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We then prove that this extended functor KEnd(-) not only descends to the category of noncommutative motives but moreover becomes co-represented by the noncommutative motive associated to the tensor algebra S[t] of the sphere spectrum S. Using this co-representability result, we then classify all the natural transformations of KEnd(-) in terms of an integer plus a fraction between polynomials with c"},"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":"1302.1214","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-02-05T21:54:53Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"45e2479844ada1c0ff56c24bb4f3f966e2cffd3e60d0c8c11507eef90ba45853","abstract_canon_sha256":"8f9db402d1173fa511882a4c8686de0eed1586dcc442ddc037620d0d8f861505"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:24.855904Z","signature_b64":"lVdORE/U2sru8quG0jYfbSeL2Rov5O9LOLJh9YGJCedgmDv2Wg484LGvOM4Qa2FJAL69UCTDFFVzXVV87DRECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15fdbb23fcfcfc4e52ccabd1f79861b2888bb71229e663f97b2520597922b360","last_reissued_at":"2026-05-18T03:34:24.855462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:24.855462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"K-theory of endomorphisms via noncommutative motives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AT","authors_text":"Andrew J. 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