{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:DPNELFMXUYC5UWGKQ456JENL5Q","short_pith_number":"pith:DPNELFMX","canonical_record":{"source":{"id":"2509.02883","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2025-09-02T23:06:07Z","cross_cats_sorted":["math.DG","math.MG"],"title_canon_sha256":"f080e39434944ba6a0c5603055962e58c648f844018b4b24b5c041bfb6380058","abstract_canon_sha256":"600facbd57f2feab36218f85c6f47a6ad9ffb3e6b77faff836b470be6abd3c40"},"schema_version":"1.0"},"canonical_sha256":"1bda459597a605da58ca873be491abec23787b3cd106a0180dadd348d2e97c35","source":{"kind":"arxiv","id":"2509.02883","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.02883","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"arxiv_version","alias_value":"2509.02883v1","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.02883","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"pith_short_12","alias_value":"DPNELFMXUYC5","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"pith_short_16","alias_value":"DPNELFMXUYC5UWGK","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"pith_short_8","alias_value":"DPNELFMX","created_at":"2026-07-05T12:03:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:DPNELFMXUYC5UWGKQ456JENL5Q","target":"record","payload":{"canonical_record":{"source":{"id":"2509.02883","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2025-09-02T23:06:07Z","cross_cats_sorted":["math.DG","math.MG"],"title_canon_sha256":"f080e39434944ba6a0c5603055962e58c648f844018b4b24b5c041bfb6380058","abstract_canon_sha256":"600facbd57f2feab36218f85c6f47a6ad9ffb3e6b77faff836b470be6abd3c40"},"schema_version":"1.0"},"canonical_sha256":"1bda459597a605da58ca873be491abec23787b3cd106a0180dadd348d2e97c35","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:03:59.196756Z","signature_b64":"ZdMp3aeENh/Bih0cjRIzGkX4UWWi9yTR7EF2OwwED6EblHk7rbJ3X5byTNf1iX7XpotrWNiA3XwRdI8d+UTKDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bda459597a605da58ca873be491abec23787b3cd106a0180dadd348d2e97c35","last_reissued_at":"2026-07-05T12:03:59.196255Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:03:59.196255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.02883","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":"2026-07-05T12:03:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dZ+BxGbCdVCEXC5kBYjX1vDazVSONE8APBLxmknQNk03lBLp/zGUSBVfXhycmP0CoGqsxQCiF+sTgc67Xv1BDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T20:23:25.056936Z"},"content_sha256":"68a473e0be68687261e5d3a27436f283fa34b35546379aeac1608b62a2304d98","schema_version":"1.0","event_id":"sha256:68a473e0be68687261e5d3a27436f283fa34b35546379aeac1608b62a2304d98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:DPNELFMXUYC5UWGKQ456JENL5Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Milnor invariants and thickness of spherical links","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.MG"],"primary_cat":"math.GT","authors_text":"Fedor Manin, Rafa{\\l} Komendarczyk, Robin Koytcheff","submitted_at":"2025-09-02T23:06:07Z","abstract_excerpt":"The ropelength of a knot or link is the minimal number of inches of 1-inch-thick rope that it takes to tie it. The relationship of this measurement to knot and link invariants has been studied by various authors. We give the first results of this type for higher-dimensional spherical links, generalizing work of the first author and Michaelides in the classical case. We find optimal asymptotic bounds on their Milnor invariants in terms of thickness, uncovering a dichotomy between a polynomial and an exponential regime. Along the way, we give a detailed treatment of these Milnor invariants and t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.02883","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/2509.02883/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":"2026-07-05T12:03:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S+VverGkvCgsH31OCDjT7pUz03nIVf5OreL/3ZKOgSlVXdmzOtnB+ljiKHgSHorjuCX1x7Q0A8RFQkLSaGPCBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T20:23:25.057434Z"},"content_sha256":"57993d18fe7202d52dfe32531be22ae25df767e73b623d9c44881b7bc1c0aaf9","schema_version":"1.0","event_id":"sha256:57993d18fe7202d52dfe32531be22ae25df767e73b623d9c44881b7bc1c0aaf9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DPNELFMXUYC5UWGKQ456JENL5Q/bundle.json","state_url":"https://pith.science