{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:ZBVILGASHLRMO2ASGTKU5HMTYV","short_pith_number":"pith:ZBVILGAS","canonical_record":{"source":{"id":"0710.3857","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"2007-10-20T17:09:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5bb3157fff6e09b4e07f1c800647176f4fd64819cb9bc06a410df9c4d5c5130e","abstract_canon_sha256":"6b6586db9d4de4f1717b077ed9e30d2c8600e5ce066d03ee9cb84e04e133c477"},"schema_version":"1.0"},"canonical_sha256":"c86a8598123ae2c7681234d54e9d93c557a2f960f884040d0538014b07a76c8e","source":{"kind":"arxiv","id":"0710.3857","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.3857","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"arxiv_version","alias_value":"0710.3857v2","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.3857","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"pith_short_12","alias_value":"ZBVILGASHLRM","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"ZBVILGASHLRMO2AS","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"ZBVILGAS","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:ZBVILGASHLRMO2ASGTKU5HMTYV","target":"record","payload":{"canonical_record":{"source":{"id":"0710.3857","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"2007-10-20T17:09:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5bb3157fff6e09b4e07f1c800647176f4fd64819cb9bc06a410df9c4d5c5130e","abstract_canon_sha256":"6b6586db9d4de4f1717b077ed9e30d2c8600e5ce066d03ee9cb84e04e133c477"},"schema_version":"1.0"},"canonical_sha256":"c86a8598123ae2c7681234d54e9d93c557a2f960f884040d0538014b07a76c8e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:46.904588Z","signature_b64":"zdYAC2MzLAS+thOwyU8FffaGvSFyocZ5O5MYYMage6pgGf8F8WQ34wJFbGq7HEdtN4IT6Q88FOd4LLwheQC8Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c86a8598123ae2c7681234d54e9d93c557a2f960f884040d0538014b07a76c8e","last_reissued_at":"2026-05-18T02:42:46.904151Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:46.904151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0710.3857","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":"2026-05-18T02:42:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sArUJ1LTmgIlvf+V0XCzkdoKnYJwY+mEmL8PZS9OpGyxmUihNljtmSSpjYiLT/fS2wAaUvFTvkocHAH3vfGyCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T18:01:36.503957Z"},"content_sha256":"fa937253fd36490d89ea0e2c67fcf2eee229997f5e4ed60d0fe77a46cab222fb","schema_version":"1.0","event_id":"sha256:fa937253fd36490d89ea0e2c67fcf2eee229997f5e4ed60d0fe77a46cab222fb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:ZBVILGASHLRMO2ASGTKU5HMTYV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalizations of Sch\\\"{o}bi's Tetrahedral Dissection","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"N. J. A. Sloane, Vinay A. Vaishampayan","submitted_at":"2007-10-20T17:09:20Z","abstract_excerpt":"Let v_1, ..., v_n be unit vectors in R^n such that v_i . v_j = -w for i != j, where -1 <w < 1/(n-1). The points Sum_{i=1..n} lambda_i v_i, where 1 >= lambda_1 >= ... >= lambda_n >= 0, form a ``Hill-simplex of the first type'', denoted by Q_n(w). It was shown by Hadwiger in 1951 that Q_n(w) is equidissectable with a cube. In 1985, Sch\\\"{o}bi gave a three-piece dissection of Q_3(w) into a triangular prism c Q_2(1/2) X I, where I denotes an interval and c = sqrt{2(w+1)/3}. The present paper generalizes Sch\\\"{o}bi's dissection to an n-piece dissection of Q_n(w) into a prism c Q_{n-1}(1/(n-1)) X I,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.3857","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":""},"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":"2026-05-18T02:42:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wgUkhO7pOgN2TvRQfmNwU4xgK0sJOigVjp6UPS7b5Smr11m40VAsDbFixKGM8ZlrExidWao0+5FaM3LAFvabDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T18:01:36.504304Z"},"content_sha256":"5fde5c98641ec97f370892b18ae6c31d6bab59c706b53c9952870ec39f79e706","schema_version":"1.0","event_id":"sha256:5fde5c98641ec97f370892b18ae6c31d6bab59c706b53c9952870ec39f79e706"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZBVILGASHLRMO2ASGTKU5HMTYV/bundle.json","state_url":"https://pith.science/pith/ZBVILGASHLRMO2ASGTKU5HMTYV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZBVILGASHLRMO2ASGTKU5HMTYV/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":"2026-06-20T18:01:36Z","links":{"resolver":"https://pith.science/pith/ZBVILGASHLRMO2ASGTKU5HMTYV","bundle":"https://pith.science/pith/ZBVILGASHLRMO2ASGTKU5HMTYV/bundle.json","state":"https://pith.science/pith/ZBVILGASHLRMO2ASGTKU5HMTYV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZBVILGASHLRMO2ASGTKU5HMTYV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:ZBVILGASHLRMO2ASGTKU5HMTYV","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":"6b6586db9d4de4f1717b077ed9e30d2c8600e5ce066d03ee9cb84e04e133c477","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.MG","submitted_at":"2007-10-20T17:09:20Z","title_canon_sha256":"5bb3157fff6e09b4e07f1c800647176f4fd64819cb9bc06a410df9c4d5c5130e"},"schema_version":"1.0","source":{"id":"0710.3857","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0710.3857","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"arxiv_version","alias_value":"0710.3857v2","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0710.3857","created_at":"2026-05-18T02:42:46Z"},{"alias_kind":"pith_short_12","alias_value":"ZBVILGASHLRM","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"ZBVILGASHLRMO2AS","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"ZBVILGAS","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:5fde5c98641ec97f370892b18ae6c31d6bab59c706b53c9952870ec39f79e706","target":"graph","created_at":"2026-05-18T02:42:46Z","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":"Let v_1, ..., v_n be unit vectors in R^n such that v_i . v_j = -w for i != j, where -1 <w < 1/(n-1). The points Sum_{i=1..n} lambda_i v_i, where 1 >= lambda_1 >= ... >= lambda_n >= 0, form a ``Hill-simplex of the first type'', denoted by Q_n(w). It was shown by Hadwiger in 1951 that Q_n(w) is equidissectable with a cube. In 1985, Sch\\\"{o}bi gave a three-piece dissection of Q_3(w) into a triangular prism c Q_2(1/2) X I, where I denotes an interval and c = sqrt{2(w+1)/3}. The present paper generalizes Sch\\\"{o}bi's dissection to an n-piece dissection of Q_n(w) into a prism c Q_{n-1}(1/(n-1)) X I,","authors_text":"N. J. A. Sloane, Vinay A. Vaishampayan","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.MG","submitted_at":"2007-10-20T17:09:20Z","title":"Generalizations of Sch\\\"{o}bi's Tetrahedral Dissection"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0710.3857","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:fa937253fd36490d89ea0e2c67fcf2eee229997f5e4ed60d0fe77a46cab222fb","target":"record","created_at":"2026-05-18T02:42:46Z","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":"6b6586db9d4de4f1717b077ed9e30d2c8600e5ce066d03ee9cb84e04e133c477","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.MG","submitted_at":"2007-10-20T17:09:20Z","title_canon_sha256":"5bb3157fff6e09b4e07f1c800647176f4fd64819cb9bc06a410df9c4d5c5130e"},"schema_version":"1.0","source":{"id":"0710.3857","kind":"arxiv","version":2}},"canonical_sha256":"c86a8598123ae2c7681234d54e9d93c557a2f960f884040d0538014b07a76c8e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c86a8598123ae2c7681234d54e9d93c557a2f960f884040d0538014b07a76c8e","first_computed_at":"2026-05-18T02:42:46.904151Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:42:46.904151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zdYAC2MzLAS+thOwyU8FffaGvSFyocZ5O5MYYMage6pgGf8F8WQ34wJFbGq7HEdtN4IT6Q88FOd4LLwheQC8Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:42:46.904588Z","signed_message":"canonical_sha256_bytes"},"source_id":"0710.3857","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa937253fd36490d89ea0e2c67fcf2eee229997f5e4ed60d0fe77a46cab222fb","sha256:5fde5c98641ec97f370892b18ae6c31d6bab59c706b53c9952870ec39f79e706"],"state_sha256":"1476fb4da3fd731f1bffa63c6b8ef1fd5bf0cb4dcfa3962d1b268a9ae84175c5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wa8H+9KzpIZw3HTXoLy18W03xVfh21slaBAn6AFgzIM3aaZ5u380AdUKT/RXf6oUzE7UzPM6yKfIWYlTt8/QDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T18:01:36.506250Z","bundle_sha256":"dde2967a52026013a42e2ffb3d9252ca7fc6198146834477110be9c283ab0f25"}}