{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7D4TASJXXNSNZB3W4KXZBJKCZ2","short_pith_number":"pith:7D4TASJX","canonical_record":{"source":{"id":"2606.08821","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-07T20:18:50Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"caf35b80736a9ea1c64e4c09d36e43b104157bd12bd81c79f41f2b69beb054a8","abstract_canon_sha256":"3b465b3100df14eef401d14687172e29fac69cec9c79389075485e11a567fe6f"},"schema_version":"1.0"},"canonical_sha256":"f8f9304937bb64dc8776e2af90a542cea761bee5de06963a81e3a8ea1e5f7784","source":{"kind":"arxiv","id":"2606.08821","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08821","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08821v1","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08821","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"pith_short_12","alias_value":"7D4TASJXXNSN","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"pith_short_16","alias_value":"7D4TASJXXNSNZB3W","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"pith_short_8","alias_value":"7D4TASJX","created_at":"2026-06-09T02:07:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7D4TASJXXNSNZB3W4KXZBJKCZ2","target":"record","payload":{"canonical_record":{"source":{"id":"2606.08821","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-07T20:18:50Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"caf35b80736a9ea1c64e4c09d36e43b104157bd12bd81c79f41f2b69beb054a8","abstract_canon_sha256":"3b465b3100df14eef401d14687172e29fac69cec9c79389075485e11a567fe6f"},"schema_version":"1.0"},"canonical_sha256":"f8f9304937bb64dc8776e2af90a542cea761bee5de06963a81e3a8ea1e5f7784","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:07:41.244041Z","signature_b64":"2RD0GovPqEWPrJZcaxDl0ZRtsyJAgKcJjHo6PFcnbPOKTSJ8NdJi3bKKuLVwP+RQ834N5+/XS7pP9PDX7lFSCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8f9304937bb64dc8776e2af90a542cea761bee5de06963a81e3a8ea1e5f7784","last_reissued_at":"2026-06-09T02:07:41.243144Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:07:41.243144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.08821","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":"2026-06-09T02:07:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b+HsNMIdWJ5vqAsnFBWMaQ5WW1vYPUCn0uGagPRll5NSxFesaBpeQ964qdWwZSHxbWCtcToL/sF7fs6pPABMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T23:45:05.372924Z"},"content_sha256":"86b7d730dbd8bce123585149a5b273bea7341af58122be82097914bd98c9abf7","schema_version":"1.0","event_id":"sha256:86b7d730dbd8bce123585149a5b273bea7341af58122be82097914bd98c9abf7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7D4TASJXXNSNZB3W4KXZBJKCZ2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Square Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.AT","authors_text":"Charles Fanning, Mehmet Aktas","submitted_at":"2026-06-07T20:18:50Z","abstract_excerpt":"Product decompositions of metric spaces are built from coordinate maps, but these maps are not part of the resulting metric space. We recover this missing coordinate structure through equivalence relations whose classes are candidate coordinate fibers, and the resulting quotient metrics reconstruct the coordinate factors. This framework characterizes exactly when a metric space admits a finite product or power presentation. We prove an equivalence of categories showing that these equivalence-relation data preserve exactly the ordered coordinate information of power presentations. For spaces wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08821","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/2606.08821/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":"2026-06-09T02:07:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KdtePRn5KUGCznQytjF3E6reObRcLOJTAS2+GqrcIn1bZPd5THhW6p+E0zuPol5c9wkMBYYmkJpzzE/18P4aDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T23:45:05.373300Z"},"content_sha256":"16ea7b45c37a09c44f95cc030a5b7d126e1254f7b26ce7537faebf9618357217","schema_version":"1.0","event_id":"sha256:16ea7b45c37a09c44f95cc030a5b7d126e1254f7b26ce7537faebf9618357217"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7D4TASJXXNSNZB3W4KXZBJKCZ2/bundle.json","state_url":"https://pith.science/pith/7D4TASJXXNSNZB3W4KXZBJKCZ2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7D4TASJXXNSNZB3W4KXZBJKCZ2/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":"2026-07-01T23:45:05Z","links":{"resolver":"https://pith.science/pith/7D4TASJXXNSNZB3W4KXZBJKCZ2","bundle":"https://pith.science/pith/7D4TASJXXNSNZB3W4KXZBJKCZ2/bundle.json","state":"https://pith.science/pith/7D4TASJXXNSNZB3W4KXZBJKCZ2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7D4TASJXXNSNZB3W4KXZBJKCZ2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7D4TASJXXNSNZB3W4KXZBJKCZ2","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":"3b465b3100df14eef401d14687172e29fac69cec9c79389075485e11a567fe6f","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-07T20:18:50Z","title_canon_sha256":"caf35b80736a9ea1c64e4c09d36e43b104157bd12bd81c79f41f2b69beb054a8"},"schema_version":"1.0","source":{"id":"2606.08821","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08821","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08821v1","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08821","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"pith_short_12","alias_value":"7D4TASJXXNSN","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"pith_short_16","alias_value":"7D4TASJXXNSNZB3W","created_at":"2026-06-09T02:07:41Z"},{"alias_kind":"pith_short_8","alias_value":"7D4TASJX","created_at":"2026-06-09T02:07:41Z"}],"graph_snapshots":[{"event_id":"sha256:16ea7b45c37a09c44f95cc030a5b7d126e1254f7b26ce7537faebf9618357217","target":"graph","created_at":"2026-06-09T02:07:41Z","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/2606.08821/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Product decompositions of metric spaces are built from coordinate maps, but these maps are not part of the resulting metric space. We recover this missing coordinate structure through equivalence relations whose classes are candidate coordinate fibers, and the resulting quotient metrics reconstruct the coordinate factors. This framework characterizes exactly when a metric space admits a finite product or power presentation. We prove an equivalence of categories showing that these equivalence-relation data preserve exactly the ordered coordinate information of power presentations. For spaces wi","authors_text":"Charles Fanning, Mehmet Aktas","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-07T20:18:50Z","title":"Square Metric Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08821","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":"sha256:86b7d730dbd8bce123585149a5b273bea7341af58122be82097914bd98c9abf7","target":"record","created_at":"2026-06-09T02:07:41Z","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":"3b465b3100df14eef401d14687172e29fac69cec9c79389075485e11a567fe6f","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-07T20:18:50Z","title_canon_sha256":"caf35b80736a9ea1c64e4c09d36e43b104157bd12bd81c79f41f2b69beb054a8"},"schema_version":"1.0","source":{"id":"2606.08821","kind":"arxiv","version":1}},"canonical_sha256":"f8f9304937bb64dc8776e2af90a542cea761bee5de06963a81e3a8ea1e5f7784","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8f9304937bb64dc8776e2af90a542cea761bee5de06963a81e3a8ea1e5f7784","first_computed_at":"2026-06-09T02:07:41.243144Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T02:07:41.243144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2RD0GovPqEWPrJZcaxDl0ZRtsyJAgKcJjHo6PFcnbPOKTSJ8NdJi3bKKuLVwP+RQ834N5+/XS7pP9PDX7lFSCg==","signature_status":"signed_v1","signed_at":"2026-06-09T02:07:41.244041Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.08821","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86b7d730dbd8bce123585149a5b273bea7341af58122be82097914bd98c9abf7","sha256:16ea7b45c37a09c44f95cc030a5b7d126e1254f7b26ce7537faebf9618357217"],"state_sha256":"88b31b97e3360b82a1362c31700bc716a0552dcb163a0c39920ac3961b2165da"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jqWsw+oNXlmoPGgmlnYlAfJhjbuVCicv85Ch+o/QnovZUoIiv/M8TuERNyJUjKDetMsx5uX0lobKAWPls/9aBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T23:45:05.375596Z","bundle_sha256":"241c1ad6fe682c143dc4c30651b40fa955a2eb9eb29e8bc46f69ff17a5dddb9f"}}