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We prove that there exist coupling constant thresholds $\\mu_o,\\mu^o\\ge0$ such that for any $\\mu\\in[-\\mu^o,\\mu_o]$ the discrete spectrum of $\\hat h_\\mu$ is empty and for any $\\mu\\in \\mathbb{R}\\setminus[-\\mu^o,\\mu_o]$ the discrete spectrum of $\\hat h_\\mu$ is a singleton $\\{e(\\mu)\\},$ and $e(\\mu)<0$ for $\\mu>\\mu_o$ and $e(\\mu)>4d^2$ for $\\"},"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":"1910.01369","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2019-10-03T09:18:31Z","cross_cats_sorted":["math-ph","math.FA","math.MP"],"title_canon_sha256":"cd1ab584042235a2bbda99806d0d346c36080f829377a2518d71812cb1f32a85","abstract_canon_sha256":"76c15b23879d979fa1b784164f6a41c119545a21191c8c48c230ee4c2ae732d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:20:20.679996Z","signature_b64":"WMEajdWE3wVnL2cTR1CfojxQkhG9CFjS9ZIHfTf9OGilF5dsepUbiyQX1lr2uV668jVeusXNNn0dyt2LifiJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6dfe699d94a35d81a72ff205caf0f42358f6a8f7e1fd0ad5a7504382ef6b726","last_reissued_at":"2026-07-05T00:20:20.679549Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:20:20.679549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Expansion of eigenvalues of rank-one perturbations of the discrete bilaplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.SP","authors_text":"Ahmad Khalkhuzhaev, Mardon Pardabaev, Shokhrukh Yu. 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