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In this work, we prove that the operator $$ \\Psi(\\textbf{z},\\textbf{a},q) \\Psi\\left(\\textbf{z}^p,\\textbf{a}^p,q^{p^2}\\right)^{-1} $$ has no poles at the primitive complex $p$-th roots of unity $q=\\zeta_p$. As a byproduct, we show that the iterated product of the operators ${\\bf M}_{\\mathcal{L}}(\\textbf{z},\\textbf{a},q )$ from the $q$-difference equation on $X$: $$ {\\bf M}_{\\mathcal{L}} (\\textbf{z} q^{(p-1)\\mathcal{L}},\\textbf{a},q) \\cdots {\\bf M}_{\\mathcal{L}} (\\t"},"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":"2412.19383","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AG","submitted_at":"2024-12-26T23:52:58Z","cross_cats_sorted":["hep-th","math-ph","math.MP","math.NT","math.RT"],"title_canon_sha256":"a47401b06196bb0450778743a23e85fff2a93ede614fcf98710a6f99717e0df9","abstract_canon_sha256":"18d557af1884df2efd9d700be74006259fd387c45ea3c12a0164ab5058bbae10"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:03.919630Z","signature_b64":"4/1oxX/5mZKk5Rgzqh52+uYuxrVHGxsCqAgyy/KhD7i8Kn+BQ876jfp196rK4nUIqDbS3QDt4JaRVotO9MVJBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8c66a2154727375f2d6e31c029afc2a40f0c58d624492a4321ed34dd83bf20c","last_reissued_at":"2026-06-03T01:05:03.919192Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:03.919192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Quantum K-theory of Quiver Varieties at Roots of Unity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["hep-th","math-ph","math.MP","math.NT","math.RT"],"primary_cat":"math.AG","authors_text":"Andrey Smirnov, Peter Koroteev","submitted_at":"2024-12-26T23:52:58Z","abstract_excerpt":"Let $\\Psi(\\textbf{z},\\textbf{a},q)$ a the fundamental solution matrix of the quantum difference equation of a Nakajima variety $X$. In this work, we prove that the operator $$ \\Psi(\\textbf{z},\\textbf{a},q) \\Psi\\left(\\textbf{z}^p,\\textbf{a}^p,q^{p^2}\\right)^{-1} $$ has no poles at the primitive complex $p$-th roots of unity $q=\\zeta_p$. As a byproduct, we show that the iterated product of the operators ${\\bf M}_{\\mathcal{L}}(\\textbf{z},\\textbf{a},q )$ from the $q$-difference equation on $X$: $$ {\\bf M}_{\\mathcal{L}} (\\textbf{z} q^{(p-1)\\mathcal{L}},\\textbf{a},q) \\cdots {\\bf M}_{\\mathcal{L}} (\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.19383","kind":"arxiv","version":4},"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/2412.19383/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":"2412.19383","created_at":"2026-06-03T01:05:03.919250+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.19383v4","created_at":"2026-06-03T01:05:03.919250+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.19383","created_at":"2026-06-03T01:05:03.919250+00:00"},{"alias_kind":"pith_short_12","alias_value":"7DDGUIKUOJZX","created_at":"2026-06-03T01:05:03.919250+00:00"},{"alias_kind":"pith_short_16","alias_value":"7DDGUIKUOJZXL4WW","created_at":"2026-06-03T01:05:03.919250+00:00"},{"alias_kind":"pith_short_8","alias_value":"7DDGUIKU","created_at":"2026-06-03T01:05:03.919250+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/7DDGUIKUOJZXL4WW4MOAFGX4FJ","json":"https://pith.science/pith/7DDGUIKUOJZXL4WW4MOAFGX4FJ.json","graph_json":"https://pith.science/api/pith-number/7DDGUIKUOJZXL4WW4MOAFGX4FJ/graph.json","events_json":"https://pith.science/api/pith-number/7DDGUIKUOJZXL4WW4MOAFGX4FJ/events.json","paper":"https://pith.science/paper/7DDGUIKU"},"agent_actions":{"view_html":"https://pith.science/pith/7DDGUIKUOJZXL4WW4MOAFGX4FJ","download_json":"https://pith.science/pith/7DDGUIKUOJZXL4WW4MOAFGX4FJ.json","view_paper":"https://pith.science/paper/7DDGUIKU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.19383&json=true","fetch_graph":"https://pith.science/api/pith-number/7DDGUIKUOJZXL4WW4MOAFGX4FJ/graph.json","fetch_events":"https://pith.science/api/pith-number/7DDGUIKUOJZXL4WW4MOAFGX4FJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7DDGUIKUOJZXL4WW4MOAFGX4FJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7DDGUIKUOJZXL4WW4MOAFGX4FJ/action/storage_attestation","attest_author":"https://pith.science/pith/7DDGUIKUOJZXL4WW4MOAFGX4FJ/action/author_attestation","sign_citation":"https://pith.science/pith/7DDGUIKUOJZXL4WW4MOAFGX4FJ/action/citation_signature","submit_replication":"https://pith.science/pith/7DDGUIKUOJZXL4WW4MOAFGX4FJ/action/replication_record"}},"created_at":"2026-06-03T01:05:03.919250+00:00","updated_at":"2026-06-03T01:05:03.919250+00:00"}