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We prove that for any $3\\le m\\le n$, the inclusion $\\Sigma^{m-2}(P_n)\\subseteq \\Sigma^{m-3}(P_n)$ is proper, but $\\Sigma^\\infty(P_n)=\\Sigma^{n-2}(P_n)$. We write down explicit character classes in each relevant $\\Sigma^{m-3}(P_n)\\setminus \\Sigma^{m-2}(P_n)$. In particular we get examples of normal subgroups $N\\le P_n$ with $P_n/N\\cong\\mathbb{Z}$ such that $N$ is of type $F_{m-3}$ but not $F_{m-2}$, for all "},"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":"1507.08597","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-30T17:51:24Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"e5a8ca22e906ee877011ee6581e4b42da2f6b6d08cf83c57edf10bcfb2255a53","abstract_canon_sha256":"0029c4001e26bd5b0de80378df98623e18f723846c77fc466e93fe00ae37cf76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:06.032092Z","signature_b64":"pWC25PNsGH+aEEQB9g+x1swRPimp9qqdcU5+GNW3+cAPRiwpwojUtPmJMzeleG83UGiE5lqe5SeriOYAJki8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"006d6f9953cea4ca15484470ebd678cbb900488e90b28631054221cdea4fc388","last_reissued_at":"2026-05-18T01:36:06.031651Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:06.031651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Separation in the BNSR-invariants of the pure braid groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Matthew C. 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