{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:2MWVZQ73QH6RDFE2537PRTO7SX","short_pith_number":"pith:2MWVZQ73","canonical_record":{"source":{"id":"1512.04386","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-14T16:16:44Z","cross_cats_sorted":[],"title_canon_sha256":"fc59d33b95be67fcb157391022b933572db4b359c187d96f8f9ae221b9022c93","abstract_canon_sha256":"a69e85ce545b0c2bce544e0f3ffbc5910e3fc84d85688e230b3305e5bf7c9700"},"schema_version":"1.0"},"canonical_sha256":"d32d5cc3fb81fd11949aeefef8cddf95f2515c311407da55fe0e0b304e0cd2d0","source":{"kind":"arxiv","id":"1512.04386","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.04386","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"arxiv_version","alias_value":"1512.04386v1","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04386","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"pith_short_12","alias_value":"2MWVZQ73QH6R","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2MWVZQ73QH6RDFE2","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2MWVZQ73","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:2MWVZQ73QH6RDFE2537PRTO7SX","target":"record","payload":{"canonical_record":{"source":{"id":"1512.04386","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-14T16:16:44Z","cross_cats_sorted":[],"title_canon_sha256":"fc59d33b95be67fcb157391022b933572db4b359c187d96f8f9ae221b9022c93","abstract_canon_sha256":"a69e85ce545b0c2bce544e0f3ffbc5910e3fc84d85688e230b3305e5bf7c9700"},"schema_version":"1.0"},"canonical_sha256":"d32d5cc3fb81fd11949aeefef8cddf95f2515c311407da55fe0e0b304e0cd2d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:22.326783Z","signature_b64":"CGJ42MEvi35YC9D0PUW/omXD9QpkUUEeqdEpvnnMCBR1vmrWsVIS32b7pGUPgTIV6gUDcYmcYv/mqiOqYECbDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d32d5cc3fb81fd11949aeefef8cddf95f2515c311407da55fe0e0b304e0cd2d0","last_reissued_at":"2026-05-18T01:24:22.326245Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:22.326245Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.04386","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":"2026-05-18T01:24:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PUsWlCdsnCc+/XjQ1g/UWLH1ZpGSsNdm11/RILt6JOIoPRsWey0h2VtswFPLYZyaSCk0a+aHL2mJWpaOi28vAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T10:04:00.450216Z"},"content_sha256":"40abb4108ba01073cf776ce03e3f736ed24a799f328cd07ab14b9c8b0b535f35","schema_version":"1.0","event_id":"sha256:40abb4108ba01073cf776ce03e3f736ed24a799f328cd07ab14b9c8b0b535f35"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:2MWVZQ73QH6RDFE2537PRTO7SX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Read-once polynomials: How many summands suffice?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Anuj Tawari, Meena Mahajan","submitted_at":"2015-12-14T16:16:44Z","abstract_excerpt":"An arithmetic read-once formula (ROF) is a formula (circuit of fan-out 1) over $+, \\times$ where each variable labels at most one leaf. Every multilinear polynomial can be expressed as the sum of ROFs. In this work, we prove, for certain multilinear polynomials, a tight lower bound on the number of summands in such an expression."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04386","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":"2026-05-18T01:24:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pGTZeCBUgM6JqfevHxEdBWFbTdISYV/11RLM0tKmoaBqXGjObiASXZm+3CkSdLtIzau7HyXniAK7ioCBjDD0Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T10:04:00.450569Z"},"content_sha256":"767ee893cbd4befe557add3f9275266e48570ae75eec13d50275fe47abd516b5","schema_version":"1.0","event_id":"sha256:767ee893cbd4befe557add3f9275266e48570ae75eec13d50275fe47abd516b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2MWVZQ73QH6RDFE2537PRTO7SX/bundle.json","state_url":"https://pith.science/pith/2MWVZQ73QH6RDFE2537PRTO7SX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2MWVZQ73QH6RDFE2537PRTO7SX/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":"2026-06-23T10:04:00Z","links":{"resolver":"https://pith.science/pith/2MWVZQ73QH6RDFE2537PRTO7SX","bundle":"https://pith.science/pith/2MWVZQ73QH6RDFE2537PRTO7SX/bundle.json","state":"https://pith.science/pith/2MWVZQ73QH6RDFE2537PRTO7SX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2MWVZQ73QH6RDFE2537PRTO7SX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2MWVZQ73QH6RDFE2537PRTO7SX","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":"a69e85ce545b0c2bce544e0f3ffbc5910e3fc84d85688e230b3305e5bf7c9700","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-14T16:16:44Z","title_canon_sha256":"fc59d33b95be67fcb157391022b933572db4b359c187d96f8f9ae221b9022c93"},"schema_version":"1.0","source":{"id":"1512.04386","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.04386","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"arxiv_version","alias_value":"1512.04386v1","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04386","created_at":"2026-05-18T01:24:22Z"},{"alias_kind":"pith_short_12","alias_value":"2MWVZQ73QH6R","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"2MWVZQ73QH6RDFE2","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"2MWVZQ73","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:767ee893cbd4befe557add3f9275266e48570ae75eec13d50275fe47abd516b5","target":"graph","created_at":"2026-05-18T01:24:22Z","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":"An arithmetic read-once formula (ROF) is a formula (circuit of fan-out 1) over $+, \\times$ where each variable labels at most one leaf. Every multilinear polynomial can be expressed as the sum of ROFs. In this work, we prove, for certain multilinear polynomials, a tight lower bound on the number of summands in such an expression.","authors_text":"Anuj Tawari, Meena Mahajan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-14T16:16:44Z","title":"Read-once polynomials: How many summands suffice?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04386","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":"sha256:40abb4108ba01073cf776ce03e3f736ed24a799f328cd07ab14b9c8b0b535f35","target":"record","created_at":"2026-05-18T01:24:22Z","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":"a69e85ce545b0c2bce544e0f3ffbc5910e3fc84d85688e230b3305e5bf7c9700","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-12-14T16:16:44Z","title_canon_sha256":"fc59d33b95be67fcb157391022b933572db4b359c187d96f8f9ae221b9022c93"},"schema_version":"1.0","source":{"id":"1512.04386","kind":"arxiv","version":1}},"canonical_sha256":"d32d5cc3fb81fd11949aeefef8cddf95f2515c311407da55fe0e0b304e0cd2d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d32d5cc3fb81fd11949aeefef8cddf95f2515c311407da55fe0e0b304e0cd2d0","first_computed_at":"2026-05-18T01:24:22.326245Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:22.326245Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CGJ42MEvi35YC9D0PUW/omXD9QpkUUEeqdEpvnnMCBR1vmrWsVIS32b7pGUPgTIV6gUDcYmcYv/mqiOqYECbDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:22.326783Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.04386","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40abb4108ba01073cf776ce03e3f736ed24a799f328cd07ab14b9c8b0b535f35","sha256:767ee893cbd4befe557add3f9275266e48570ae75eec13d50275fe47abd516b5"],"state_sha256":"d8f15c82e45ce62763103a675802a706936ca805177011a2d0e776622ea256f7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zjuhzJElzza3r0HVJP1pLrjydPHILR2/MH4N7OYIJJsvAwAGEN8veuR2Rz7rXxiLUjMFBS/RP8W71kvglhYRAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T10:04:00.452650Z","bundle_sha256":"6098d8ffe16aa4fe6ebf7aba19b62f874f65fd68f8d9424e0408a2b006459a6b"}}