{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:KI4UQ7Z2SSNR7KAO6IUNRIK4OS","short_pith_number":"pith:KI4UQ7Z2","schema_version":"1.0","canonical_sha256":"5239487f3a949b1fa80ef228d8a15c74834e9debd26867228dd6d6c4d7e629df","source":{"kind":"arxiv","id":"1309.3792","version":1},"attestation_state":"computed","paper":{"title":"Exact Complexity: The Spectral Decomposition of Intrinsic Computation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","nlin.CD","nlin.CG"],"primary_cat":"cond-mat.stat-mech","authors_text":"Christopher J. Ellison, James P. Crutchfield, Paul M. Riechers","submitted_at":"2013-09-15T20:06:56Z","abstract_excerpt":"We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectrum of the mixed-state presentation of a process's epsilon-machine causal-state dynamic. Measures include correlation functions, power spectra, past-future mutual information, transient and synchronization informations, and many others. As a result, a direct and complete analysis of intrinsic computation is now available for the temporal organizatio"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1309.3792","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2013-09-15T20:06:56Z","cross_cats_sorted":["cs.IT","math.IT","nlin.CD","nlin.CG"],"title_canon_sha256":"460af8ad311a0f4cf764987d2f491cf743344a601444da4bcba76ab226c45b40","abstract_canon_sha256":"f20dc90f59f53daf08970db1214a05d99f620819bf9928678bde543c3ccd3170"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:20.529240Z","signature_b64":"udMa/1YdOyjowZ29yMTPxfiskuctUGNBhdAa4DzUb4+jVCawGc2m0vWf+N1r8EwopV6tWzu8cGHIEumnO5qVDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5239487f3a949b1fa80ef228d8a15c74834e9debd26867228dd6d6c4d7e629df","last_reissued_at":"2026-05-18T03:13:20.528693Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:20.528693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact Complexity: The Spectral Decomposition of Intrinsic Computation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","nlin.CD","nlin.CG"],"primary_cat":"cond-mat.stat-mech","authors_text":"Christopher J. Ellison, James P. Crutchfield, Paul M. Riechers","submitted_at":"2013-09-15T20:06:56Z","abstract_excerpt":"We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectrum of the mixed-state presentation of a process's epsilon-machine causal-state dynamic. Measures include correlation functions, power spectra, past-future mutual information, transient and synchronization informations, and many others. As a result, a direct and complete analysis of intrinsic computation is now available for the temporal organizatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3792","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1309.3792","created_at":"2026-05-18T03:13:20.528784+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.3792v1","created_at":"2026-05-18T03:13:20.528784+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.3792","created_at":"2026-05-18T03:13:20.528784+00:00"},{"alias_kind":"pith_short_12","alias_value":"KI4UQ7Z2SSNR","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_16","alias_value":"KI4UQ7Z2SSNR7KAO","created_at":"2026-05-18T12:27:49.015174+00:00"},{"alias_kind":"pith_short_8","alias_value":"KI4UQ7Z2","created_at":"2026-05-18T12:27:49.015174+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS","json":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS.json","graph_json":"https://pith.science/api/pith-number/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/graph.json","events_json":"https://pith.science/api/pith-number/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/events.json","paper":"https://pith.science/paper/KI4UQ7Z2"},"agent_actions":{"view_html":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS","download_json":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS.json","view_paper":"https://pith.science/paper/KI4UQ7Z2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.3792&json=true","fetch_graph":"https://pith.science/api/pith-number/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/graph.json","fetch_events":"https://pith.science/api/pith-number/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/action/storage_attestation","attest_author":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/action/author_attestation","sign_citation":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/action/citation_signature","submit_replication":"https://pith.science/pith/KI4UQ7Z2SSNR7KAO6IUNRIK4OS/action/replication_record"}},"created_at":"2026-05-18T03:13:20.528784+00:00","updated_at":"2026-05-18T03:13:20.528784+00:00"}