{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:GRBDL4GLW4KH3NJJKDRU2HVDDE","short_pith_number":"pith:GRBDL4GL","canonical_record":{"source":{"id":"1603.02605","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-03-08T17:58:40Z","cross_cats_sorted":[],"title_canon_sha256":"31440a2f929f8fe470b6d047465e0b19b2fac5bcd9b03ae2d519cd12ef4646a1","abstract_canon_sha256":"f988d036011b2956230d72c76b598d5ae99a27dde2bc4fed85072d9471ff241d"},"schema_version":"1.0"},"canonical_sha256":"344235f0cbb7147db52950e34d1ea319054b797565acf32de4bc68522aee5bcb","source":{"kind":"arxiv","id":"1603.02605","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02605","created_at":"2026-05-18T01:19:22Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02605v1","created_at":"2026-05-18T01:19:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02605","created_at":"2026-05-18T01:19:22Z"},{"alias_kind":"pith_short_12","alias_value":"GRBDL4GLW4KH","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"GRBDL4GLW4KH3NJJ","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"GRBDL4GL","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:GRBDL4GLW4KH3NJJKDRU2HVDDE","target":"record","payload":{"canonical_record":{"source":{"id":"1603.02605","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-03-08T17:58:40Z","cross_cats_sorted":[],"title_canon_sha256":"31440a2f929f8fe470b6d047465e0b19b2fac5bcd9b03ae2d519cd12ef4646a1","abstract_canon_sha256":"f988d036011b2956230d72c76b598d5ae99a27dde2bc4fed85072d9471ff241d"},"schema_version":"1.0"},"canonical_sha256":"344235f0cbb7147db52950e34d1ea319054b797565acf32de4bc68522aee5bcb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:22.807410Z","signature_b64":"IM48KpWuKxoPGg1yBvWLDCxF/l9m/uFP5ecF5ZnL65ezQW7TNBlCoKBlm4uK3FrZhYFM/7+07qr4fYrnCkfUBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"344235f0cbb7147db52950e34d1ea319054b797565acf32de4bc68522aee5bcb","last_reissued_at":"2026-05-18T01:19:22.806705Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:22.806705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.02605","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":"2026-05-18T01:19:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nToFA50K7TqhKSuym3ML33MKUQPCnmwEgeLPMfWdjQbs7eBJSg3W7u6NKGqyv9qIY3U+7Yx6lilzLqz6Pk0QCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T11:34:25.309482Z"},"content_sha256":"0e7376331d9c686e71b7edb28e41061279ca02e5d7dcc307630694739013f6c2","schema_version":"1.0","event_id":"sha256:0e7376331d9c686e71b7edb28e41061279ca02e5d7dcc307630694739013f6c2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:GRBDL4GLW4KH3NJJKDRU2HVDDE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sums of read-once formulas: 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":"2016-03-08T17:58:40Z","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":"1603.02605","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":"2026-05-18T01:19:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"USxFb9KGE/lyPHjSy/XJtdKFHbzb4KbSVrxbtubcOt1xRo+BLKhhHiUWwWPt4+8I+6HWyX5VVUfWxt/S92fxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T11:34:25.309819Z"},"content_sha256":"f91a9e6c517a66927656cf7dff0433b338d6219b6fd6883e5e7f634252e73be0","schema_version":"1.0","event_id":"sha256:f91a9e6c517a66927656cf7dff0433b338d6219b6fd6883e5e7f634252e73be0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GRBDL4GLW4KH3NJJKDRU2HVDDE/bundle.json","state_url":"https://pith.science/pith/GRBDL4GLW4KH3NJJKDRU2HVDDE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GRBDL4GLW4KH3NJJKDRU2HVDDE/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":"2026-06-23T11:34:25Z","links":{"resolver":"https://pith.science/pith/GRBDL4GLW4KH3NJJKDRU2HVDDE","bundle":"https://pith.science/pith/GRBDL4GLW4KH3NJJKDRU2HVDDE/bundle.json","state":"https://pith.science/pith/GRBDL4GLW4KH3NJJKDRU2HVDDE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GRBDL4GLW4KH3NJJKDRU2HVDDE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:GRBDL4GLW4KH3NJJKDRU2HVDDE","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":"f988d036011b2956230d72c76b598d5ae99a27dde2bc4fed85072d9471ff241d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-03-08T17:58:40Z","title_canon_sha256":"31440a2f929f8fe470b6d047465e0b19b2fac5bcd9b03ae2d519cd12ef4646a1"},"schema_version":"1.0","source":{"id":"1603.02605","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.02605","created_at":"2026-05-18T01:19:22Z"},{"alias_kind":"arxiv_version","alias_value":"1603.02605v1","created_at":"2026-05-18T01:19:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02605","created_at":"2026-05-18T01:19:22Z"},{"alias_kind":"pith_short_12","alias_value":"GRBDL4GLW4KH","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"GRBDL4GLW4KH3NJJ","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"GRBDL4GL","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:f91a9e6c517a66927656cf7dff0433b338d6219b6fd6883e5e7f634252e73be0","target":"graph","created_at":"2026-05-18T01:19:22Z","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":"2016-03-08T17:58:40Z","title":"Sums of read-once formulas: How many summands suffice?"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02605","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":"sha256:0e7376331d9c686e71b7edb28e41061279ca02e5d7dcc307630694739013f6c2","target":"record","created_at":"2026-05-18T01:19:22Z","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":"f988d036011b2956230d72c76b598d5ae99a27dde2bc4fed85072d9471ff241d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-03-08T17:58:40Z","title_canon_sha256":"31440a2f929f8fe470b6d047465e0b19b2fac5bcd9b03ae2d519cd12ef4646a1"},"schema_version":"1.0","source":{"id":"1603.02605","kind":"arxiv","version":1}},"canonical_sha256":"344235f0cbb7147db52950e34d1ea319054b797565acf32de4bc68522aee5bcb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"344235f0cbb7147db52950e34d1ea319054b797565acf32de4bc68522aee5bcb","first_computed_at":"2026-05-18T01:19:22.806705Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:22.806705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IM48KpWuKxoPGg1yBvWLDCxF/l9m/uFP5ecF5ZnL65ezQW7TNBlCoKBlm4uK3FrZhYFM/7+07qr4fYrnCkfUBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:22.807410Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.02605","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e7376331d9c686e71b7edb28e41061279ca02e5d7dcc307630694739013f6c2","sha256:f91a9e6c517a66927656cf7dff0433b338d6219b6fd6883e5e7f634252e73be0"],"state_sha256":"4fa15af44a6f0943d8c0bb71e50c58c0329c652bd94010acd2eb718d457658ff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TuVvSdhZGTkIKv6f2V3SXc+1dUMixBzj5AUqRPUTHA+VPr+jCAcvGEMUDi7vVfzZiKfd0oznQvQJyM/mkif1Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T11:34:25.311689Z","bundle_sha256":"39de14b0f9bd988362859b074729cdb3610eed043e4da51ab055d0764ac9eed6"}}