{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NTNBOYH245K33AFSSPCCUVP52L","short_pith_number":"pith:NTNBOYH2","canonical_record":{"source":{"id":"2607.26363","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.MG","submitted_at":"2026-07-29T00:40:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"78516b3efd62d082d839bf9927286be2eab941a2e3f1fb64b97a8bf74728f245","abstract_canon_sha256":"36ac2adbf81f61fb24653ff851094afb13b004871f515e4d31e54d9849cd5a13"},"schema_version":"1.0"},"canonical_sha256":"6cda1760fae755bd80b293c42a55fdd2fa21d2abb65cd381789986efa4407114","source":{"kind":"arxiv","id":"2607.26363","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26363","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26363v1","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26363","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"pith_short_12","alias_value":"NTNBOYH245K3","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"pith_short_16","alias_value":"NTNBOYH245K33AFS","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"pith_short_8","alias_value":"NTNBOYH2","created_at":"2026-07-30T01:18:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NTNBOYH245K33AFSSPCCUVP52L","target":"record","payload":{"canonical_record":{"source":{"id":"2607.26363","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.MG","submitted_at":"2026-07-29T00:40:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"78516b3efd62d082d839bf9927286be2eab941a2e3f1fb64b97a8bf74728f245","abstract_canon_sha256":"36ac2adbf81f61fb24653ff851094afb13b004871f515e4d31e54d9849cd5a13"},"schema_version":"1.0"},"canonical_sha256":"6cda1760fae755bd80b293c42a55fdd2fa21d2abb65cd381789986efa4407114","receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6cda1760fae755bd80b293c42a55fdd2fa21d2abb65cd381789986efa4407114","last_reissued_at":"2026-07-30T01:18:12.526780Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:18:12.526780Z"},"source_kind":"arxiv","source_id":"2607.26363","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":"2026-07-30T01:18:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U8X0frdlDtoca20VFnK13awRnjUsmEqdp3aBWZJa0ibtpXmpd6vlATsp0sbHJ6Z8h6K+g6HbVhLYS4s12tKlCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T00:03:27.010720Z"},"content_sha256":"484d78e317f8cc3257cd1ca3b143b6a7146bdbb5e145f71d4138b6a927de43e8","schema_version":"1.0","event_id":"sha256:484d78e317f8cc3257cd1ca3b143b6a7146bdbb5e145f71d4138b6a927de43e8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NTNBOYH245K33AFSSPCCUVP52L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Neoplatonic solids","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Matthew Ellison, Peter Doyle","submitted_at":"2026-07-29T00:40:20Z","abstract_excerpt":"A \\emph{6-net} is a simplicial triangulation of the $2$-sphere with maximum degree $\\leq 6$. Experiments suggest that every $6$-net admits a unique realization as an undented Euclidean polyhedron built from unit equilateral triangles, and a unique realization as an ideal equilateral hyperbolic polyhedron. We call these \\emph{neoplatonic solids} and \\emph{ideal neoplatonics}.\n  A net is \\emph{prime} if every 3-cycle bounds a face. A computer-assisted proof shows that every prime $6$-net with $v \\leq 50$ has a unique realization as a convex ideal neoplatonic. Numerical homotopy from this realiza"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26363","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/2607.26363/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":"2026-07-30T01:18:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qL/5GPNjnYOLIJISCwaN5QzQ8hkYoYI5C9IiFPr9a6sDn5bdZGRtSRkB7Xq5BPxy/yvaJaZ6Gl3TbcQ9U34dAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T00:03:27.011216Z"},"content_sha256":"5adbc48e62fedb67c5bc2c0e69ee80263987bae811c1ac39e698c644c6b33ccf","schema_version":"1.0","event_id":"sha256:5adbc48e62fedb67c5bc2c0e69ee80263987bae811c1ac39e698c644c6b33ccf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NTNBOYH245K33AFSSPCCUVP52L/bundle.json","state_url":"https://pith.science