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In this paper I show that if $K$ and $L$ are anabelomorphic $p$-adic fields i.e. $K$ and $L$ have topologically isomorphic absolute Galois groups, then the stacks of Langlands parameters (for the fields $K$ and $L$) considered in [Fargues and Scholze, 2024], are also isomorphic (Theorem 2.2.1). This leads to Conjecture 3.3.1 which provides a precise relationship between the main conjecture of [Fargues and Scholze, 2024] and anabelomorphy of $p$-adic fields considered in [Joshi, 2020a]. I establish my conjecture fo"},"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":"2606.29478","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-28T16:12:46Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"88d838115bd578b37461bb57e96135e1e931588782ee204ebd025766d82a02ba","abstract_canon_sha256":"6021181302ed21ea6801a86328f2fb31884bd2cdb5813f0001a0f3dd02eddc7a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:18:08.103406Z","signature_b64":"ns5KajdroMoOENX+v8FeacFSuIG0mND//AwSsQUeB4VcFkavyAPCXGoHrNig4zlFr7gde9sW7SxWn+UX+0CiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"614de9e0fa3b700f82e142bed110275ea8d016f6cf70d838d3a09102e66debc7","last_reissued_at":"2026-06-30T01:18:08.102785Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:18:08.102785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Categorical Local Langlands Correspondence and Anabelomorphy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Kirti Joshi","submitted_at":"2026-06-28T16:12:46Z","abstract_excerpt":"Let $G/\\mathbb{Q}_p$ be a connected, split, reductive group over $\\mathbb{Q}_p$. 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