{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:PULIL7UIOEMT7GJA7SGKM7MYF5","short_pith_number":"pith:PULIL7UI","canonical_record":{"source":{"id":"2605.15031","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-14T16:27:06Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"7de0a153954adacefbc2741926036bd3f9abc39656b8d3a2707fdd56e4c9b78f","abstract_canon_sha256":"072c08411ec8594960ad27691be51e1dbc96bfb542d3e20bdab5b7d1547279b4"},"schema_version":"1.0"},"canonical_sha256":"7d1685fe8871193f9920fc8ca67d982f6d09fbcc1e2146b895ad916c354fa0cf","source":{"kind":"arxiv","id":"2605.15031","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.15031","created_at":"2026-05-17T23:38:54Z"},{"alias_kind":"arxiv_version","alias_value":"2605.15031v1","created_at":"2026-05-17T23:38:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.15031","created_at":"2026-05-17T23:38:54Z"},{"alias_kind":"pith_short_12","alias_value":"PULIL7UIOEMT","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"PULIL7UIOEMT7GJA","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"PULIL7UI","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:PULIL7UIOEMT7GJA7SGKM7MYF5","target":"record","payload":{"canonical_record":{"source":{"id":"2605.15031","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-14T16:27:06Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"7de0a153954adacefbc2741926036bd3f9abc39656b8d3a2707fdd56e4c9b78f","abstract_canon_sha256":"072c08411ec8594960ad27691be51e1dbc96bfb542d3e20bdab5b7d1547279b4"},"schema_version":"1.0"},"canonical_sha256":"7d1685fe8871193f9920fc8ca67d982f6d09fbcc1e2146b895ad916c354fa0cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:38:54.583469Z","signature_b64":"uY6fwMwnDhptLFooGTOiVf0+u3/AT5oFZmw1ygNn4wACGvqnNKIQeE5U0IZWKb6wRjjacWuCSWWgasI/EmXZCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d1685fe8871193f9920fc8ca67d982f6d09fbcc1e2146b895ad916c354fa0cf","last_reissued_at":"2026-05-17T23:38:54.582768Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:38:54.582768Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.15031","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":"2026-05-17T23:38:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KqFS0+ycdU+WW4dv0HdEZ/JAEs0jGOoLKwwsmMFS5bC4mgIOwepIvGoSUXYt9S8e/2cZcB3vR79AbZYVu0ayDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T23:18:24.674882Z"},"content_sha256":"b62ccd1e2d09f5bb94dc7dcec0dfbe6791383439b2f0d29811e1690339d06b80","schema_version":"1.0","event_id":"sha256:b62ccd1e2d09f5bb94dc7dcec0dfbe6791383439b2f0d29811e1690339d06b80"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:PULIL7UIOEMT7GJA7SGKM7MYF5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal submanifolds confined in space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Tobias Holck Colding, William P. Minicozzi II","submitted_at":"2026-05-14T16:27:06Z","abstract_excerpt":"Already in $\\bf{R}^4$, there are many known examples of minimal hypersurfaces, yet few structural results. We show that minimal submanifolds, of any dimension, that are confined in space are very restricted. It is well-known that the half-space theorem fails already for hypersurfaces in $\\bf{R}^4$, where there are many examples contained in a slab. In $\\bf{R}^3$ the height of the catenoid grows at a logarithmic rate, whereas in higher dimension the height of the catenoid remains bounded. We will see that even in high dimensions, minimal submanifolds that are confined in space must satisfy stro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.15031","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":"2026-05-17T23:38:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KDdm1Ebvrfp2QgizcUjLMhOraD7lKAjpvHjQyCyxoJLXR1nnFIvQCFiAXSVnOsBYGZkj8g2lMPO2p0sE526FAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T23:18:24.675482Z"},"content_sha256":"33184c2caa5aaab375927af6fc2df6e22532933526145326b359dbe92b73727f","schema_version":"1.0","event_id":"sha256:33184c2caa5aaab375927af6fc2df6e22532933526145326b359dbe92b73727f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PULIL7UIOEMT7GJA7SGKM7MYF5/bundle.json","state_url":"https://pith.science