/pith/NTNBOYH245K33AFSSPCCUVP52L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NTNBOYH245K33AFSSPCCUVP52L/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":"2026-08-21T00:03:27Z","links":{"resolver":"https://pith.science/pith/NTNBOYH245K33AFSSPCCUVP52L","bundle":"https://pith.science/pith/NTNBOYH245K33AFSSPCCUVP52L/bundle.json","state":"https://pith.science/pith/NTNBOYH245K33AFSSPCCUVP52L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NTNBOYH245K33AFSSPCCUVP52L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NTNBOYH245K33AFSSPCCUVP52L","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":"36ac2adbf81f61fb24653ff851094afb13b004871f515e4d31e54d9849cd5a13","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.MG","submitted_at":"2026-07-29T00:40:20Z","title_canon_sha256":"78516b3efd62d082d839bf9927286be2eab941a2e3f1fb64b97a8bf74728f245"},"schema_version":"1.0","source":{"id":"2607.26363","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26363","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26363v1","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26363","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"pith_short_12","alias_value":"NTNBOYH245K3","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"pith_short_16","alias_value":"NTNBOYH245K33AFS","created_at":"2026-07-30T01:18:12Z"},{"alias_kind":"pith_short_8","alias_value":"NTNBOYH2","created_at":"2026-07-30T01:18:12Z"}],"graph_snapshots":[{"event_id":"sha256:5adbc48e62fedb67c5bc2c0e69ee80263987bae811c1ac39e698c644c6b33ccf","target":"graph","created_at":"2026-07-30T01:18:12Z","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/2607.26363/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A \\emph{6-net} is a simplicial triangulation of the $2$-sphere with maximum degree $\\leq 6$. Experiments suggest that every $6$-net admits a unique realization as an undented Euclidean polyhedron built from unit equilateral triangles, and a unique realization as an ideal equilateral hyperbolic polyhedron. We call these \\emph{neoplatonic solids} and \\emph{ideal neoplatonics}.\n  A net is \\emph{prime} if every 3-cycle bounds a face. A computer-assisted proof shows that every prime $6$-net with $v \\leq 50$ has a unique realization as a convex ideal neoplatonic. Numerical homotopy from this realiza","authors_text":"Matthew Ellison, Peter Doyle","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.MG","submitted_at":"2026-07-29T00:40:20Z","title":"Neoplatonic solids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26363","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":"sha256:484d78e317f8cc3257cd1ca3b143b6a7146bdbb5e145f71d4138b6a927de43e8","target":"record","created_at":"2026-07-30T01:18:12Z","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":"36ac2adbf81f61fb24653ff851094afb13b004871f515e4d31e54d9849cd5a13","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.MG","submitted_at":"2026-07-29T00:40:20Z","title_canon_sha256":"78516b3efd62d082d839bf9927286be2eab941a2e3f1fb64b97a8bf74728f245"},"schema_version":"1.0","source":{"id":"2607.26363","kind":"arxiv","version":1}},"canonical_sha256":"6cda1760fae755bd80b293c42a55fdd2fa21d2abb65cd381789986efa4407114","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6cda1760fae755bd80b293c42a55fdd2fa21d2abb65cd381789986efa4407114","first_computed_at":"2026-07-30T01:18:12.526780Z","kind":"pith_receipt","last_reissued_at":"2026-07-30T01:18:12.526780Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2607.26363","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:484d78e317f8cc3257cd1ca3b143b6a7146bdbb5e145f71d4138b6a927de43e8","sha256:5adbc48e62fedb67c5bc2c0e69ee80263987bae811c1ac39e698c644c6b33ccf"],"state_sha256":"cc91bc9eec8066e24c582af8031c78686cb76a6ff842e5d2bab5d239cb9befcf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xCtA8g5Bq0yKassPXRsIBpiiK3cZkYg3SRn+yWEhGnakqmrfxgL6PoeGmARo9qlnweAR0s/otXiLvSOau17iCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T00:03:27.015204Z","bundle_sha256":"95bc7f2cd59c389ff621e412baa095cbaa6aaa6cc6f08f837f1ff36745c3a3e5"}}