{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:QOJN4NNFCKWDNZRL2LXWS6VJUJ","short_pith_number":"pith:QOJN4NNF","canonical_record":{"source":{"id":"1607.02087","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-07-07T17:20:41Z","cross_cats_sorted":[],"title_canon_sha256":"9989a8711bd223d587f433d183bacbb92192a16735c95f943bdf8a92996fc9f5","abstract_canon_sha256":"ac770ce653198ed73d26e55273a92d743511f3f7c35192af13a35904fb94a89c"},"schema_version":"1.0"},"canonical_sha256":"8392de35a512ac36e62bd2ef697aa9a2490df6e92b76d3be076c372f3fdae769","source":{"kind":"arxiv","id":"1607.02087","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.02087","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"arxiv_version","alias_value":"1607.02087v1","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.02087","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"pith_short_12","alias_value":"QOJN4NNFCKWD","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"QOJN4NNFCKWDNZRL","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"QOJN4NNF","created_at":"2026-05-18T12:30:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:QOJN4NNFCKWDNZRL2LXWS6VJUJ","target":"record","payload":{"canonical_record":{"source":{"id":"1607.02087","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-07-07T17:20:41Z","cross_cats_sorted":[],"title_canon_sha256":"9989a8711bd223d587f433d183bacbb92192a16735c95f943bdf8a92996fc9f5","abstract_canon_sha256":"ac770ce653198ed73d26e55273a92d743511f3f7c35192af13a35904fb94a89c"},"schema_version":"1.0"},"canonical_sha256":"8392de35a512ac36e62bd2ef697aa9a2490df6e92b76d3be076c372f3fdae769","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:17.503492Z","signature_b64":"Q7/LXrUKVJa/DObzIUMzVIEMU0i/QrZcnQiJBiH5zaGBNoVD9ICl198v5+EwFCdxRCKgHvwGYGiamX/LjshUBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8392de35a512ac36e62bd2ef697aa9a2490df6e92b76d3be076c372f3fdae769","last_reissued_at":"2026-05-18T00:48:17.502882Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:17.502882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.02087","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":"2026-05-18T00:48:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HBDrkW0gjoeYkZLzL2qOFzx4JAccNnruBSdJGPqlitCwQ+lyAxVbgc3irUAMDNp2WVO5dYeQcMIDAEoX2j6oCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:23:50.597191Z"},"content_sha256":"bf9b1d06766f74fa4484f7796c37177ce38fe007de1700ea2f55d80918ba964b","schema_version":"1.0","event_id":"sha256:bf9b1d06766f74fa4484f7796c37177ce38fe007de1700ea2f55d80918ba964b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:QOJN4NNFCKWDNZRL2LXWS6VJUJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimising Dirichlet eigenvalues on cuboids of unit measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Katie Gittins, Michiel van den Berg","submitted_at":"2016-07-07T17:20:41Z","abstract_excerpt":"We consider the minimisation of Dirichlet eigenvalues $\\lambda_k$, $k \\in \\N$, of the Laplacian on cuboids of unit measure in $\\R^3$. We prove that any sequence of optimal cuboids in $\\R^3$ converges to a cube of unit measure in the sense of Hausdorff as $k \\rightarrow \\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02087","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":""},"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":"2026-05-18T00:48:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eJphpYtsgbp8w2o2EXM7FqFeOzU0/2QuHRjenKf8GnrcB9Thm3VgOkkfFUgMRPlJxcAmCzlEWM5vEnfazKw7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T05:23:50.597774Z"},"content_sha256":"7b809fde91455d94ed38bd2842d77d5a8371440c87cbed5b0edaba9b73ae3c03","schema_version":"1.0","event_id":"sha256:7b809fde91455d94ed38bd2842d77d5a8371440c87cbed5b0edaba9b73ae3c03"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QOJN4NNFCKWDNZRL2LXWS6VJUJ/bundle.json","state_url":"https://pith.science/pith/QOJN4NNFCKWDNZRL2LXWS6VJUJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QOJN4NNFCKWDNZRL2LXWS6VJUJ/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":"2026-05-28T05:23:50Z","links":{"resolver":"https://pith.science/pith/QOJN4NNFCKWDNZRL2LXWS6VJUJ","bundle":"https://pith.science/pith/QOJN4NNFCKWDNZRL2LXWS6VJUJ/bundle.json","state":"https://pith.science/pith/QOJN4NNFCKWDNZRL2LXWS6VJUJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QOJN4NNFCKWDNZRL2LXWS6VJUJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:QOJN4NNFCKWDNZRL2LXWS6VJUJ","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":"ac770ce653198ed73d26e55273a92d743511f3f7c35192af13a35904fb94a89c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-07-07T17:20:41Z","title_canon_sha256":"9989a8711bd223d587f433d183bacbb92192a16735c95f943bdf8a92996fc9f5"},"schema_version":"1.0","source":{"id":"1607.02087","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.02087","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"arxiv_version","alias_value":"1607.02087v1","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.02087","created_at":"2026-05-18T00:48:17Z"},{"alias_kind":"pith_short_12","alias_value":"QOJN4NNFCKWD","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"QOJN4NNFCKWDNZRL","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"QOJN4NNF","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:7b809fde91455d94ed38bd2842d77d5a8371440c87cbed5b0edaba9b73ae3c03","target":"graph","created_at":"2026-05-18T00:48:17Z","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":"We consider the minimisation of Dirichlet eigenvalues $\\lambda_k$, $k \\in \\N$, of the Laplacian on cuboids of unit measure in $\\R^3$. We prove that any sequence of optimal cuboids in $\\R^3$ converges to a cube of unit measure in the sense of Hausdorff as $k \\rightarrow \\infty$.","authors_text":"Katie Gittins, Michiel van den Berg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-07-07T17:20:41Z","title":"Minimising Dirichlet eigenvalues on cuboids of unit measure"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02087","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:bf9b1d06766f74fa4484f7796c37177ce38fe007de1700ea2f55d80918ba964b","target":"record","created_at":"2026-05-18T00:48:17Z","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":"ac770ce653198ed73d26e55273a92d743511f3f7c35192af13a35904fb94a89c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-07-07T17:20:41Z","title_canon_sha256":"9989a8711bd223d587f433d183bacbb92192a16735c95f943bdf8a92996fc9f5"},"schema_version":"1.0","source":{"id":"1607.02087","kind":"arxiv","version":1}},"canonical_sha256":"8392de35a512ac36e62bd2ef697aa9a2490df6e92b76d3be076c372f3fdae769","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8392de35a512ac36e62bd2ef697aa9a2490df6e92b76d3be076c372f3fdae769","first_computed_at":"2026-05-18T00:48:17.502882Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:48:17.502882Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q7/LXrUKVJa/DObzIUMzVIEMU0i/QrZcnQiJBiH5zaGBNoVD9ICl198v5+EwFCdxRCKgHvwGYGiamX/LjshUBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:48:17.503492Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.02087","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bf9b1d06766f74fa4484f7796c37177ce38fe007de1700ea2f55d80918ba964b","sha256:7b809fde91455d94ed38bd2842d77d5a8371440c87cbed5b0edaba9b73ae3c03"],"state_sha256":"46e25c281942aacf30ad885c4431aaa1310324ca1106a06b554f2a6f6089c7bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g5rO9KXtJb6o8nkH0Hgf8cNYVcSNkX6kOtChRwb+5cXis+D2kHZ05YGETNsjcoanER0MUFyiiz3C9qTjWMyPAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T05:23:50.600510Z","bundle_sha256":"efd5550e45796250c55d2103f7a2a0dd46b071d055fca9e1d2151c60f3a5564a"}}