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As a consequence, the Kervaire invariant element $\\theta_5$ is contained in the strictly defined 4-fold Toda bracket $\\langle 2, \\theta_4, \\theta_4, 2\\rangle$.\n  Our result has a geometric corollary: the 61-sphere has a unique smooth structure and it is the last odd dimensional case - the only ones are $S^1, S^3, S^5$ and $S^{61}$.\n  Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential $d_3(D_3) = B_3$. We prove this differential by introducing a new technique based on the algebrai"},"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":"1601.02184","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-10T07:29:46Z","cross_cats_sorted":["math.DG","math.GT"],"title_canon_sha256":"a2d1bacb5026b40ce2ab624a5b5a7788f61bf69ed440e9667d6c7cc212a7b956","abstract_canon_sha256":"c278012ae510b7ffc2dda4410900b1f6a3b8ab71a9ded686b51935bdab1af3bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:24.076170Z","signature_b64":"/qboIODS0ab6hlmKu0hgbIqX/hEwzW9Dvjkge4izzncxA6/FTYA2QiAMYmaJRLq0FAOK4QY72FSPkhDRH5o7CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de3fe481ad7b7ca649a979202c3436d3351fa5be0785b12e71e22620c4c8a660","last_reissued_at":"2026-05-18T00:42:24.075705Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:24.075705Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The triviality of the 61-stem in the stable homotopy groups of spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.GT"],"primary_cat":"math.AT","authors_text":"Guozhen Wang, Zhouli Xu","submitted_at":"2016-01-10T07:29:46Z","abstract_excerpt":"We prove that the 2-primary $\\pi_{61}$ is zero. 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