{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:SW6ZZXOV6ZAC5XHHNVTFVQDNON","short_pith_number":"pith:SW6ZZXOV","canonical_record":{"source":{"id":"2605.16240","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T17:48:24Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"77bbdb5dcbb564500479e646e2fa253e32be4a8170c0013cc56db8a7a90eade3","abstract_canon_sha256":"48fdc696dd9abddc4660923951ef49946974a9c30c480b3a424e75f7c2a61e6e"},"schema_version":"1.0"},"canonical_sha256":"95bd9cddd5f6402edce76d665ac06d737070f621b7e9d2fe7ff5226ed8bccc06","source":{"kind":"arxiv","id":"2605.16240","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.16240","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"arxiv_version","alias_value":"2605.16240v2","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.16240","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"pith_short_12","alias_value":"SW6ZZXOV6ZAC","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"pith_short_16","alias_value":"SW6ZZXOV6ZAC5XHH","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"pith_short_8","alias_value":"SW6ZZXOV","created_at":"2026-05-20T00:05:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:SW6ZZXOV6ZAC5XHHNVTFVQDNON","target":"record","payload":{"canonical_record":{"source":{"id":"2605.16240","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T17:48:24Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"77bbdb5dcbb564500479e646e2fa253e32be4a8170c0013cc56db8a7a90eade3","abstract_canon_sha256":"48fdc696dd9abddc4660923951ef49946974a9c30c480b3a424e75f7c2a61e6e"},"schema_version":"1.0"},"canonical_sha256":"95bd9cddd5f6402edce76d665ac06d737070f621b7e9d2fe7ff5226ed8bccc06","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:05:48.088214Z","signature_b64":"+4XzIENeW7fBsyLlAjroKGT+569wPcuwcihikIVfewkbloeoBvuxanCfGJMsB9SXuuRfSx1vY0v1buiF6JeWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95bd9cddd5f6402edce76d665ac06d737070f621b7e9d2fe7ff5226ed8bccc06","last_reissued_at":"2026-05-20T00:05:48.087543Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:05:48.087543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.16240","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-20T00:05:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mLCI6cGTb/g5g9WyyeMaR+cmg7RifBcYutgiSFWMaZNopFTYoBF3BCJ/8jvawAgzZb008pJPu85KuhFJ9iw7Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T00:29:17.364800Z"},"content_sha256":"11d081717b8add38886c98f2874e8d645592ffe214756b64e8b5194bc26e344a","schema_version":"1.0","event_id":"sha256:11d081717b8add38886c98f2874e8d645592ffe214756b64e8b5194bc26e344a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:SW6ZZXOV6ZAC5XHHNVTFVQDNON","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Evaluation of two determinants involving $q$-integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Zhi-Wei Sun","submitted_at":"2026-05-15T17:48:24Z","abstract_excerpt":"The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities: $$\\det\\left[\\left[\\left\\lfloor\\frac{aj-(a+1)k}n\\right\\rfloor\\right]_q\\right]_{1\\leqslant j,k\\leqslant n}=-\\left(\\frac{a(a+1)}n\\right)q^{(1-3n)/2}$$ and $$\\det\\left[\\left[\\left\\lceil\\frac{(a+1)j-ak}n\\right\\rceil\\right]_q\\right]_{1\\leqslant j,k\\leqslant n}=\\left(\\frac{a(a+1)}n\\right)q^{(n-1)/2},$$ where $(\\frac{\\cdot}n)$ denotes the Jacobi symbol."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16240","kind":"arxiv","version":2},"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/2605.16240/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":"2026-05-20T00:05:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EukfwU3pk3owiXv0bRfmYOO/JUKXiuAmOTqx5HoFP2JEkfbS5Hkgtbec5bNdds7hnOdxThvoCNp8EDN+z17kBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T00:29:17.365387Z"},"content_sha256":"f2c56e4a13d12143b4df020506e5cf6f35fcfbcfc2930874c0b13440ac556a80","schema_version":"1.0","event_id":"sha256:f2c56e4a13d12143b4df020506e5cf6f35fcfbcfc2930874c0b13440ac556a80"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/bundle.json","state_url":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