXOACADCOE2I3A7G/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":"2026-05-18T20:03:44Z","links":{"resolver":"https://pith.science/pith/2MSZS5PRAMEXOACADCOE2I3A7G","bundle":"https://pith.science/pith/2MSZS5PRAMEXOACADCOE2I3A7G/bundle.json","state":"https://pith.science/pith/2MSZS5PRAMEXOACADCOE2I3A7G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2MSZS5PRAMEXOACADCOE2I3A7G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2MSZS5PRAMEXOACADCOE2I3A7G","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":"d75dc86c1844e7a8dd84391413281390a657c75edf6c0027e3995c1dc7b68d8d","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-05-12T20:26:37Z","title_canon_sha256":"2033f25ac2c4fe99f6388d8642007658c5c673bb3d92d64cc7094117ca7b4ffc"},"schema_version":"1.0","source":{"id":"2605.12717","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12717","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12717v1","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12717","created_at":"2026-05-18T03:09:49Z"},{"alias_kind":"pith_short_12","alias_value":"2MSZS5PRAMEX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"2MSZS5PRAMEXOACA","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"2MSZS5PR","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:96fc55cfd96647aaabb9e9bd9c005d4d21a9b7e7c886e82ad746d190de804f98","target":"graph","created_at":"2026-05-18T03:09:49Z","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":"Our main result is that, surprisingly, there is a simple rule that does satisfy long-run IP: the angular mean, the spherical analog of the arithmetic mean."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That long-run average proportionality is acceptable even when per-batch outcomes can deviate substantially, and that voter preferences remain fixed across batches."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"The angular mean of voter scoring vectors satisfies long-run individual proportionality for sequential linear ranking decisions."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings."}],"snapshot_sha256":"c8a0a0cf58106f9c53006213f855fd3b6aca16c14a34bcadcccf662a3b1d7e15"},"formal_canon":{"evidence_count":1,"snapshot_sha256":"57ef7755063a80e809e2cef5b1a1a2f6623d8549d7389426ed99fa04dc846fa6"},"paper":{"abstract_excerpt":"AI alignment and participatory design motivate a new democratic design problem: how to collectively choose a decision rule to use repeatedly. We study this problem for linear ranking rules, which repeatedly rank items $x_j$ within batches $X=(x_1,\\dots,x_m)\\in(\\mathbb{R}^d)^m$, where each item's ranking is dictated by its score $\\langle \\theta^*,x_j\\rangle$ according to a fixed scoring vector $\\theta^*$. Given voters' preferred scoring vectors $\\theta^{(1)},\\dots,\\theta^{(n)}$ and their population fractions $\\alpha^{(1)},\\dots,\\alpha^{(n)}$, we ask how to choose a collective vector $\\theta^*$ ","authors_text":"Bailey Flanigan, Carmel Baharav, Maximilian T. Wittmann, Niclas Boehmer","cross_cats":["cs.AI"],"headline":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-05-12T20:26:37Z","title":"The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions"},"references":{"count":50,"internal_anchors":0,"resolved_work":50,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"On the convergence of gradient descent for finding the Riemannian center of mass","work_id":"63fa1ffd-f1c8-440a-ad4c-688565a2a94d","year":2013},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"The moral machine experiment","work_id":"f3db6ff3-741e-4e73-9145-61daa7f87012","year":2018},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"On the stability of moral preferences: A problem with computational elicitation methods","work_id":"2aa99587-1a15-4b27-ace5-d9eff8d192c7","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Optimal budget aggregation with single-peaked preferences","work_id":"6fe369b5-00d3-4d1b-9767-8e6df37f8ff6","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Justified representation for perpetual voting","work_id":"3a0ce6bc-8d7b-46eb-aa02-c68e86208775","year":2021}],"snapshot_sha256":"bf3a5ee429ad6fde8e84e430e1590463b2bd60f14745e807fca9c078b49f6696"},"source":{"id":"2605.12717","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T19:52:24.096293Z","id":"ce602db4-2870-4808-94e8-ad1673f33dcb","model_set":{"reader":"grok-4.3"},"one_line_summary":"The angular mean of voter scoring vectors satisfies long-run individual proportionality for sequential linear ranking decisions.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings.","strongest_claim":"Our main result is that, surprisingly, there is a simple rule that does satisfy long-run IP: the angular mean, the spherical analog of the arithmetic mean.","weakest_assumption":"That long-run average proportionality is acceptable even when per-batch outcomes can deviate substantially, and that voter preferences remain fixed across batches."}},"verdict_id":"ce602db4-2870-4808-94e8-ad1673f33dcb"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a682ed7ef8527e584fca6fc616402eaf71034a842666c76e8c5da529b6e1491a","target":"record","created_at":"2026-05-18T03:09:49Z","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":"d75dc86c1844e7a8dd84391413281390a657c75edf6c0027e3995c1dc7b68d8d","cross_cats_sorted":["cs.AI"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.GT","submitted_at":"2026-05-12T20:26:37Z","title_canon_sha256":"2033f25ac2c4fe99f6388d8642007658c5c673bb3d92d64cc7094117ca7b4ffc"},"schema_version":"1.0","source":{"id":"2605.12717","kind":"arxiv","version":1}},"canonical_sha256":"d3259975f10309770040189c4d2360f990301ce34659fd7fb2556c9e10e25da3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d3259975f10309770040189c4d2360f990301ce34659fd7fb2556c9e10e25da3","first_computed_at":"2026-05-18T03:09:49.458299Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:49.458299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"T+VNz+7mND02Rmy8R2WDH71RuKw//dV3ilTfvnwNXMSFOK27j4GZ9LzHSs2ShrKEbwgWYM4s06sRFCNxBgd6Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:49.458984Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12717","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a682ed7ef8527e584fca6fc616402eaf71034a842666c76e8c5da529b6e1491a","sha256:96fc55cfd96647aaabb9e9bd9c005d4d21a9b7e7c886e82ad746d190de804f98"],"state_sha256":"1d727f8b919a15608ecd5dd48fa4a81183114275ec8d09755cae05122fb91699"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZtJhR75F6UbYj2jORywc3jxtKzgnThrs/C8weBinGVQIFOirZSnHuNAIcfnQzhuRAjO3FCw2CKnFIGBGcA0uAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-18T20:03:44.820077Z","bundle_sha256":"262fb01f6ecd32a52c87ba7d04441c77565474d79eda4742b806b879e3b888cb"}}