{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:RFUS2BU4DSVZXUAB52GSNMRNVI","short_pith_number":"pith:RFUS2BU4","canonical_record":{"source":{"id":"2607.09376","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-10T12:58:11Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1e7d5fe4a9b22a4fb310514df5a7ec7cffcc651247253bd31af2440b469dd8e4","abstract_canon_sha256":"fb21b51f62b7361ca34bdb6bcb55e7d405e81958310f4696467095ca83ba62cd"},"schema_version":"1.0"},"canonical_sha256":"89692d069c1cab9bd001ee8d26b22daa1614f90a6c0ca861fe69e78e76e506f5","source":{"kind":"arxiv","id":"2607.09376","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.09376","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"arxiv_version","alias_value":"2607.09376v1","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.09376","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"pith_short_12","alias_value":"RFUS2BU4DSVZ","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"pith_short_16","alias_value":"RFUS2BU4DSVZXUAB","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"pith_short_8","alias_value":"RFUS2BU4","created_at":"2026-07-13T01:19:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:RFUS2BU4DSVZXUAB52GSNMRNVI","target":"record","payload":{"canonical_record":{"source":{"id":"2607.09376","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-10T12:58:11Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1e7d5fe4a9b22a4fb310514df5a7ec7cffcc651247253bd31af2440b469dd8e4","abstract_canon_sha256":"fb21b51f62b7361ca34bdb6bcb55e7d405e81958310f4696467095ca83ba62cd"},"schema_version":"1.0"},"canonical_sha256":"89692d069c1cab9bd001ee8d26b22daa1614f90a6c0ca861fe69e78e76e506f5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-13T01:19:08.025946Z","signature_b64":"14eX5lV+sZaPJCVd/+f4VKVvohGlBTFwfD+tq80Aw6dmRa3N6c4N6qbdBZKfbn7ni15zVF4FTprVH0bTrdd1Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89692d069c1cab9bd001ee8d26b22daa1614f90a6c0ca861fe69e78e76e506f5","last_reissued_at":"2026-07-13T01:19:08.023944Z","signature_status":"signed_v1","first_computed_at":"2026-07-13T01:19:08.023944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.09376","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-13T01:19:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bowa5Bw2wu/bArAFc0EMuIk/ddICPsSCmq4EH0mOdTFNW3OYT7ZqZg7mjQJT+0AVLww56Wo0j4DRIozOEu55BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T20:16:49.643706Z"},"content_sha256":"c754fcfa379741a2f46df870c05f798e45a7f5bca629d502179041fbadbbb60e","schema_version":"1.0","event_id":"sha256:c754fcfa379741a2f46df870c05f798e45a7f5bca629d502179041fbadbbb60e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:RFUS2BU4DSVZXUAB52GSNMRNVI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hankel determinants of weighted binary sums of digits","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Bartosz Sobolewski, Maciej Ulas","submitted_at":"2026-07-10T12:58:11Z","abstract_excerpt":"Let $s_\\mathbf{w}$ be the weighted binary sum-of-digits function associated with an arbitrary sequence of complex weights $\\mathbf{w}=(w_j)_{j\\geq 0}$. We investigate Hankel determinants $\\mathcal{H}_\\mathbf{w}(n) = \\det [s_{\\mathbf{w}}(i+j)]_{0\\leq i,j<n}$ and derive a general recursion that allows us to effectively compute $\\mathcal{H}_\\mathbf{w}(n)$ for all $n$. Applying it to the ordinary binary sum-of-digits, that is, $w_j=1$, we express $\\mathcal{H}_\\mathbf{w}(n)$ in a closed form for several sequences of indices, including the remarkably simple $$ \\mathcal{H}_\\mathbf{w}(\\lceil 2^{k+2}/3"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09376","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.09376/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":"2026-07-13T01:19:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l+zMz6i6CBTxFM7Z/uHfM35Hlz+mLMFG2w2lohpIpI3mGN1XCsFXoRrkEI+SwKfkhJnBRztsqXS4+NwBaFeNDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T20:16:49.646751Z"},"content_sha256":"e96003c53a014507f053953f9f8946fb55ddbc29fd4c54d7a2c0294006cf857d","schema_version":"1.0","event_id":"sha256:e96003c53a014507f053953f9f8946fb55ddbc29fd4c54d7a2c0294006cf857d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RFUS2BU4DSVZXUAB52GSNMRNVI/bundle.json","state_url":"https://pith.science/pith