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\\phi^5 = 0.\n  Restricting ourselves to the space of symmetric solutions \\psi for which \\psi(x) = \\psi(-x), we find a local centre-stable manifold, in a neighborhood of \\phi(x, 1), for this wave equation in the weighted Sobolev space <x>^{-1} \\dot H^1 "},"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":"1212.2285","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-12-11T03:06:38Z","cross_cats_sorted":[],"title_canon_sha256":"319c3b75833a7e8da5b423700ed0019a5575f5dedaf192be753c24dcdca85c5e","abstract_canon_sha256":"63f6fd89ecda1e420f5ee65ebce4c54a8e8f4473e1d8f9d02c1bf79bf210f3cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:45.403005Z","signature_b64":"8bZ6YPMFs2iOqxgi1rsupbPMgm5fwJ/sC16kso6O4metDFD/W/mzcYf/PLyVhL+Elsp7LgB35IFZ9x8324gNAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8325048f701b72fa27952de41eab75b158be49f2ab29e37e80e5d0c24c1959df","last_reissued_at":"2026-05-18T03:38:45.402142Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:45.402142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Centre-Stable Manifold for the Energy-Critical Wave Equation in R^3 in the Symmetric Setting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Marius Beceanu","submitted_at":"2012-12-11T03:06:38Z","abstract_excerpt":"Consider the focusing semilinear wave equation in R^3 with energy-critical nonlinearity\n  \\partial_t^2 \\psi - 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