{"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"}