{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:YDQ5PITELBJINZ545OORKR5DSL","short_pith_number":"pith:YDQ5PITE","canonical_record":{"source":{"id":"1307.4809","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-18T00:57:58Z","cross_cats_sorted":[],"title_canon_sha256":"9073a6389efafb8491b586cb33731398f39ca80a8228fa4702d21baac65e6626","abstract_canon_sha256":"0f22c376ee2a99dfc3bb23b33e7cba27a2e931229e6524fe44ef981273503bb1"},"schema_version":"1.0"},"canonical_sha256":"c0e1d7a264585286e7bceb9d1547a392c48bbb46503793af172198ee8ce7e955","source":{"kind":"arxiv","id":"1307.4809","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.4809","created_at":"2026-05-18T01:31:20Z"},{"alias_kind":"arxiv_version","alias_value":"1307.4809v2","created_at":"2026-05-18T01:31:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.4809","created_at":"2026-05-18T01:31:20Z"},{"alias_kind":"pith_short_12","alias_value":"YDQ5PITELBJI","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"YDQ5PITELBJINZ54","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"YDQ5PITE","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:YDQ5PITELBJINZ545OORKR5DSL","target":"record","payload":{"canonical_record":{"source":{"id":"1307.4809","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-18T00:57:58Z","cross_cats_sorted":[],"title_canon_sha256":"9073a6389efafb8491b586cb33731398f39ca80a8228fa4702d21baac65e6626","abstract_canon_sha256":"0f22c376ee2a99dfc3bb23b33e7cba27a2e931229e6524fe44ef981273503bb1"},"schema_version":"1.0"},"canonical_sha256":"c0e1d7a264585286e7bceb9d1547a392c48bbb46503793af172198ee8ce7e955","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:20.037765Z","signature_b64":"EwtZ7k63129bAPpSlEkD6H6frbBgRPDZR4WCjB3YC/VALvUpcceR2tygtzjn3YnC7X68oV0mvLt18mETGOn+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0e1d7a264585286e7bceb9d1547a392c48bbb46503793af172198ee8ce7e955","last_reissued_at":"2026-05-18T01:31:20.037074Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:20.037074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.4809","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1wWjQrRWAFVkR4JRo+gLqQLs61i8K+ybgYutpXMxGpicXIfDUvNjgs9t2B6unB8FRY/DIaneJiSg3hOTpSOpAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T00:47:24.511322Z"},"content_sha256":"a382d16df22362fbb1e94bd87cb83a69d01839db9306263fc17abcd644e43438","schema_version":"1.0","event_id":"sha256:a382d16df22362fbb1e94bd87cb83a69d01839db9306263fc17abcd644e43438"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:YDQ5PITELBJINZ545OORKR5DSL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Grothendieck's theorem for absolutely summing multilinear operators is optimal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Daniel Pellegrino, Juan B. Seoane-Sepulveda","submitted_at":"2013-07-18T00:57:58Z","abstract_excerpt":"Grothendieck's theorem asserts that every continuous linear operator from $\\ell_{1}$ to $\\ell_{2}$ is absolutely $\\left( 1;1\\right) $-summing. In this note we prove that the optimal constant $g_{m}$ so that every continuous $m$-linear operator from $\\ell_{1}\\times\\cdots\\times\\ell_{1}$ to $\\ell_{2}$ is absolutely $\\left( g_{m};1\\right) $-summing is $\\frac{2}{m+1}$. We also show that if $g_{m}<\\frac{2}{m+1}$ there is $\\mathfrak{c}$ dimensional linear space composed by continuous non absolutely $\\left( g_{m};1\\right) $-summing $m$-linear operators from $\\ell_{1}\\times\\cdots\\times\\ell_{1}$ to $\\el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.4809","kind":"arxiv","version":2},"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":"2026-05-18T01:31:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w6foyJ9Lmqe53kF43gxhEigW9SpC53VjoO1ZWcTCuPTaASdcxTItqpL1T1EzMuRIUNYw51MUTizApKvZ3IOeCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T00:47:24.511675Z"},"content_sha256":"55860efd733dd7ce8a05075e9b7a8e99f31835ea17adfc42257cf599c925036f","schema_version":"1.0","event_id":"sha256:55860efd733dd7ce8a05075e9b7a8e99f31835ea17adfc42257cf599c925036f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YDQ5PITELBJINZ545OORKR5DSL/bundle.json","state_url":"https://pith.science/pith/YDQ5PITELBJINZ545OORKR5DSL