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Relative to the decomposition $H=PH\\oplus P^{\\perp}H$, write \\[ T_{t}=\\begin{pmatrix}C_{t} & V^{*}_{t}\\\\ V_{t} & D_{t} \\end{pmatrix}, \\] where $C_{t}=PT_{t}P|_{PH}$, $V_{t}=P^{\\perp}T_{t}P|_{PH}$, and $D_{t}=P^{\\perp}T_{t}P^{\\perp}|_{P^{\\perp}H}$. The compressed family $\\left(C_{t}\\right)$ consists of positive contractions but need not form a semigroup. Its defect is given by \\[ C_{s+t}-C_{s}C_{t}=V^{*}_{s}V_{t} \\] while the complementary block satisfies \\[ D_{s+t}-D_{s}D_{t}=V_{s"},"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":"2606.12561","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-10T18:12:57Z","cross_cats_sorted":[],"title_canon_sha256":"c995205a58314e993344a67403b4bb185a5248276b6316963ceaeb95eb598be9","abstract_canon_sha256":"3a03d6606249f2411a57913adecd71400b07a665075998ee09da70fc4c56f860"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T00:07:54.724288Z","signature_b64":"lX0odFBJOqsymHsNppDRTIEpd+gTDhEpa1kYcgDbqahBnI0Mjd8+yXPtdcN5TcJHlBFw2phSkyOAjAjYwZp2Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9104926fa230058f6084a331a98e47cd1b235138c1f9945f8dd3461b19ff47c9","last_reissued_at":"2026-06-12T00:07:54.723871Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T00:07:54.723871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Compression Covariance and Tangent kernels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"James Tian","submitted_at":"2026-06-10T18:12:57Z","abstract_excerpt":"Let $A\\geq0$ be self-adjoint on a Hilbert space $H$, let $T_{t}=e^{-tA}$, and let $P$ be an orthogonal projection. Relative to the decomposition $H=PH\\oplus P^{\\perp}H$, write \\[ T_{t}=\\begin{pmatrix}C_{t} & V^{*}_{t}\\\\ V_{t} & D_{t} \\end{pmatrix}, \\] where $C_{t}=PT_{t}P|_{PH}$, $V_{t}=P^{\\perp}T_{t}P|_{PH}$, and $D_{t}=P^{\\perp}T_{t}P^{\\perp}|_{P^{\\perp}H}$. The compressed family $\\left(C_{t}\\right)$ consists of positive contractions but need not form a semigroup. 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