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The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$ \\sum_{n=0}^\\infty\\frac{\\binom{4n}{2n}{\\binom{2n}{n}}^2}{2^{8n}3^{2n}}\\,(8n+1) =\\frac{2\\sqrt{3}}{\\pi}, $$ of the two supercongruences $$ S(p-1)\\equiv p\\biggl(\\frac{-"},"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":"1803.01830","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-03-05T18:50:28Z","cross_cats_sorted":["math.CA","math.CO","math.QA"],"title_canon_sha256":"6dce36801b3982354b9c1a3ead56632dfc08941f208c9f023867cc6801e06c33","abstract_canon_sha256":"ffb687438073982a401b7617e74aaf28d6dae871c4fc0242a81111baacf08523"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:07.409332Z","signature_b64":"eRIkoAUdRAw8ggTdNQb5EuNxlOU4MPznfhRUiWKfb/g2rKDkil0YS2gkpHRE7WEUp0VTmEiBkLe2kw3Nbm5YBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d00055d8ab3c592211b15c6998cd826a6db9851c02f30dfba17b584efd94524","last_reissued_at":"2026-05-17T23:54:07.408858Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:07.408858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A $q$-microscope for supercongruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO","math.QA"],"primary_cat":"math.NT","authors_text":"Victor J. 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