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Moreover, this promotes the known isomorphism $\\pi_{\\ast,\\ast} C\\tau \\cong \\mathrm{Ext}^{\\ast,\\ast}_{BP_{\\ast}BP}(BP_{\\ast},BP_{\\ast})$ to an isomorphism of rings which also preserves higher products.\n  We then consider the closed symmetric monoidal categor"},"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":"1701.04877","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2017-01-17T21:37:43Z","cross_cats_sorted":[],"title_canon_sha256":"15f38eb09f32d5e62b38bb8cba62fb42f51415937b60b225163cba46c4f92e5e","abstract_canon_sha256":"eda5824029b909d6460ebfa7f6ea145c4f2efb6197ab02aa26d196735cc91d49"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:33.612681Z","signature_b64":"56Tv84j/2r9ZBi5WdckbW0J2XIawrgk68MMvi+46OB2C8aLmM5uoPDL71SO9DcmzjgoLY1MCpwDLo4bWhPNSBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff8a695e49965cec9b9c9547905381a552d76bb64f97a68a0165271dc6613407","last_reissued_at":"2026-05-18T00:52:33.612127Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:33.612127Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Motivic Cofiber of $\\tau$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Bogdan Gheorghe","submitted_at":"2017-01-17T21:37:43Z","abstract_excerpt":"Consider the Tate twist $\\tau \\in H^{0,1}(S^{0,0})$ in the mod 2 cohomology of the motivic sphere. 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