/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":"2026-05-23T00:29:17Z","links":{"resolver":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON","bundle":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/bundle.json","state":"https://pith.science/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SW6ZZXOV6ZAC5XHHNVTFVQDNON/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:SW6ZZXOV6ZAC5XHHNVTFVQDNON","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":"48fdc696dd9abddc4660923951ef49946974a9c30c480b3a424e75f7c2a61e6e","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T17:48:24Z","title_canon_sha256":"77bbdb5dcbb564500479e646e2fa253e32be4a8170c0013cc56db8a7a90eade3"},"schema_version":"1.0","source":{"id":"2605.16240","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.16240","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"arxiv_version","alias_value":"2605.16240v2","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.16240","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"pith_short_12","alias_value":"SW6ZZXOV6ZAC","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"pith_short_16","alias_value":"SW6ZZXOV6ZAC5XHH","created_at":"2026-05-20T00:05:48Z"},{"alias_kind":"pith_short_8","alias_value":"SW6ZZXOV","created_at":"2026-05-20T00:05:48Z"}],"graph_snapshots":[{"event_id":"sha256:f2c56e4a13d12143b4df020506e5cf6f35fcfbcfc2930874c0b13440ac556a80","target":"graph","created_at":"2026-05-20T00:05:48Z","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/2605.16240/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The $q$-analogue of an integer $m$ is given by $[m]_q=(1-q^m)/(1-q)$. Let $a$ be an integer, and let $n$ be a positive odd integer. Via discrete Fourier transforms, we establish the following two identities: $$\\det\\left[\\left[\\left\\lfloor\\frac{aj-(a+1)k}n\\right\\rfloor\\right]_q\\right]_{1\\leqslant j,k\\leqslant n}=-\\left(\\frac{a(a+1)}n\\right)q^{(1-3n)/2}$$ and $$\\det\\left[\\left[\\left\\lceil\\frac{(a+1)j-ak}n\\right\\rceil\\right]_q\\right]_{1\\leqslant j,k\\leqslant n}=\\left(\\frac{a(a+1)}n\\right)q^{(n-1)/2},$$ where $(\\frac{\\cdot}n)$ denotes the Jacobi symbol.","authors_text":"Zhi-Wei Sun","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T17:48:24Z","title":"Evaluation of two determinants involving $q$-integers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.16240","kind":"arxiv","version":2},"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":"sha256:11d081717b8add38886c98f2874e8d645592ffe214756b64e8b5194bc26e344a","target":"record","created_at":"2026-05-20T00:05:48Z","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":"48fdc696dd9abddc4660923951ef49946974a9c30c480b3a424e75f7c2a61e6e","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-15T17:48:24Z","title_canon_sha256":"77bbdb5dcbb564500479e646e2fa253e32be4a8170c0013cc56db8a7a90eade3"},"schema_version":"1.0","source":{"id":"2605.16240","kind":"arxiv","version":2}},"canonical_sha256":"95bd9cddd5f6402edce76d665ac06d737070f621b7e9d2fe7ff5226ed8bccc06","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95bd9cddd5f6402edce76d665ac06d737070f621b7e9d2fe7ff5226ed8bccc06","first_computed_at":"2026-05-20T00:05:48.087543Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:05:48.087543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+4XzIENeW7fBsyLlAjroKGT+569wPcuwcihikIVfewkbloeoBvuxanCfGJMsB9SXuuRfSx1vY0v1buiF6JeWBA==","signature_status":"signed_v1","signed_at":"2026-05-20T00:05:48.088214Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.16240","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:11d081717b8add38886c98f2874e8d645592ffe214756b64e8b5194bc26e344a","sha256:f2c56e4a13d12143b4df020506e5cf6f35fcfbcfc2930874c0b13440ac556a80"],"state_sha256":"d57a54533264a2e7ddff090d0e1ed78b010cb4842773b16a39d8c5d1f8d49e47"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jcvTRAlko6vorIyhFN6skFV8BNskw9FyIONZCrfjPfCOBDYCYQ4yZp4yBhm+hKYCXYdK7XbHtoEf+1lT+x5XDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T00:29:17.369529Z","bundle_sha256":"f1bdf35e969aa728d8969b29c19bd811c82134caca5f18a190125ca12171a11b"}}