/RFUS2BU4DSVZXUAB52GSNMRNVI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RFUS2BU4DSVZXUAB52GSNMRNVI/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":"2026-08-11T20:16:49Z","links":{"resolver":"https://pith.science/pith/RFUS2BU4DSVZXUAB52GSNMRNVI","bundle":"https://pith.science/pith/RFUS2BU4DSVZXUAB52GSNMRNVI/bundle.json","state":"https://pith.science/pith/RFUS2BU4DSVZXUAB52GSNMRNVI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RFUS2BU4DSVZXUAB52GSNMRNVI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:RFUS2BU4DSVZXUAB52GSNMRNVI","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":"fb21b51f62b7361ca34bdb6bcb55e7d405e81958310f4696467095ca83ba62cd","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-10T12:58:11Z","title_canon_sha256":"1e7d5fe4a9b22a4fb310514df5a7ec7cffcc651247253bd31af2440b469dd8e4"},"schema_version":"1.0","source":{"id":"2607.09376","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.09376","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"arxiv_version","alias_value":"2607.09376v1","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.09376","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"pith_short_12","alias_value":"RFUS2BU4DSVZ","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"pith_short_16","alias_value":"RFUS2BU4DSVZXUAB","created_at":"2026-07-13T01:19:08Z"},{"alias_kind":"pith_short_8","alias_value":"RFUS2BU4","created_at":"2026-07-13T01:19:08Z"}],"graph_snapshots":[{"event_id":"sha256:e96003c53a014507f053953f9f8946fb55ddbc29fd4c54d7a2c0294006cf857d","target":"graph","created_at":"2026-07-13T01:19:08Z","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/2607.09376/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $s_\\mathbf{w}$ be the weighted binary sum-of-digits function associated with an arbitrary sequence of complex weights $\\mathbf{w}=(w_j)_{j\\geq 0}$. We investigate Hankel determinants $\\mathcal{H}_\\mathbf{w}(n) = \\det [s_{\\mathbf{w}}(i+j)]_{0\\leq i,j<n}$ and derive a general recursion that allows us to effectively compute $\\mathcal{H}_\\mathbf{w}(n)$ for all $n$. Applying it to the ordinary binary sum-of-digits, that is, $w_j=1$, we express $\\mathcal{H}_\\mathbf{w}(n)$ in a closed form for several sequences of indices, including the remarkably simple $$ \\mathcal{H}_\\mathbf{w}(\\lceil 2^{k+2}/3","authors_text":"Bartosz Sobolewski, Maciej Ulas","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-10T12:58:11Z","title":"Hankel determinants of weighted binary sums of digits"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.09376","kind":"arxiv","version":1},"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:c754fcfa379741a2f46df870c05f798e45a7f5bca629d502179041fbadbbb60e","target":"record","created_at":"2026-07-13T01:19:08Z","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":"fb21b51f62b7361ca34bdb6bcb55e7d405e81958310f4696467095ca83ba62cd","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-07-10T12:58:11Z","title_canon_sha256":"1e7d5fe4a9b22a4fb310514df5a7ec7cffcc651247253bd31af2440b469dd8e4"},"schema_version":"1.0","source":{"id":"2607.09376","kind":"arxiv","version":1}},"canonical_sha256":"89692d069c1cab9bd001ee8d26b22daa1614f90a6c0ca861fe69e78e76e506f5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"89692d069c1cab9bd001ee8d26b22daa1614f90a6c0ca861fe69e78e76e506f5","first_computed_at":"2026-07-13T01:19:08.023944Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-13T01:19:08.023944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"14eX5lV+sZaPJCVd/+f4VKVvohGlBTFwfD+tq80Aw6dmRa3N6c4N6qbdBZKfbn7ni15zVF4FTprVH0bTrdd1Dg==","signature_status":"signed_v1","signed_at":"2026-07-13T01:19:08.025946Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.09376","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c754fcfa379741a2f46df870c05f798e45a7f5bca629d502179041fbadbbb60e","sha256:e96003c53a014507f053953f9f8946fb55ddbc29fd4c54d7a2c0294006cf857d"],"state_sha256":"07aa9055f8f08449ba30de81bd36e564a489d47f61a372f7c3c3062c2661cb0a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R9yKC1BZY2Xo+78nlHH2tHV5bIEouiubnNfuQ2VKjl85pVhu0ega6j9k+MYS0wkveLOcqBOC1diTef/ODmsJAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T20:16:49.657765Z","bundle_sha256":"d46e6321fbbbb130a13515e6bbdafb5159fb75e47d986364cc146c786d12bbcb"}}