{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:OBGWJLXM6VHFLC6IV2O7Y7SUN2","short_pith_number":"pith:OBGWJLXM","canonical_record":{"source":{"id":"2601.18258","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2026-01-26T08:30:52Z","cross_cats_sorted":[],"title_canon_sha256":"610ee81e7b92d76b4cc183018ccdf910e8f8e11f5bebbb2151919163e4b17bac","abstract_canon_sha256":"272ca3f8cab560ea9d7c202be86e1b6aef9d7b83876f667acc929b2ff18066d6"},"schema_version":"1.0"},"canonical_sha256":"704d64aeecf54e558bc8ae9dfc7e546e8da8a1bf8d7ce727652c21cdf6ae3d4f","source":{"kind":"arxiv","id":"2601.18258","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.18258","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"arxiv_version","alias_value":"2601.18258v3","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.18258","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"pith_short_12","alias_value":"OBGWJLXM6VHF","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"pith_short_16","alias_value":"OBGWJLXM6VHFLC6I","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"pith_short_8","alias_value":"OBGWJLXM","created_at":"2026-05-20T00:04:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:OBGWJLXM6VHFLC6IV2O7Y7SUN2","target":"record","payload":{"canonical_record":{"source":{"id":"2601.18258","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2026-01-26T08:30:52Z","cross_cats_sorted":[],"title_canon_sha256":"610ee81e7b92d76b4cc183018ccdf910e8f8e11f5bebbb2151919163e4b17bac","abstract_canon_sha256":"272ca3f8cab560ea9d7c202be86e1b6aef9d7b83876f667acc929b2ff18066d6"},"schema_version":"1.0"},"canonical_sha256":"704d64aeecf54e558bc8ae9dfc7e546e8da8a1bf8d7ce727652c21cdf6ae3d4f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:04:23.319466Z","signature_b64":"s7vnXgA71V++Q9M6kbZAP2prbcyZrQbExLp+AkSqJs9PY/erDAA0aW7aKXwOhtHwpbBwtLpWa2vVtWdsEVAiAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"704d64aeecf54e558bc8ae9dfc7e546e8da8a1bf8d7ce727652c21cdf6ae3d4f","last_reissued_at":"2026-05-20T00:04:23.318741Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:04:23.318741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2601.18258","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T00:04:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kpeDjt+afJANYgYOWTttdpkaqZLzHmQms5BDp7NoKFAGc0gK15DAQx7bggFs5KheTqYSiad/QUiNW1rx1plcAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T18:31:27.612378Z"},"content_sha256":"cef0c5a24ce726ec48c6c0f53263d878bcb817737c5eb0d11023e64ddcc93f88","schema_version":"1.0","event_id":"sha256:cef0c5a24ce726ec48c6c0f53263d878bcb817737c5eb0d11023e64ddcc93f88"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:OBGWJLXM6VHFLC6IV2O7Y7SUN2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Duality, Reconstruction, and Structural Toolkit Theorems in Algebraic Phase Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Joe Gildea","submitted_at":"2026-01-26T08:30:52Z","abstract_excerpt":"We study finite-depth reconstruction frameworks based on representation theory and show that non-rigid reconstruction behaviour is naturally accompanied by intrinsic structural boundaries. Within the finite-depth setting considered in this paper, reconstruction is controlled up to boundary equivalence by the associated filtered representation data together with boundary stratification.\n  We show that algebraic phases satisfying the axioms of Algebraic Phase Theory (APT) are reconstructible up to intrinsic phase equivalence from their filtered representation categories together with their bound"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.18258","kind":"arxiv","version":3},"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/2601.18258/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":"2026-05-20T00:04:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RyDRefPsMUo1F+qR3JepPlXmTgMdhTAlrSutr0X67LaKuMY5OICDdeJPD0TWcTh2vuE4T6Hm1159EW58QFMIAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T18:31:27.612802Z"},"content_sha256":"9a8475a379c70f170091ac0ddacd13463675a1bf3e36494fa8bc06c494816c6d","schema_version":"1.0","event_id":"sha256:9a8475a379c70f170091ac0ddacd13463675a1bf3e36494fa8bc06c494816c6d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OBGWJLXM6VHFLC6IV2O7Y7SUN2/bundle.json","state_url":"https://pith.science/