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We show that completion of $R({\\mathbb{G}})_G$-modules at $\\mathcal{F}$ agrees with algebraic completion at the ideal $I_{\\mathbb{G}}(\\mathcal{F})=\\bigcap_{H\\in\\mathcal{F}}\\mathrm{ker} (\\pi_0R({\\mathbb{G}})^{ G}\\to \\pi_0R({\\mathbb{G}})^{ H}).$ For ${\\mathbb{G}}=\\mu_{\\mathbb{P}^\\infty}$ over $\\mathrm{KU}$ this recovers the family completion theorem of Adams, Haeberly, Jackowski, and May, and for the trivi"},"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":"2608.02390","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-08-03T15:31:27Z","cross_cats_sorted":[],"title_canon_sha256":"0af4acc0edc2a3469436a95abbce18f8327b7029a6f89d13effbf7b028f5fd6b","abstract_canon_sha256":"c58052b5319167f980469b1675599e9c39db17ef19a11d3633b3a8184e2adbc2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:12:07.374114Z","signature_b64":"FeqKxYK8J8Mz4B9tcQ+GOfjqLxisPjX4ZmGIgm+qX6obqD/7KdM2VmfyQOducQtUcLWEjIKVEhteFsGwJtRQBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16d0a9a555c840d99485da73e46c4161d667242b263329c0efcd3c38188da2e7","last_reissued_at":"2026-08-04T02:12:07.372456Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:12:07.372456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A family completion theorem for tempered cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Leonard Tokic","submitted_at":"2026-08-03T15:31:27Z","abstract_excerpt":"Let ${\\mathbb{G}}$ be an oriented $\\mathbb{P}$-divisible group over a noetherian $\\mathbb{E}_\\infty$-ring $R$, let $G$ be a finite group, and let $\\mathcal{F}$ be a family of subgroups of $G$. 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