/pith/PULIL7UIOEMT7GJA7SGKM7MYF5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PULIL7UIOEMT7GJA7SGKM7MYF5/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":"2026-05-26T23:18:24Z","links":{"resolver":"https://pith.science/pith/PULIL7UIOEMT7GJA7SGKM7MYF5","bundle":"https://pith.science/pith/PULIL7UIOEMT7GJA7SGKM7MYF5/bundle.json","state":"https://pith.science/pith/PULIL7UIOEMT7GJA7SGKM7MYF5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PULIL7UIOEMT7GJA7SGKM7MYF5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:PULIL7UIOEMT7GJA7SGKM7MYF5","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":"072c08411ec8594960ad27691be51e1dbc96bfb542d3e20bdab5b7d1547279b4","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-14T16:27:06Z","title_canon_sha256":"7de0a153954adacefbc2741926036bd3f9abc39656b8d3a2707fdd56e4c9b78f"},"schema_version":"1.0","source":{"id":"2605.15031","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.15031","created_at":"2026-05-17T23:38:54Z"},{"alias_kind":"arxiv_version","alias_value":"2605.15031v1","created_at":"2026-05-17T23:38:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.15031","created_at":"2026-05-17T23:38:54Z"},{"alias_kind":"pith_short_12","alias_value":"PULIL7UIOEMT","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"PULIL7UIOEMT7GJA","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"PULIL7UI","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:33184c2caa5aaab375927af6fc2df6e22532933526145326b359dbe92b73727f","target":"graph","created_at":"2026-05-17T23:38:54Z","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":"Already in $\\bf{R}^4$, there are many known examples of minimal hypersurfaces, yet few structural results. We show that minimal submanifolds, of any dimension, that are confined in space are very restricted. It is well-known that the half-space theorem fails already for hypersurfaces in $\\bf{R}^4$, where there are many examples contained in a slab. In $\\bf{R}^3$ the height of the catenoid grows at a logarithmic rate, whereas in higher dimension the height of the catenoid remains bounded. We will see that even in high dimensions, minimal submanifolds that are confined in space must satisfy stro","authors_text":"Tobias Holck Colding, William P. Minicozzi II","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-14T16:27:06Z","title":"Minimal submanifolds confined in space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.15031","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:b62ccd1e2d09f5bb94dc7dcec0dfbe6791383439b2f0d29811e1690339d06b80","target":"record","created_at":"2026-05-17T23:38:54Z","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":"072c08411ec8594960ad27691be51e1dbc96bfb542d3e20bdab5b7d1547279b4","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-14T16:27:06Z","title_canon_sha256":"7de0a153954adacefbc2741926036bd3f9abc39656b8d3a2707fdd56e4c9b78f"},"schema_version":"1.0","source":{"id":"2605.15031","kind":"arxiv","version":1}},"canonical_sha256":"7d1685fe8871193f9920fc8ca67d982f6d09fbcc1e2146b895ad916c354fa0cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7d1685fe8871193f9920fc8ca67d982f6d09fbcc1e2146b895ad916c354fa0cf","first_computed_at":"2026-05-17T23:38:54.582768Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:38:54.582768Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uY6fwMwnDhptLFooGTOiVf0+u3/AT5oFZmw1ygNn4wACGvqnNKIQeE5U0IZWKb6wRjjacWuCSWWgasI/EmXZCQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:38:54.583469Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.15031","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b62ccd1e2d09f5bb94dc7dcec0dfbe6791383439b2f0d29811e1690339d06b80","sha256:33184c2caa5aaab375927af6fc2df6e22532933526145326b359dbe92b73727f"],"state_sha256":"b0af2e563195672a9ce24d589635b863c6a2b008cfa7baf1576be43c384ce67c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yLMtfBaqw0I3gasdEHB8fHCFLSdVuuELH04B0RaXGJLzYAjMfZ0MR3mQJmc4FO+tj74hg7m7xTklN0iyU84iBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T23:18:24.677864Z","bundle_sha256":"dec569972f085da5fa6061a21f3b81366d1e545eeb8731603ea1224c65932f6a"}}