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Let $\\mathcal{N}(G)$ be the lattice of normal subgroups ordered by inclusion, regarded as an abstract lattice. Define $\\operatorname{LatAut}(G) := \\operatorname{Aut}(\\mathcal{N}(G))$. The \\emph{LatAut tower} is the sequence defined by $G_0 = G$, $G_{n+1} = \\operatorname{LatAut}(G_n)$.\n  Let $G$ be a \\emph{tower group} if $G \\cong \\prod_{k \\geq 3} S_k^{a_k}$ with finitely many $a_k \\neq 0$. We establish the following for tower groups.\n  \\emph{Product Formula.} $\\operatorname{LatAut}\\!\\bigl(\\prod_{k \\geq 3} S_k^{a_k}\\bigr) \\cong S_{a_4} \\times S_B$, where $B = \\sum_{k ","authors_text":"Sonukumar, Vinay Madhusudanan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-20T11:00:00Z","title":"Termination of the Lattice-Automorphism Tower for Direct Products of Symmetric Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21025","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:224e348a51eea0936cda5f42ffa841292c68fca335db721ab665e2cf6133984b","target":"record","created_at":"2026-05-21T01:05:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"08aea0c90345d67e61e2393cc355abc81e2c3d269784e59ca9140e89071ee37e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-05-20T11:00:00Z","title_canon_sha256":"826a36eac87c0f2194d06dc6918b3c8835c127e15fee7d2effc2c15d3ca52750"},"schema_version":"1.0","source":{"id":"2605.21025","kind":"arxiv","version":1}},"canonical_sha256":"72f2f0c3f8695512b4eef605bcf94903de4bfd913531418975a6ad9de2f70cc5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"72f2f0c3f8695512b4eef605bcf94903de4bfd913531418975a6ad9de2f70cc5","first_computed_at":"2026-05-21T01:05:33.005569Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:05:33.005569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R8yzcEDmil6VKHYMHy0Tl8+5C8f+DCecCAnvvooUWasjmyZhaCI0oQ/fkA4sifJycn0VI9Q/sAjzzSI8leuEDw==","signature_status":"signed_v1","signed_at":"2026-05-21T01:05:33.006424Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.21025","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:224e348a51eea0936cda5f42ffa841292c68fca335db721ab665e2cf6133984b","sha256:1da7d7c76a26a76be876ae05ff3a983c0f5d0e1529db4f806d250a8b6773bcf2"],"state_sha256":"6ba57dcd586292a5df48d03b0ad87b76e828e72a7e2abe7d1a46af22c62ccbe1"}