{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:BYBCMID4IMT7DL6OG5NXDVZ3GX","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":"98d7e9f7e8b3f661af0083d59bcf58f054741bd99771a0fd7516ee41b516e3e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-08-07T14:37:14Z","title_canon_sha256":"8ea819222061e932505c843491532d5bafd0014e4457e573d57071a231e7133d"},"schema_version":"1.0","source":{"id":"0808.1038","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0808.1038","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"arxiv_version","alias_value":"0808.1038v1","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.1038","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"pith_short_12","alias_value":"BYBCMID4IMT7","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"BYBCMID4IMT7DL6O","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"BYBCMID4","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:a9394ac6aca0dada1ecf120ee3ab131282ae31e6d0411709ad114377b6c9805f","target":"graph","created_at":"2026-05-18T02:15:43Z","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"},"paper":{"abstract_excerpt":"The absolute logarithmic Weil height is well defined on the group of units of the algebraic closure of the rational numbers, modulo roots of unity, and induces a metric topology on this group. We show that the completion of this metric space is a Banach space over the field of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of L1 of a certain totally disconnected, locally compact space, equipped with a certain measure satisfying an invariance property with respect to the absolute Galois group.","authors_text":"Daniel Allcock, Jeffrey D. Vaaler","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-08-07T14:37:14Z","title":"A Banach space determined by the Weil height"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.1038","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":"sha256:5959e5d0e7e41fae4385b794a9f87f3d47e336ca973689bb175a115d6440c44a","target":"record","created_at":"2026-05-18T02:15:43Z","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":"98d7e9f7e8b3f661af0083d59bcf58f054741bd99771a0fd7516ee41b516e3e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2008-08-07T14:37:14Z","title_canon_sha256":"8ea819222061e932505c843491532d5bafd0014e4457e573d57071a231e7133d"},"schema_version":"1.0","source":{"id":"0808.1038","kind":"arxiv","version":1}},"canonical_sha256":"0e0226207c4327f1afce375b71d73b35e2822cbc89328b49f273e464b724ec64","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e0226207c4327f1afce375b71d73b35e2822cbc89328b49f273e464b724ec64","first_computed_at":"2026-05-18T02:15:43.794671Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:15:43.794671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8ThbOhxDPd/z1CHh8z4ApJZ/TLLlEKTMsMX+yIuDx4Jod2o+MquF0eEFHj3XPV0XUbd+ZSVCOGv2AKJXmHRPDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:15:43.795343Z","signed_message":"canonical_sha256_bytes"},"source_id":"0808.1038","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5959e5d0e7e41fae4385b794a9f87f3d47e336ca973689bb175a115d6440c44a","sha256:a9394ac6aca0dada1ecf120ee3ab131282ae31e6d0411709ad114377b6c9805f"],"state_sha256":"5221ad5470188139f687c5f4ce6d9e469234ce7b1145b725c455b99daf77974e"}