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In this paper we compute the lowest mod 2 \\'{e}tale homological obstruction class to the existence of a $K$-rational point on $X$, and show that it is the cup product of the form $$ o_{n+1} = [a_0]\\cup\\cdots\\cup[a_n]. $$\n  Our computation is an \\'{e}tale-homotopy analogue of the topological fact that Stiefel-Whitney classes are the homological obstructions to find a section to the unit sphere bundle of a real v"},"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":"1902.03404","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-02-09T10:48:55Z","cross_cats_sorted":[],"title_canon_sha256":"cc967f13c0affb151da8c1fd994243a35f90d145a40de05c12eb6bea0bb3487c","abstract_canon_sha256":"e79997fcaf04a2cfa45100f3de8a91a09be2b456f46a60e896e905f5676c239a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:20.957459Z","signature_b64":"mH4lJdFRqlzsPG/ogK7wlpo+6sk/TV0KB7bbOJW5aKyeevw4sm+8pVHDPBAN+Mbd0HwXpovioY7H9+OwFPXfAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"452dd5b5dba8e1db4837b5b026c6a01983bc0855540680edc36f71b309b9d7ce","last_reissued_at":"2026-05-17T23:54:20.956888Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:20.956888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"\\'{E}tale Homotopy Obstructions of Arithmetic Spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Edo Arad, Shachar Carmeli, Tomer M. 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