{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:CJKXQOWV2NYTHNBLLAR5LHB2H2","short_pith_number":"pith:CJKXQOWV","canonical_record":{"source":{"id":"2606.18569","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2026-06-17T00:41:19Z","cross_cats_sorted":[],"title_canon_sha256":"ed523f3f65141613f09ce5bc0f9fe9568c604810c8e5afaf6bb57913dcb277f7","abstract_canon_sha256":"aecc4a3d0c568ea2fe8c95406fc73ef36248dc4d780c103c2f0bb1ad51e13a50"},"schema_version":"1.0"},"canonical_sha256":"1255783ad5d37133b42b5823d59c3a3ebe53c9b2f059ea9f9a9ce2f69d429afb","source":{"kind":"arxiv","id":"2606.18569","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.18569","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.18569v1","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.18569","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"pith_short_12","alias_value":"CJKXQOWV2NYT","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"pith_short_16","alias_value":"CJKXQOWV2NYTHNBL","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"pith_short_8","alias_value":"CJKXQOWV","created_at":"2026-06-19T16:11:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:CJKXQOWV2NYTHNBLLAR5LHB2H2","target":"record","payload":{"canonical_record":{"source":{"id":"2606.18569","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2026-06-17T00:41:19Z","cross_cats_sorted":[],"title_canon_sha256":"ed523f3f65141613f09ce5bc0f9fe9568c604810c8e5afaf6bb57913dcb277f7","abstract_canon_sha256":"aecc4a3d0c568ea2fe8c95406fc73ef36248dc4d780c103c2f0bb1ad51e13a50"},"schema_version":"1.0"},"canonical_sha256":"1255783ad5d37133b42b5823d59c3a3ebe53c9b2f059ea9f9a9ce2f69d429afb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:11:41.138371Z","signature_b64":"yxUzmtLtDTYYRWzEbkFg6+TJQbEb9yzeQkmgsb8NG4/GrDSzQt5CgF1oYp67+cR1rap/GuCYwuMb6/1rIn92Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1255783ad5d37133b42b5823d59c3a3ebe53c9b2f059ea9f9a9ce2f69d429afb","last_reissued_at":"2026-06-19T16:11:41.138022Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:11:41.138022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.18569","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":"2026-06-19T16:11:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qc1sQ161JuUW8SUGKh+vlVcrAekPtlKQVREoyHqWRJDxEopj5oR3ZLQnnsGVkzyUOqiXIUYMBr9Y5VlZzifWCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-13T15:07:23.293765Z"},"content_sha256":"591d17737463d4484b37c2937c7e190ff4ddc2727ef263ea629195a251911630","schema_version":"1.0","event_id":"sha256:591d17737463d4484b37c2937c7e190ff4ddc2727ef263ea629195a251911630"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:CJKXQOWV2NYTHNBLLAR5LHB2H2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tangent Spheres and Integer Distances","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"David Eppstein","submitted_at":"2026-06-17T00:41:19Z","abstract_excerpt":"The Erd\\H{o}s-Anning theorem states that any point set for which all distances are integers, in a Euclidean space of any dimension, must be either finite or collinear. We prove the same result in hyperbolic space of any dimension. A quantitative form of our result also extends for the first time to Euclidean spaces of dimension greater than two: if a set of points with integer distances in $\\mathbb{E}^D$ or $\\mathbb{H}^D$ has a subset of $D+1$ points in general position whose diameter is $d$, then the whole set has size $O(D(d+1)^D)$. To prove these results we formulate a lemma that, if the gr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18569","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/2606.18569/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":"2026-06-19T16:11:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZCTDSml0Kcz0hQUIijHh7Nn+aGlEsA6WPoOvH2vSgSS7cDUFH9kGPvASxsm2GxVXOe8YqilkKrjXExNvdjyyBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-13T15:07:23.294123Z"},"content_sha256":"ba19322cb03bb8865935f79806c22db6225254cfc7b8951bf36682b121fba825","schema_version":"1.0","event_id":"sha256:ba19322cb03bb8865935f79806c22db6225254cfc7b8951bf36682b121fba825"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CJKXQOWV2NYTHNBLLAR5LHB2H2/bundle.json","state_url":"https://pith.science