/pith/DPNELFMXUYC5UWGKQ456JENL5Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DPNELFMXUYC5UWGKQ456JENL5Q/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":"2026-08-22T20:23:25Z","links":{"resolver":"https://pith.science/pith/DPNELFMXUYC5UWGKQ456JENL5Q","bundle":"https://pith.science/pith/DPNELFMXUYC5UWGKQ456JENL5Q/bundle.json","state":"https://pith.science/pith/DPNELFMXUYC5UWGKQ456JENL5Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DPNELFMXUYC5UWGKQ456JENL5Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:DPNELFMXUYC5UWGKQ456JENL5Q","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":"600facbd57f2feab36218f85c6f47a6ad9ffb3e6b77faff836b470be6abd3c40","cross_cats_sorted":["math.DG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2025-09-02T23:06:07Z","title_canon_sha256":"f080e39434944ba6a0c5603055962e58c648f844018b4b24b5c041bfb6380058"},"schema_version":"1.0","source":{"id":"2509.02883","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.02883","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"arxiv_version","alias_value":"2509.02883v1","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.02883","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"pith_short_12","alias_value":"DPNELFMXUYC5","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"pith_short_16","alias_value":"DPNELFMXUYC5UWGK","created_at":"2026-07-05T12:03:59Z"},{"alias_kind":"pith_short_8","alias_value":"DPNELFMX","created_at":"2026-07-05T12:03:59Z"}],"graph_snapshots":[{"event_id":"sha256:57993d18fe7202d52dfe32531be22ae25df767e73b623d9c44881b7bc1c0aaf9","target":"graph","created_at":"2026-07-05T12:03:59Z","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/2509.02883/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The ropelength of a knot or link is the minimal number of inches of 1-inch-thick rope that it takes to tie it. The relationship of this measurement to knot and link invariants has been studied by various authors. We give the first results of this type for higher-dimensional spherical links, generalizing work of the first author and Michaelides in the classical case. We find optimal asymptotic bounds on their Milnor invariants in terms of thickness, uncovering a dichotomy between a polynomial and an exponential regime. Along the way, we give a detailed treatment of these Milnor invariants and t","authors_text":"Fedor Manin, Rafa{\\l} Komendarczyk, Robin Koytcheff","cross_cats":["math.DG","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2025-09-02T23:06:07Z","title":"Milnor invariants and thickness of spherical links"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.02883","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":"sha256:68a473e0be68687261e5d3a27436f283fa34b35546379aeac1608b62a2304d98","target":"record","created_at":"2026-07-05T12:03:59Z","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":"600facbd57f2feab36218f85c6f47a6ad9ffb3e6b77faff836b470be6abd3c40","cross_cats_sorted":["math.DG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2025-09-02T23:06:07Z","title_canon_sha256":"f080e39434944ba6a0c5603055962e58c648f844018b4b24b5c041bfb6380058"},"schema_version":"1.0","source":{"id":"2509.02883","kind":"arxiv","version":1}},"canonical_sha256":"1bda459597a605da58ca873be491abec23787b3cd106a0180dadd348d2e97c35","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1bda459597a605da58ca873be491abec23787b3cd106a0180dadd348d2e97c35","first_computed_at":"2026-07-05T12:03:59.196255Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:03:59.196255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZdMp3aeENh/Bih0cjRIzGkX4UWWi9yTR7EF2OwwED6EblHk7rbJ3X5byTNf1iX7XpotrWNiA3XwRdI8d+UTKDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T12:03:59.196756Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.02883","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:68a473e0be68687261e5d3a27436f283fa34b35546379aeac1608b62a2304d98","sha256:57993d18fe7202d52dfe32531be22ae25df767e73b623d9c44881b7bc1c0aaf9"],"state_sha256":"b4c811860331bdd63059a4da6563a601ad7af52760e3adb2afd2beeb1503e4ed"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G0uAJiq9xi76VKByIcWhb7f2PLZBSKJfua+xOwZcefq+iCyLlfMD1ran5W4Hq7QF7tbeYaW5xrFhCcS/Lf7pDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T20:23:25.061376Z","bundle_sha256":"1c442fee4a7d83d3bdf6a96e53918b395c263190b346b1f4ad89fca209ac0d8a"}}