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For a nonempty set $S$ of simple nodes and an integer $\\ell\\ge 0$, let $H_{S,\\ell}$ be the hyperplane on which the exponents of the roots meeting $S$ sum to $|S|-\\ell$. We prove that every proper-support hyperplane $H_{S,\\ell}$ is a genuine polar divisor at a generic point, while exact homogeneity leaves only the unshif"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.18945","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-07-21T10:27:57Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"4a722bd6b585db2d223669daa8896f45e8bc88e6c0517a34b0ad8417905c674b","abstract_canon_sha256":"e72f0c3492d7975564c0e509929e8db920e5bd84f766733bab0f5947ad0de319"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-22T01:23:21.010101Z","signature_b64":"Kb0a7RLprWtEX3Cm2VQhohLaPP+mQDD9RlRQasINsphV8abNEyIryswYFm0Do2fpTZkWyzL8CzoGZL2AapPeAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80fd9cd6fd7171cb0ecbe3f2523399f6d4c8fd5cb0daa33a120a3f479c81a5fc","last_reissued_at":"2026-07-22T01:23:21.009258Z","signature_status":"signed_v1","first_computed_at":"2026-07-22T01:23:21.009258Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generic polar divisors and flag residues for root-system zeta functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.RT","authors_text":"Jonas Matuzas","submitted_at":"2026-07-21T10:27:57Z","abstract_excerpt":"Let $\\Phi$ be an irreducible crystallographic root system and let $Z_\\Phi(\\mathbf{s})$ be the untwisted Komori-Matsumoto-Tsumura zeta function of $\\Phi$, with one complex exponent for each positive coroot; its diagonal specialization is the single-variable Witten zeta function. For a nonempty set $S$ of simple nodes and an integer $\\ell\\ge 0$, let $H_{S,\\ell}$ be the hyperplane on which the exponents of the roots meeting $S$ sum to $|S|-\\ell$. We prove that every proper-support hyperplane $H_{S,\\ell}$ is a genuine polar divisor at a generic point, while exact homogeneity leaves only the unshif"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.18945","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/2607.18945/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.18945","created_at":"2026-07-22T01:23:21.009694+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.18945v1","created_at":"2026-07-22T01:23:21.009694+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.18945","created_at":"2026-07-22T01:23:21.009694+00:00"},{"alias_kind":"pith_short_12","alias_value":"QD6ZZVX5OFY4","created_at":"2026-07-22T01:23:21.009694+00:00"},{"alias_kind":"pith_short_16","alias_value":"QD6ZZVX5OFY4WDWL","created_at":"2026-07-22T01:23:21.009694+00:00"},{"alias_kind":"pith_short_8","alias_value":"QD6ZZVX5","created_at":"2026-07-22T01:23:21.009694+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63","json":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63.json","graph_json":"https://pith.science/api/pith-number/QD6ZZVX5OFY4WDWL4PZFEM4Z63/graph.json","events_json":"https://pith.science/api/pith-number/QD6ZZVX5OFY4WDWL4PZFEM4Z63/events.json","paper":"https://pith.science/paper/QD6ZZVX5"},"agent_actions":{"view_html":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63","download_json":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63.json","view_paper":"https://pith.science/paper/QD6ZZVX5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.18945&json=true","fetch_graph":"https://pith.science/api/pith-number/QD6ZZVX5OFY4WDWL4PZFEM4Z63/graph.json","fetch_events":"https://pith.science/api/pith-number/QD6ZZVX5OFY4WDWL4PZFEM4Z63/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63/action/storage_attestation","attest_author":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63/action/author_attestation","sign_citation":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63/action/citation_signature","submit_replication":"https://pith.science/pith/QD6ZZVX5OFY4WDWL4PZFEM4Z63/action/replication_record"}},"created_at":"2026-07-22T01:23:21.009694+00:00","updated_at":"2026-07-22T01:23:21.009694+00:00"}