{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:DL6Q4EKGC7DMPOUEQZWVFVHLW4","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":"ae07d064a838a3fb25f618f1ed6814146d8c6e4b0973ba39e3a1ec501047a72c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-09-21T13:34:00Z","title_canon_sha256":"c764bbaf8b1c24bb33ef1cd2efd5b39ebf4feb5a0b938a756e7bbac1e2889b71"},"schema_version":"1.0","source":{"id":"1609.06552","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.06552","created_at":"2026-05-18T01:04:07Z"},{"alias_kind":"arxiv_version","alias_value":"1609.06552v1","created_at":"2026-05-18T01:04:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.06552","created_at":"2026-05-18T01:04:07Z"},{"alias_kind":"pith_short_12","alias_value":"DL6Q4EKGC7DM","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"DL6Q4EKGC7DMPOUE","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"DL6Q4EKG","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:0a381ee25098025407ebedc48f52df038721ca99d0ae8dd723f9a5500ce335ea","target":"graph","created_at":"2026-05-18T01:04:07Z","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":"If $S$ is a given regular $d$-simplex of edge length $a$ in the $d$-dimensional Euclidean space $\\mathcal{E}$, then the distances $t_1$, $\\cdots$, $t_{d+1}$ of an arbitrary point in $\\mathcal{E}$ to the vertices of $S$ are related by the elegant relation $$(d+1)\\left( a^4+t_1^4+\\cdots+t_{d+1}^4\\right)=\\left( a^2+t_1^2+\\cdots+t_{d+1}^2\\right)^2.$$ The purpose of this paper is to prove that this is essentially the only relation that exists among $t_1,\\cdots,t_{d+1}.$ The proof uses tools from analysis, algebra, and geometry.","authors_text":"Bach Nguyen, Mostafa Hayajneh, Mowaffaq Hajja, Shadi Shaqaqha","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-09-21T13:34:00Z","title":"Distances from the vertices of a regular simplex"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.06552","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":"sha256:3ece5e7b74d187b735c63c7c0255349877372af5c9bdf011549268ed1ef9117f","target":"record","created_at":"2026-05-18T01:04:07Z","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":"ae07d064a838a3fb25f618f1ed6814146d8c6e4b0973ba39e3a1ec501047a72c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-09-21T13:34:00Z","title_canon_sha256":"c764bbaf8b1c24bb33ef1cd2efd5b39ebf4feb5a0b938a756e7bbac1e2889b71"},"schema_version":"1.0","source":{"id":"1609.06552","kind":"arxiv","version":1}},"canonical_sha256":"1afd0e114617c6c7ba84866d52d4ebb70a46ad328064022ed68b26dc7bec3d3b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1afd0e114617c6c7ba84866d52d4ebb70a46ad328064022ed68b26dc7bec3d3b","first_computed_at":"2026-05-18T01:04:07.965707Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:07.965707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Cs8Idx2VB/eIDf0aN6Xq6ZLRy53lx/57OnPZCpTljcEhDLiox+JFvqMe9XN1LplHcK8CnhBx9gYWw07QjCQABA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:07.966529Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.06552","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ece5e7b74d187b735c63c7c0255349877372af5c9bdf011549268ed1ef9117f","sha256:0a381ee25098025407ebedc48f52df038721ca99d0ae8dd723f9a5500ce335ea"],"state_sha256":"8527cfa173b2f6814557cb26ac273be882c59e8b0666d73e5e5e84ed6f400171"}