{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:SGCATDHIHMYUQGZCGZ7LFJLXFA","short_pith_number":"pith:SGCATDHI","canonical_record":{"source":{"id":"0903.2716","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-16T10:23:11Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"27704195124b258116509daca5cd9de3e5863944486432175b82de24797b2ac4","abstract_canon_sha256":"a3631ccccdccc68eafef5dc7b48128806a05ec8a5908205ace2c8b954d2f9056"},"schema_version":"1.0"},"canonical_sha256":"9184098ce83b31481b22367eb2a577280e67b0633c42e35d33cc916f2c1f28c0","source":{"kind":"arxiv","id":"0903.2716","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.2716","created_at":"2026-05-18T02:14:28Z"},{"alias_kind":"arxiv_version","alias_value":"0903.2716v3","created_at":"2026-05-18T02:14:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.2716","created_at":"2026-05-18T02:14:28Z"},{"alias_kind":"pith_short_12","alias_value":"SGCATDHIHMYU","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"SGCATDHIHMYUQGZC","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"SGCATDHI","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:SGCATDHIHMYUQGZCGZ7LFJLXFA","target":"record","payload":{"canonical_record":{"source":{"id":"0903.2716","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-16T10:23:11Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"27704195124b258116509daca5cd9de3e5863944486432175b82de24797b2ac4","abstract_canon_sha256":"a3631ccccdccc68eafef5dc7b48128806a05ec8a5908205ace2c8b954d2f9056"},"schema_version":"1.0"},"canonical_sha256":"9184098ce83b31481b22367eb2a577280e67b0633c42e35d33cc916f2c1f28c0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:14:28.079924Z","signature_b64":"i5JOpCIfMMc+H63LfZQWim2zq47/XDUVlZdMF720Xp0e1FRrCVMwm4pBIhA6KxO1dHlk9KG/hpAvkGhYSqy/AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9184098ce83b31481b22367eb2a577280e67b0633c42e35d33cc916f2c1f28c0","last_reissued_at":"2026-05-18T02:14:28.079447Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:14:28.079447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.2716","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":"2026-05-18T02:14:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+lCOLbQAr62D65lwSERd79285hmQLAiLH03olFsyQeteAhW1V9lyxfQLVoxkYnAprN2r1XKST+S7MYIpXDG1CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T13:31:44.959682Z"},"content_sha256":"28c58b73cd6aed2a57feb2e2d36257642cc83f555ce0c30f117f34b1ac579a28","schema_version":"1.0","event_id":"sha256:28c58b73cd6aed2a57feb2e2d36257642cc83f555ce0c30f117f34b1ac579a28"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:SGCATDHIHMYUQGZCGZ7LFJLXFA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"H\\\"older-continuous rough paths by Fourier normal ordering","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"J. Unterberger","submitted_at":"2009-03-16T10:23:11Z","abstract_excerpt":"We construct in this article an explicit geometric rough path over arbitrary $d$-dimensional paths with finite $1/\\alpha$-variation for any $\\alpha\\in(0,1)$. The method may be coined as 'Fourier normal ordering', since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest Fourier frequencies. In doing so, there appear non-trivial tree combinatorics, which are best understood by using the structure of the Hopf algebra of decorated rooted trees (in connection with the Chen or multiplicative property) and o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.2716","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":""},"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":"2026-05-18T02:14:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"naU7dt+34X/ZtRa6NaykClG7b8R1zzwoqreEE0abuwuA1kKq5x7CCXQ1gtIJWEcmMl1IheUfEecfu7JyJMzVAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T13:31:44.960028Z"},"content_sha256":"ea1e36ed69b1f3732d7b29f38dabf5c8eece1e0dfe294e7c61549b542cbbc31d","schema_version":"1.0","event_id":"sha256:ea1e36ed69b1f3732d7b29f38dabf5c8eece1e0dfe294e7c61549b542cbbc31d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SGCATDHIHMYUQGZCGZ7LFJLXFA/bundle.json