{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:GF53I3WF3YJF6K6E4KOJFYJDJQ","short_pith_number":"pith:GF53I3WF","canonical_record":{"source":{"id":"2607.02130","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2026-07-02T13:06:56Z","cross_cats_sorted":["cs.CG","math.GR"],"title_canon_sha256":"4b349fa1a54ba1dbd819d8ee93908e576f83f7038134efd68f428e8aa163ce68","abstract_canon_sha256":"6c73833d02f9b7ff75339d11bbe9b86b4eec3d913ce34f8d2da2256db0c07613"},"schema_version":"1.0"},"canonical_sha256":"317bb46ec5de125f2bc4e29c92e1234c02d783b99b29f29242d4f0d21a49f91d","source":{"kind":"arxiv","id":"2607.02130","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02130","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02130v1","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02130","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"pith_short_12","alias_value":"GF53I3WF3YJF","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"pith_short_16","alias_value":"GF53I3WF3YJF6K6E","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"pith_short_8","alias_value":"GF53I3WF","created_at":"2026-07-03T01:17:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:GF53I3WF3YJF6K6E4KOJFYJDJQ","target":"record","payload":{"canonical_record":{"source":{"id":"2607.02130","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2026-07-02T13:06:56Z","cross_cats_sorted":["cs.CG","math.GR"],"title_canon_sha256":"4b349fa1a54ba1dbd819d8ee93908e576f83f7038134efd68f428e8aa163ce68","abstract_canon_sha256":"6c73833d02f9b7ff75339d11bbe9b86b4eec3d913ce34f8d2da2256db0c07613"},"schema_version":"1.0"},"canonical_sha256":"317bb46ec5de125f2bc4e29c92e1234c02d783b99b29f29242d4f0d21a49f91d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-03T01:17:43.021768Z","signature_b64":"NNvTRHei/nJjOQWdfqmmXqDaTI6aa0xlvizseFhNNxCb010jHdlVvBnK67Kb1/HjQrCdgJFc0VCgkr2csuOeBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"317bb46ec5de125f2bc4e29c92e1234c02d783b99b29f29242d4f0d21a49f91d","last_reissued_at":"2026-07-03T01:17:43.021362Z","signature_status":"signed_v1","first_computed_at":"2026-07-03T01:17:43.021362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.02130","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":"2026-07-03T01:17:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vPa5gp8p7BWBK42nNtixggDmdZSvTgUHnByWuy/ghTEPirn2T7d0gK4OTLVI4OHmvlKPrh3xqgrsfRuWpff1DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:30:21.556593Z"},"content_sha256":"19d71102aab2de5d836b63d86950476d0f16d910cda0268e9e8d0ac8f6c53e27","schema_version":"1.0","event_id":"sha256:19d71102aab2de5d836b63d86950476d0f16d910cda0268e9e8d0ac8f6c53e27"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:GF53I3WF3YJF6K6E4KOJFYJDJQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An algorithmic approach for computing fundamental domains of crystallographic groups","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.CG","math.GR"],"primary_cat":"math.MG","authors_text":"Alice C. Niemeyer, Lukas Schnelle, Reymond Akpanya","submitted_at":"2026-07-02T13:06:56Z","abstract_excerpt":"A crystallographic group is a discrete subgroup of the Euclidean group $\\operatorname{E}(n)$ that has a compact fundamental domain. Since such a crystallographic group $\\Gamma$ is infinite, computing fundamental domains of $\\Gamma$ is algorithmically challenging. We address this difficulty by targeting the computation of Dirichlet cells that can form fundamental domains of $\\Gamma$. We show that the half-spaces defining such a Dirichlet cell can be derived from elements of $\\Gamma$ acting on $\\mathbb{R}^n$ that can be expressed as words of bounded length in a suitable generating set. Based on "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02130","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/2607.02130/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":"2026-07-03T01:17:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Icbmf5Xmsxh7KzF58Y2RYOeo/Hp92L6tdx76ltAlm4HAAwI+u5jlF1n9cEubLUmBnC55H60TKU/ksg7sOYLhAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T16:30:21.557084Z"},"content_sha256":"c2f2f7332a3b961fbc6f8a318cc8285e9ba80c8a2ceb393f7910f71bb86aaea5","schema_version":"1.0","event_id":"sha256:c2f2f7332a3b961fbc6f8a318cc8285e9ba80c8a2ceb393f7910f71bb86aaea5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GF53I3WF3YJF6K6E4KOJFYJDJQ/bundle.json","state_url":"https://pith.science