{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:VPQXSIF7R2X3LWBBONAYYHITH2","short_pith_number":"pith:VPQXSIF7","canonical_record":{"source":{"id":"2408.02812","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-08-05T20:26:19Z","cross_cats_sorted":[],"title_canon_sha256":"ad299a50a41d87f97f70975a43e65be8aa8692dadfbf82aa297cbf539fa8b014","abstract_canon_sha256":"6655f264a824baad4596ebb8ad8a55dead77940f45bfa063a64f3b4a92325524"},"schema_version":"1.0"},"canonical_sha256":"abe17920bf8eafb5d82173418c1d133e99944e3d55761e5d11b35adeeadfc8d7","source":{"kind":"arxiv","id":"2408.02812","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.02812","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"arxiv_version","alias_value":"2408.02812v2","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.02812","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"pith_short_12","alias_value":"VPQXSIF7R2X3","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"pith_short_16","alias_value":"VPQXSIF7R2X3LWBB","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"pith_short_8","alias_value":"VPQXSIF7","created_at":"2026-06-09T02:07:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:VPQXSIF7R2X3LWBBONAYYHITH2","target":"record","payload":{"canonical_record":{"source":{"id":"2408.02812","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-08-05T20:26:19Z","cross_cats_sorted":[],"title_canon_sha256":"ad299a50a41d87f97f70975a43e65be8aa8692dadfbf82aa297cbf539fa8b014","abstract_canon_sha256":"6655f264a824baad4596ebb8ad8a55dead77940f45bfa063a64f3b4a92325524"},"schema_version":"1.0"},"canonical_sha256":"abe17920bf8eafb5d82173418c1d133e99944e3d55761e5d11b35adeeadfc8d7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:07:00.017895Z","signature_b64":"h1vAUPXXpcn4QH+seEVVUTnLEYl83CHsY2tcY/XvgSxSLc9tTD1NEQHBR2+vw5WZYyatOjza0pXJ1jPbkdcjCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abe17920bf8eafb5d82173418c1d133e99944e3d55761e5d11b35adeeadfc8d7","last_reissued_at":"2026-06-09T02:07:00.016862Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:07:00.016862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2408.02812","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":"2026-06-09T02:07:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fXg6xRUG8Xcm/Lok4Dh6AvZ63jN5L+Nszur/4bEYewknZvb+3ZpTIkq6vhI+K/bs8C3mguBUM6b59lYMybXeBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:15:16.619910Z"},"content_sha256":"efcc0bcd5cb279528f5b7daf816c36249a5f40315240bee1c12e5bb08a98ef84","schema_version":"1.0","event_id":"sha256:efcc0bcd5cb279528f5b7daf816c36249a5f40315240bee1c12e5bb08a98ef84"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:VPQXSIF7R2X3LWBBONAYYHITH2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Continuous Terminal Embeddings of Sets of Positive Reach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Mark Iwen, Rafael Chiclana, Simone Brugiapaglia, Tim Hoheisel","submitted_at":"2024-08-05T20:26:19Z","abstract_excerpt":"In this paper we prove the existence of H\\\"{o}lder continuous terminal embeddings of any desired $X \\subseteq \\mathbb{R}^d$ into $\\mathbb{R}^{m}$ with $m=\\mathcal{O}(\\varepsilon^{-2}\\omega(S_X)^2)$, for arbitrarily small distortion $\\varepsilon$, where $\\omega(S_X)$ denotes the Gaussian width of the unit secants of $X$. More specifically, when $X$ is a finite set we provide terminal embeddings that are locally $\\frac{1}{2}$-H\\\"{o}lder almost everywhere, and when $X$ is infinite with positive reach we give terminal embeddings that are locally $\\frac{1}{4}$-H\\\"{o}lder everywhere sufficiently clo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.02812","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.02812/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":"2026-06-09T02:07:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LpLmKR/lRKGcJlo4rHlwYKraor7AfeiLAiVx8lz67W5egpI84GwwEcfYzqeEOCcJyzGSdHR6Nl2SORiX3dguBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:15:16.620276Z"},"content_sha256":"6d7a2496204445f897897e33153d9e321b9bb7c396a31bb07cfe89e65c866965","schema_version":"1.0","event_id":"sha256:6d7a2496204445f897897e33153d9e321b9bb7c396a31bb07cfe89e65c866965"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VPQXSIF7R2X3LWBBONAYYHITH2/bundle.json","state_url":"https://pith.science/pith/VPQXSIF7R2X3LWBBONAYYHITH2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VPQXSIF7R2X3LWBBONAYYHITH2/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":"2026-06-25T08:15:16Z","links":{"resolver":"https://pith.science/pith/VPQXSIF7R2X3LWBBONAYYHITH2","bundle":"https://pith.science/pith/VPQXSIF7R2X3LWBBONAYYHITH2/bundle.json","state":"https://pith.science/pith/VPQXSIF7R2X3LWBBONAYYHITH2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VPQXSIF7R2X3LWBBONAYYHITH2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:VPQXSIF7R2X3LWBBONAYYHITH2","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":"6655f264a824baad4596ebb8ad8a55dead77940f45bfa063a64f3b4a92325524","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-08-05T20:26:19Z","title_canon_sha256":"ad299a50a41d87f97f70975a43e65be8aa8692dadfbf82aa297cbf539fa8b014"},"schema_version":"1.0","source":{"id":"2408.02812","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2408.02812","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"arxiv_version","alias_value":"2408.02812v2","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.02812","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"pith_short_12","alias_value":"VPQXSIF7R2X3","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"pith_short_16","alias_value":"VPQXSIF7R2X3LWBB","created_at":"2026-06-09T02:07:00Z"},{"alias_kind":"pith_short_8","alias_value":"VPQXSIF7","created_at":"2026-06-09T02:07:00Z"}],"graph_snapshots":[{"event_id":"sha256:6d7a2496204445f897897e33153d9e321b9bb7c396a31bb07cfe89e65c866965","target":"graph","created_at":"2026-06-09T02:07:00Z","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/2408.02812/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we prove the existence of H\\\"{o}lder continuous terminal embeddings of any desired $X \\subseteq \\mathbb{R}^d$ into $\\mathbb{R}^{m}$ with $m=\\mathcal{O}(\\varepsilon^{-2}\\omega(S_X)^2)$, for arbitrarily small distortion $\\varepsilon$, where $\\omega(S_X)$ denotes the Gaussian width of the unit secants of $X$. More specifically, when $X$ is a finite set we provide terminal embeddings that are locally $\\frac{1}{2}$-H\\\"{o}lder almost everywhere, and when $X$ is infinite with positive reach we give terminal embeddings that are locally $\\frac{1}{4}$-H\\\"{o}lder everywhere sufficiently clo","authors_text":"Mark Iwen, Rafael Chiclana, Simone Brugiapaglia, Tim Hoheisel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-08-05T20:26:19Z","title":"On Continuous Terminal Embeddings of Sets of Positive Reach"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.02812","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:efcc0bcd5cb279528f5b7daf816c36249a5f40315240bee1c12e5bb08a98ef84","target":"record","created_at":"2026-06-09T02:07:00Z","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":"6655f264a824baad4596ebb8ad8a55dead77940f45bfa063a64f3b4a92325524","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2024-08-05T20:26:19Z","title_canon_sha256":"ad299a50a41d87f97f70975a43e65be8aa8692dadfbf82aa297cbf539fa8b014"},"schema_version":"1.0","source":{"id":"2408.02812","kind":"arxiv","version":2}},"canonical_sha256":"abe17920bf8eafb5d82173418c1d133e99944e3d55761e5d11b35adeeadfc8d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"abe17920bf8eafb5d82173418c1d133e99944e3d55761e5d11b35adeeadfc8d7","first_computed_at":"2026-06-09T02:07:00.016862Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T02:07:00.016862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h1vAUPXXpcn4QH+seEVVUTnLEYl83CHsY2tcY/XvgSxSLc9tTD1NEQHBR2+vw5WZYyatOjza0pXJ1jPbkdcjCw==","signature_status":"signed_v1","signed_at":"2026-06-09T02:07:00.017895Z","signed_message":"canonical_sha256_bytes"},"source_id":"2408.02812","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:efcc0bcd5cb279528f5b7daf816c36249a5f40315240bee1c12e5bb08a98ef84","sha256:6d7a2496204445f897897e33153d9e321b9bb7c396a31bb07cfe89e65c866965"],"state_sha256":"292f841f0d57db2b1536a5790bb0cbbb7c847e4a3c731fa260ecb2bd3ad43216"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vLh9yyIYqbVB/2TFHG7K/AU2yymLM2MIItD2BgzOE5e9Rz0Yjj6z5KZ5PZiWEgUqtFLynGP9ZAYI0P9ww74CDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T08:15:16.622226Z","bundle_sha256":"7c6a64126a0630360041bf89494996bb2914e040e685c5995d96894ba09db4c8"}}