","state_url":"https://pith.science/pith/SGCATDHIHMYUQGZCGZ7LFJLXFA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SGCATDHIHMYUQGZCGZ7LFJLXFA/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":"2026-06-27T13:31:44Z","links":{"resolver":"https://pith.science/pith/SGCATDHIHMYUQGZCGZ7LFJLXFA","bundle":"https://pith.science/pith/SGCATDHIHMYUQGZCGZ7LFJLXFA/bundle.json","state":"https://pith.science/pith/SGCATDHIHMYUQGZCGZ7LFJLXFA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SGCATDHIHMYUQGZCGZ7LFJLXFA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:SGCATDHIHMYUQGZCGZ7LFJLXFA","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":"a3631ccccdccc68eafef5dc7b48128806a05ec8a5908205ace2c8b954d2f9056","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-16T10:23:11Z","title_canon_sha256":"27704195124b258116509daca5cd9de3e5863944486432175b82de24797b2ac4"},"schema_version":"1.0","source":{"id":"0903.2716","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.2716","created_at":"2026-05-18T02:14:28Z"},{"alias_kind":"arxiv_version","alias_value":"0903.2716v3","created_at":"2026-05-18T02:14:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.2716","created_at":"2026-05-18T02:14:28Z"},{"alias_kind":"pith_short_12","alias_value":"SGCATDHIHMYU","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"SGCATDHIHMYUQGZC","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"SGCATDHI","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:ea1e36ed69b1f3732d7b29f38dabf5c8eece1e0dfe294e7c61549b542cbbc31d","target":"graph","created_at":"2026-05-18T02:14:28Z","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":"We construct in this article an explicit geometric rough path over arbitrary $d$-dimensional paths with finite $1/\\alpha$-variation for any $\\alpha\\in(0,1)$. The method may be coined as 'Fourier normal ordering', since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest Fourier frequencies. In doing so, there appear non-trivial tree combinatorics, which are best understood by using the structure of the Hopf algebra of decorated rooted trees (in connection with the Chen or multiplicative property) and o","authors_text":"J. Unterberger","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-16T10:23:11Z","title":"H\\\"older-continuous rough paths by Fourier normal ordering"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.2716","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":"sha256:28c58b73cd6aed2a57feb2e2d36257642cc83f555ce0c30f117f34b1ac579a28","target":"record","created_at":"2026-05-18T02:14:28Z","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":"a3631ccccdccc68eafef5dc7b48128806a05ec8a5908205ace2c8b954d2f9056","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-16T10:23:11Z","title_canon_sha256":"27704195124b258116509daca5cd9de3e5863944486432175b82de24797b2ac4"},"schema_version":"1.0","source":{"id":"0903.2716","kind":"arxiv","version":3}},"canonical_sha256":"9184098ce83b31481b22367eb2a577280e67b0633c42e35d33cc916f2c1f28c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9184098ce83b31481b22367eb2a577280e67b0633c42e35d33cc916f2c1f28c0","first_computed_at":"2026-05-18T02:14:28.079447Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:14:28.079447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i5JOpCIfMMc+H63LfZQWim2zq47/XDUVlZdMF720Xp0e1FRrCVMwm4pBIhA6KxO1dHlk9KG/hpAvkGhYSqy/AA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:14:28.079924Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.2716","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:28c58b73cd6aed2a57feb2e2d36257642cc83f555ce0c30f117f34b1ac579a28","sha256:ea1e36ed69b1f3732d7b29f38dabf5c8eece1e0dfe294e7c61549b542cbbc31d"],"state_sha256":"e02547baedd016daeee94cb90176ed9feb927f483414201692c625709bf76393"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FB08V3WImHOoqNb5+WCM5p+qQo/U3E8PTxtD+BM+lSpoLx8bzox5wLC2Ff5ftBsosyUrJlBiCnMqNP36pK6yAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T13:31:44.961913Z","bundle_sha256":"e785bd8ad1cb73b79b8a26f7c490b848fa213fe485f18fae41ed1e36467dd6a4"}}