/pith/GF53I3WF3YJF6K6E4KOJFYJDJQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GF53I3WF3YJF6K6E4KOJFYJDJQ/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":"2026-08-03T16:30:21Z","links":{"resolver":"https://pith.science/pith/GF53I3WF3YJF6K6E4KOJFYJDJQ","bundle":"https://pith.science/pith/GF53I3WF3YJF6K6E4KOJFYJDJQ/bundle.json","state":"https://pith.science/pith/GF53I3WF3YJF6K6E4KOJFYJDJQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GF53I3WF3YJF6K6E4KOJFYJDJQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GF53I3WF3YJF6K6E4KOJFYJDJQ","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":"6c73833d02f9b7ff75339d11bbe9b86b4eec3d913ce34f8d2da2256db0c07613","cross_cats_sorted":["cs.CG","math.GR"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2026-07-02T13:06:56Z","title_canon_sha256":"4b349fa1a54ba1dbd819d8ee93908e576f83f7038134efd68f428e8aa163ce68"},"schema_version":"1.0","source":{"id":"2607.02130","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02130","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02130v1","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02130","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"pith_short_12","alias_value":"GF53I3WF3YJF","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"pith_short_16","alias_value":"GF53I3WF3YJF6K6E","created_at":"2026-07-03T01:17:43Z"},{"alias_kind":"pith_short_8","alias_value":"GF53I3WF","created_at":"2026-07-03T01:17:43Z"}],"graph_snapshots":[{"event_id":"sha256:c2f2f7332a3b961fbc6f8a318cc8285e9ba80c8a2ceb393f7910f71bb86aaea5","target":"graph","created_at":"2026-07-03T01:17:43Z","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/2607.02130/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A crystallographic group is a discrete subgroup of the Euclidean group $\\operatorname{E}(n)$ that has a compact fundamental domain. Since such a crystallographic group $\\Gamma$ is infinite, computing fundamental domains of $\\Gamma$ is algorithmically challenging. We address this difficulty by targeting the computation of Dirichlet cells that can form fundamental domains of $\\Gamma$. We show that the half-spaces defining such a Dirichlet cell can be derived from elements of $\\Gamma$ acting on $\\mathbb{R}^n$ that can be expressed as words of bounded length in a suitable generating set. Based on ","authors_text":"Alice C. Niemeyer, Lukas Schnelle, Reymond Akpanya","cross_cats":["cs.CG","math.GR"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2026-07-02T13:06:56Z","title":"An algorithmic approach for computing fundamental domains of crystallographic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02130","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":"sha256:19d71102aab2de5d836b63d86950476d0f16d910cda0268e9e8d0ac8f6c53e27","target":"record","created_at":"2026-07-03T01:17:43Z","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":"6c73833d02f9b7ff75339d11bbe9b86b4eec3d913ce34f8d2da2256db0c07613","cross_cats_sorted":["cs.CG","math.GR"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.MG","submitted_at":"2026-07-02T13:06:56Z","title_canon_sha256":"4b349fa1a54ba1dbd819d8ee93908e576f83f7038134efd68f428e8aa163ce68"},"schema_version":"1.0","source":{"id":"2607.02130","kind":"arxiv","version":1}},"canonical_sha256":"317bb46ec5de125f2bc4e29c92e1234c02d783b99b29f29242d4f0d21a49f91d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"317bb46ec5de125f2bc4e29c92e1234c02d783b99b29f29242d4f0d21a49f91d","first_computed_at":"2026-07-03T01:17:43.021362Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:43.021362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NNvTRHei/nJjOQWdfqmmXqDaTI6aa0xlvizseFhNNxCb010jHdlVvBnK67Kb1/HjQrCdgJFc0VCgkr2csuOeBg==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:43.021768Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.02130","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19d71102aab2de5d836b63d86950476d0f16d910cda0268e9e8d0ac8f6c53e27","sha256:c2f2f7332a3b961fbc6f8a318cc8285e9ba80c8a2ceb393f7910f71bb86aaea5"],"state_sha256":"d4c6cd3a7f407aac851e4030681cba15c1bee5de9418993e917458c1a92e8ebe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xxkwql9jihD6IcCeLVl28mTRSLKy0beNUNIZ5WT7HSmjwlsJUPkKY3zKZY/u6dDw8sbXMzy6QBYBoPlN0UNJCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T16:30:21.560524Z","bundle_sha256":"bfb352287d1916a415592e145160a530e9d1b41b67ea12ea0ad6d06782659bc6"}}