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We establish an equivalence of cyclotomic $\\mathbb{E}_\\infty$-rings $\\mathrm{THH}(\\mathbb{Z})^{\\wedge}_p \\simeq \\mathrm{sh}(\\mathrm{j}_p^{\\mathrm{triv}})$ and an equivalence of $\\mathbb{E}_\\infty$-rings $\\mathrm{TP}(\\mathbb{Z})^{\\wedge}_p \\simeq \\mathrm{j}_p^{\\mathrm{t}\\mathrm{S}^1}$. We also record a few applications of this: a new perspective, with some new information, on the description of $\\mathrm{TC}(\\mathbb{Z})^{\\wedge}_p$ as a spectrum; height $1$ analogues of the fiber squares of Antieau"},"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":"2505.02218","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-05-04T19:01:32Z","cross_cats_sorted":["math.AG","math.KT"],"title_canon_sha256":"647ab3c059bde19b5ff14372308767a2516534826ef406de7c12cf4c8ca76858","abstract_canon_sha256":"256ae28e293ae11d5655be87e91d142256e1ff35663840a44f4baedb372700d2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:58:33.848038Z","signature_b64":"cdLN1pbljiEuoxM8IqJXI2oKe4kg85i4LxRwAzx28t/rG6eLeKTcpU8NjKuKSwyVIjBqG9rqQ8ny9FHCiSfxDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f4d269f2464b4d612ab3211649b11cb7924e97754ed3cffb7714c9d9dad8f0f","last_reissued_at":"2026-07-05T10:58:33.847630Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:58:33.847630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"THH(Z) and the image of J","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.KT"],"primary_cat":"math.AT","authors_text":"Arpon Raksit, Sanath K. 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