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YDQ5PITELBJINZ545OORKR5DSL/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":"2026-06-26T00:47:24Z","links":{"resolver":"https://pith.science/pith/YDQ5PITELBJINZ545OORKR5DSL","bundle":"https://pith.science/pith/YDQ5PITELBJINZ545OORKR5DSL/bundle.json","state":"https://pith.science/pith/YDQ5PITELBJINZ545OORKR5DSL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YDQ5PITELBJINZ545OORKR5DSL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:YDQ5PITELBJINZ545OORKR5DSL","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":"0f22c376ee2a99dfc3bb23b33e7cba27a2e931229e6524fe44ef981273503bb1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-18T00:57:58Z","title_canon_sha256":"9073a6389efafb8491b586cb33731398f39ca80a8228fa4702d21baac65e6626"},"schema_version":"1.0","source":{"id":"1307.4809","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.4809","created_at":"2026-05-18T01:31:20Z"},{"alias_kind":"arxiv_version","alias_value":"1307.4809v2","created_at":"2026-05-18T01:31:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.4809","created_at":"2026-05-18T01:31:20Z"},{"alias_kind":"pith_short_12","alias_value":"YDQ5PITELBJI","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"YDQ5PITELBJINZ54","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"YDQ5PITE","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:55860efd733dd7ce8a05075e9b7a8e99f31835ea17adfc42257cf599c925036f","target":"graph","created_at":"2026-05-18T01:31:20Z","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":"Grothendieck's theorem asserts that every continuous linear operator from $\\ell_{1}$ to $\\ell_{2}$ is absolutely $\\left( 1;1\\right) $-summing. In this note we prove that the optimal constant $g_{m}$ so that every continuous $m$-linear operator from $\\ell_{1}\\times\\cdots\\times\\ell_{1}$ to $\\ell_{2}$ is absolutely $\\left( g_{m};1\\right) $-summing is $\\frac{2}{m+1}$. We also show that if $g_{m}<\\frac{2}{m+1}$ there is $\\mathfrak{c}$ dimensional linear space composed by continuous non absolutely $\\left( g_{m};1\\right) $-summing $m$-linear operators from $\\ell_{1}\\times\\cdots\\times\\ell_{1}$ to $\\el","authors_text":"Daniel Pellegrino, Juan B. Seoane-Sepulveda","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-18T00:57:58Z","title":"Grothendieck's theorem for absolutely summing multilinear operators is optimal"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.4809","kind":"arxiv","version":2},"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:a382d16df22362fbb1e94bd87cb83a69d01839db9306263fc17abcd644e43438","target":"record","created_at":"2026-05-18T01:31:20Z","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":"0f22c376ee2a99dfc3bb23b33e7cba27a2e931229e6524fe44ef981273503bb1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-07-18T00:57:58Z","title_canon_sha256":"9073a6389efafb8491b586cb33731398f39ca80a8228fa4702d21baac65e6626"},"schema_version":"1.0","source":{"id":"1307.4809","kind":"arxiv","version":2}},"canonical_sha256":"c0e1d7a264585286e7bceb9d1547a392c48bbb46503793af172198ee8ce7e955","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0e1d7a264585286e7bceb9d1547a392c48bbb46503793af172198ee8ce7e955","first_computed_at":"2026-05-18T01:31:20.037074Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:20.037074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EwtZ7k63129bAPpSlEkD6H6frbBgRPDZR4WCjB3YC/VALvUpcceR2tygtzjn3YnC7X68oV0mvLt18mETGOn+CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:20.037765Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.4809","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a382d16df22362fbb1e94bd87cb83a69d01839db9306263fc17abcd644e43438","sha256:55860efd733dd7ce8a05075e9b7a8e99f31835ea17adfc42257cf599c925036f"],"state_sha256":"4c0221ae31918ff432f160d7d8d0e5d85c86f7e59c4a0850885f7dae077d4fd7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E7QUk1TKqWr74KOoaoBcu8iTuQ5Oo1p0tZInG1Yzc6gf0hs6v9awZ4nlLBRiJ/qghQzq57f6jBZIT0NrDSjwCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T00:47:24.513812Z","bundle_sha256":"a90383e8bf20512261ba15c7c427bb14e4d13da6fbff1ca0b6a1cf06dd846642"}}