{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:FN4ON4ACBEGQAJB4OB2VG7CK3Y","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":"fee9ef7aaad0254ca9e14ce59d741cccff1224516b06a069185203fec748d31c","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-06T11:12:36Z","title_canon_sha256":"a8c92f0c79f2f60b3999c0ba09f16e5a2c2c410325c610cff52f03df5af8dcfe"},"schema_version":"1.0","source":{"id":"2605.04765","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.04765","created_at":"2026-06-10T01:10:02Z"},{"alias_kind":"arxiv_version","alias_value":"2605.04765v2","created_at":"2026-06-10T01:10:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.04765","created_at":"2026-06-10T01:10:02Z"},{"alias_kind":"pith_short_12","alias_value":"FN4ON4ACBEGQ","created_at":"2026-06-10T01:10:02Z"},{"alias_kind":"pith_short_16","alias_value":"FN4ON4ACBEGQAJB4","created_at":"2026-06-10T01:10:02Z"},{"alias_kind":"pith_short_8","alias_value":"FN4ON4AC","created_at":"2026-06-10T01:10:02Z"}],"graph_snapshots":[{"event_id":"sha256:ab504808d9bb39237f35f0f5ab128d4e0a8ed6ebd3377441ec29dda0de2c0665","target":"graph","created_at":"2026-06-10T01:10:02Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We establish a convergence theorem showing that the trigonometric interpolant converges at the rate O(n^{-min(r+β,d)}) in the supremum norm on the original interval, where r is the smoothness of the target function, d the number of Gram polynomials, and β ∈ [0,1] a Fourier-decay parameter."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The blending continuation of Gram polynomials can be constructed with the stated flexibility while preserving the controlled boundary behavior and without introducing uncontrolled errors that would invalidate the convergence analysis."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"GenFC generalizes FC-Gram via flexible Gram polynomial blending, proving O(n^{-min(r+β,d)}) convergence and showing better accuracy than prior versions in numerical tests."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A generalized FC-Gram framework adds flexibility to Gram polynomial blending and proves convergence rates of O(n to the minus min of r plus beta and d) for non-periodic functions."}],"snapshot_sha256":"f5547f1f13e0d6e67e2e93bf98d64e7c49837080ae0f06ebd4eed91c48c19dee"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-20T11:34:43.324498Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_title_agreement","ran_at":"2026-05-19T22:01:29.176049Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T14:11:40.207049Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2605.04765/integrity.json","findings":[],"snapshot_sha256":"759834e52e7d2c7b5d2bbbf49217070d3f0b4c35bd3b2bf71e34137d868572a9","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The FC-Gram algorithm constructs high-order trigonometric approximations of nonperiodic functions by periodically extending them to a larger interval, with the quality of the blending continuation of Gram polynomials over the extension interval directly governing the approximation accuracy. We introduce GenFC, a generalized FC-Gram framework in which the continuation of each Gram polynomial is shaped by a cutoff function satisfying prescribed boundary flatness conditions. We establish a convergence theorem showing that for any such family the GenFC approximation error satisfies $O(n^{-\\min(r+\\","authors_text":"Akash Anand, Prakash Nainwal","cross_cats":["cs.NA"],"headline":"A generalized FC-Gram framework adds flexibility to Gram polynomial blending and proves convergence rates of O(n to the minus min of r plus beta and d) for non-periodic functions.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-06T11:12:36Z","title":"A Generalized FC-Gram Approximation Framework with Analysis and Applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.04765","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-08T15:47:49.606749Z","id":"aaefa962-93b4-461d-80b4-4360bcb74ff8","model_set":{"reader":"grok-4.3"},"one_line_summary":"GenFC generalizes FC-Gram via flexible Gram polynomial blending, proving O(n^{-min(r+β,d)}) convergence and showing better accuracy than prior versions in numerical tests.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A generalized FC-Gram framework adds flexibility to Gram polynomial blending and proves convergence rates of O(n to the minus min of r plus beta and d) for non-periodic functions.","strongest_claim":"We establish a convergence theorem showing that the trigonometric interpolant converges at the rate O(n^{-min(r+β,d)}) in the supremum norm on the original interval, where r is the smoothness of the target function, d the number of Gram polynomials, and β ∈ [0,1] a Fourier-decay parameter.","weakest_assumption":"The blending continuation of Gram polynomials can be constructed with the stated flexibility while preserving the controlled boundary behavior and without introducing uncontrolled errors that would invalidate the convergence analysis."}},"verdict_id":"aaefa962-93b4-461d-80b4-4360bcb74ff8"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a2056cdb5de3a3bde10f35831314ddc9498977c1d9c4e06711368b2b24df7c86","target":"record","created_at":"2026-06-10T01:10:02Z","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":"fee9ef7aaad0254ca9e14ce59d741cccff1224516b06a069185203fec748d31c","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-06T11:12:36Z","title_canon_sha256":"a8c92f0c79f2f60b3999c0ba09f16e5a2c2c410325c610cff52f03df5af8dcfe"},"schema_version":"1.0","source":{"id":"2605.04765","kind":"arxiv","version":2}},"canonical_sha256":"2b78e6f002090d00243c7075537c4ade13deae0492263513768974a4f2e6e542","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b78e6f002090d00243c7075537c4ade13deae0492263513768974a4f2e6e542","first_computed_at":"2026-06-10T01:10:02.753849Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:10:02.753849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O2rU7JWEJo9AM8oWunROOtDjgCtafW6Kgd2MYwBR3FyH90R6+s6kwyrO/okuQQk+/K0m7fpHKqpFdyGCpbT2Aw==","signature_status":"signed_v1","signed_at":"2026-06-10T01:10:02.754682Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.04765","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a2056cdb5de3a3bde10f35831314ddc9498977c1d9c4e06711368b2b24df7c86","sha256:ab504808d9bb39237f35f0f5ab128d4e0a8ed6ebd3377441ec29dda0de2c0665"],"state_sha256":"aabbc598690a5eaa0da2e3bbad6de4384b6f3ffb1a2544c4befba7ba2da06530"}