pith/OBGWJLXM6VHFLC6IV2O7Y7SUN2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OBGWJLXM6VHFLC6IV2O7Y7SUN2/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":"2026-06-10T18:31:27Z","links":{"resolver":"https://pith.science/pith/OBGWJLXM6VHFLC6IV2O7Y7SUN2","bundle":"https://pith.science/pith/OBGWJLXM6VHFLC6IV2O7Y7SUN2/bundle.json","state":"https://pith.science/pith/OBGWJLXM6VHFLC6IV2O7Y7SUN2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OBGWJLXM6VHFLC6IV2O7Y7SUN2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:OBGWJLXM6VHFLC6IV2O7Y7SUN2","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":"272ca3f8cab560ea9d7c202be86e1b6aef9d7b83876f667acc929b2ff18066d6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2026-01-26T08:30:52Z","title_canon_sha256":"610ee81e7b92d76b4cc183018ccdf910e8f8e11f5bebbb2151919163e4b17bac"},"schema_version":"1.0","source":{"id":"2601.18258","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2601.18258","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"arxiv_version","alias_value":"2601.18258v3","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2601.18258","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"pith_short_12","alias_value":"OBGWJLXM6VHF","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"pith_short_16","alias_value":"OBGWJLXM6VHFLC6I","created_at":"2026-05-20T00:04:23Z"},{"alias_kind":"pith_short_8","alias_value":"OBGWJLXM","created_at":"2026-05-20T00:04:23Z"}],"graph_snapshots":[{"event_id":"sha256:9a8475a379c70f170091ac0ddacd13463675a1bf3e36494fa8bc06c494816c6d","target":"graph","created_at":"2026-05-20T00:04:23Z","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/2601.18258/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study finite-depth reconstruction frameworks based on representation theory and show that non-rigid reconstruction behaviour is naturally accompanied by intrinsic structural boundaries. Within the finite-depth setting considered in this paper, reconstruction is controlled up to boundary equivalence by the associated filtered representation data together with boundary stratification.\n  We show that algebraic phases satisfying the axioms of Algebraic Phase Theory (APT) are reconstructible up to intrinsic phase equivalence from their filtered representation categories together with their bound","authors_text":"Joe Gildea","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2026-01-26T08:30:52Z","title":"Duality, Reconstruction, and Structural Toolkit Theorems in Algebraic Phase Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2601.18258","kind":"arxiv","version":3},"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:cef0c5a24ce726ec48c6c0f53263d878bcb817737c5eb0d11023e64ddcc93f88","target":"record","created_at":"2026-05-20T00:04:23Z","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":"272ca3f8cab560ea9d7c202be86e1b6aef9d7b83876f667acc929b2ff18066d6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2026-01-26T08:30:52Z","title_canon_sha256":"610ee81e7b92d76b4cc183018ccdf910e8f8e11f5bebbb2151919163e4b17bac"},"schema_version":"1.0","source":{"id":"2601.18258","kind":"arxiv","version":3}},"canonical_sha256":"704d64aeecf54e558bc8ae9dfc7e546e8da8a1bf8d7ce727652c21cdf6ae3d4f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"704d64aeecf54e558bc8ae9dfc7e546e8da8a1bf8d7ce727652c21cdf6ae3d4f","first_computed_at":"2026-05-20T00:04:23.318741Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:23.318741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s7vnXgA71V++Q9M6kbZAP2prbcyZrQbExLp+AkSqJs9PY/erDAA0aW7aKXwOhtHwpbBwtLpWa2vVtWdsEVAiAg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:23.319466Z","signed_message":"canonical_sha256_bytes"},"source_id":"2601.18258","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cef0c5a24ce726ec48c6c0f53263d878bcb817737c5eb0d11023e64ddcc93f88","sha256:9a8475a379c70f170091ac0ddacd13463675a1bf3e36494fa8bc06c494816c6d"],"state_sha256":"f1d76c7665c320d250b45b53578bbc3b5e20100b1d0c5880862fed7cf91a61cf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"32pe1Cy9pNsInEiRxJyobAD0EPE4kI3NbwtxmwSklLFWEJnuQFKz7U0D5MLT7YHjIhOhpkPUh/OAGA2HTj84CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T18:31:27.614861Z","bundle_sha256":"a1623ac4f0194b5a76e5caa16882b427264460d62f5a2f80af7f921249fd016d"}}