/pith/CJKXQOWV2NYTHNBLLAR5LHB2H2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CJKXQOWV2NYTHNBLLAR5LHB2H2/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":"2026-07-13T15:07:23Z","links":{"resolver":"https://pith.science/pith/CJKXQOWV2NYTHNBLLAR5LHB2H2","bundle":"https://pith.science/pith/CJKXQOWV2NYTHNBLLAR5LHB2H2/bundle.json","state":"https://pith.science/pith/CJKXQOWV2NYTHNBLLAR5LHB2H2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CJKXQOWV2NYTHNBLLAR5LHB2H2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:CJKXQOWV2NYTHNBLLAR5LHB2H2","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":"aecc4a3d0c568ea2fe8c95406fc73ef36248dc4d780c103c2f0bb1ad51e13a50","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2026-06-17T00:41:19Z","title_canon_sha256":"ed523f3f65141613f09ce5bc0f9fe9568c604810c8e5afaf6bb57913dcb277f7"},"schema_version":"1.0","source":{"id":"2606.18569","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.18569","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.18569v1","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.18569","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"pith_short_12","alias_value":"CJKXQOWV2NYT","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"pith_short_16","alias_value":"CJKXQOWV2NYTHNBL","created_at":"2026-06-19T16:11:41Z"},{"alias_kind":"pith_short_8","alias_value":"CJKXQOWV","created_at":"2026-06-19T16:11:41Z"}],"graph_snapshots":[{"event_id":"sha256:ba19322cb03bb8865935f79806c22db6225254cfc7b8951bf36682b121fba825","target":"graph","created_at":"2026-06-19T16:11:41Z","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/2606.18569/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Erd\\H{o}s-Anning theorem states that any point set for which all distances are integers, in a Euclidean space of any dimension, must be either finite or collinear. We prove the same result in hyperbolic space of any dimension. A quantitative form of our result also extends for the first time to Euclidean spaces of dimension greater than two: if a set of points with integer distances in $\\mathbb{E}^D$ or $\\mathbb{H}^D$ has a subset of $D+1$ points in general position whose diameter is $d$, then the whole set has size $O(D(d+1)^D)$. To prove these results we formulate a lemma that, if the gr","authors_text":"David Eppstein","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2026-06-17T00:41:19Z","title":"Tangent Spheres and Integer Distances"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.18569","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":"sha256:591d17737463d4484b37c2937c7e190ff4ddc2727ef263ea629195a251911630","target":"record","created_at":"2026-06-19T16:11:41Z","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":"aecc4a3d0c568ea2fe8c95406fc73ef36248dc4d780c103c2f0bb1ad51e13a50","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2026-06-17T00:41:19Z","title_canon_sha256":"ed523f3f65141613f09ce5bc0f9fe9568c604810c8e5afaf6bb57913dcb277f7"},"schema_version":"1.0","source":{"id":"2606.18569","kind":"arxiv","version":1}},"canonical_sha256":"1255783ad5d37133b42b5823d59c3a3ebe53c9b2f059ea9f9a9ce2f69d429afb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1255783ad5d37133b42b5823d59c3a3ebe53c9b2f059ea9f9a9ce2f69d429afb","first_computed_at":"2026-06-19T16:11:41.138022Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:11:41.138022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yxUzmtLtDTYYRWzEbkFg6+TJQbEb9yzeQkmgsb8NG4/GrDSzQt5CgF1oYp67+cR1rap/GuCYwuMb6/1rIn92Dg==","signature_status":"signed_v1","signed_at":"2026-06-19T16:11:41.138371Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.18569","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:591d17737463d4484b37c2937c7e190ff4ddc2727ef263ea629195a251911630","sha256:ba19322cb03bb8865935f79806c22db6225254cfc7b8951bf36682b121fba825"],"state_sha256":"5ab12bfb47c7a259b67e3fcce573cac9f79252c017711bb7456fc543479d9b89"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QQekARPRhO1wNLgfS45h5jtczjXv+Xmye+loHwcMCE0iu2ftkvsQKS1j5K4NdHyZ4VmN7qNPi8LBBHmVReC5CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-13T15:07:23.296120Z","bundle_sha256":"feb474708a9cda62d8e99f7405fa461b899af1ee7daebae4e43d9891d8b13e4f"}}