{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:RWBUZCOGBYTEO2TDTFMOILYKBG","short_pith_number":"pith:RWBUZCOG","canonical_record":{"source":{"id":"1410.7789","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-28T20:20:27Z","cross_cats_sorted":[],"title_canon_sha256":"f7c488375613c157fd702e3cb8533e002bdea3a29e5a1e69fc6b575ad1b6f1d3","abstract_canon_sha256":"34faf86ba97e887d6cf45dadc204767057a1b3a21fd6047908502d0e50e1d94a"},"schema_version":"1.0"},"canonical_sha256":"8d834c89c60e26476a639958e42f0a098fc276a00653d64169997a714b95c632","source":{"kind":"arxiv","id":"1410.7789","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.7789","created_at":"2026-05-18T00:22:40Z"},{"alias_kind":"arxiv_version","alias_value":"1410.7789v2","created_at":"2026-05-18T00:22:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7789","created_at":"2026-05-18T00:22:40Z"},{"alias_kind":"pith_short_12","alias_value":"RWBUZCOGBYTE","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"RWBUZCOGBYTEO2TD","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"RWBUZCOG","created_at":"2026-05-18T12:28:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:RWBUZCOGBYTEO2TDTFMOILYKBG","target":"record","payload":{"canonical_record":{"source":{"id":"1410.7789","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-28T20:20:27Z","cross_cats_sorted":[],"title_canon_sha256":"f7c488375613c157fd702e3cb8533e002bdea3a29e5a1e69fc6b575ad1b6f1d3","abstract_canon_sha256":"34faf86ba97e887d6cf45dadc204767057a1b3a21fd6047908502d0e50e1d94a"},"schema_version":"1.0"},"canonical_sha256":"8d834c89c60e26476a639958e42f0a098fc276a00653d64169997a714b95c632","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:40.648555Z","signature_b64":"A5wefzuuG9SYEbKBAmj/L9J7DbshaM9FzE76bYkq9xoDycZ8kHJBONY/6fJNpAovk44lSsKWaBqw60O4V+UYBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d834c89c60e26476a639958e42f0a098fc276a00653d64169997a714b95c632","last_reissued_at":"2026-05-18T00:22:40.647920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:40.647920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.7789","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F4O66w6IIerjHOMC3JbDGxhErTk2dCtXcOOz3iUwa48z7cSN0fiAcNzYySLZaUPDz+Gj3FuKwGS9njd/uPZgCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T15:05:10.077496Z"},"content_sha256":"98a65c689f78ab72f01b1de0799fa361036c3ee7c7b237430ed6fe7cdbe51747","schema_version":"1.0","event_id":"sha256:98a65c689f78ab72f01b1de0799fa361036c3ee7c7b237430ed6fe7cdbe51747"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:RWBUZCOGBYTEO2TDTFMOILYKBG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Birch's theorem with shifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sam Chow","submitted_at":"2014-10-28T20:20:27Z","abstract_excerpt":"Let $f_1, ..., f_R$ be rational forms of degree $d \\ge 2$ in $n > \\sigma + R(R+1)(d-1)2^{d-1}$ variables, where $\\sigma$ is the dimension of the affine variety cut out by the condition $\\mathrm{rank}(\\nabla f_k)_{k=1}^R < R$. Assume that $\\mathbf{f} = \\mathbf{0}$ has a nonsingular real solution, and that the forms $(1,...,1) \\cdot \\nabla f_k$ are linearly independent. Let $\\boldsymbol{\\tau} \\in \\mathbb{R}^R$, let $\\mu$ be an irrational real number, and let $\\eta$ be a positive real number. We consider the values taken by $\\mathbf{f}(x_1 + \\mu, ..., x_n + \\mu)$ for integers $x_1, ..., x_n$. We "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7789","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l8D7IAOCUpcCCqjNFC69P5YZMI4NAChsXUWDgJMlCzYCrn96z2fRm5GGgBx8R+WcGOi8GohJOHrOHClb3xFUDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T15:05:10.077841Z"},"content_sha256":"3f38113fc89869ac7ced251034ded1536a96473a2d7d4ed8eab8214c2bdd671a","schema_version":"1.0","event_id":"sha256:3f38113fc89869ac7ced251034ded1536a96473a2d7d4ed8eab8214c2bdd671a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RWBUZCOGBYTEO2TDTFMOILYKBG/bundle.json","state_url":"https://pith.science/pith/RWBUZCOGBYTEO2TDTFMOILYKBG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RWBUZCOGBYTEO2TDTFMOILYKBG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T15:05:10Z","links":{"resolver":"https://pith.science/pith/RWBUZCOGBYTEO2TDTFMOILYKBG","bundle":"https://pith.science/pith/RWBUZCOGBYTEO2TDTFMOILYKBG/bundle.json","state":"https://pith.science/pith/RWBUZCOGBYTEO2TDTFMOILYKBG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RWBUZCOGBYTEO2TDTFMOILYKBG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:RWBUZCOGBYTEO2TDTFMOILYKBG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"34faf86ba97e887d6cf45dadc204767057a1b3a21fd6047908502d0e50e1d94a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-28T20:20:27Z","title_canon_sha256":"f7c488375613c157fd702e3cb8533e002bdea3a29e5a1e69fc6b575ad1b6f1d3"},"schema_version":"1.0","source":{"id":"1410.7789","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.7789","created_at":"2026-05-18T00:22:40Z"},{"alias_kind":"arxiv_version","alias_value":"1410.7789v2","created_at":"2026-05-18T00:22:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.7789","created_at":"2026-05-18T00:22:40Z"},{"alias_kind":"pith_short_12","alias_value":"RWBUZCOGBYTE","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"RWBUZCOGBYTEO2TD","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"RWBUZCOG","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:3f38113fc89869ac7ced251034ded1536a96473a2d7d4ed8eab8214c2bdd671a","target":"graph","created_at":"2026-05-18T00:22:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $f_1, ..., f_R$ be rational forms of degree $d \\ge 2$ in $n > \\sigma + R(R+1)(d-1)2^{d-1}$ variables, where $\\sigma$ is the dimension of the affine variety cut out by the condition $\\mathrm{rank}(\\nabla f_k)_{k=1}^R < R$. Assume that $\\mathbf{f} = \\mathbf{0}$ has a nonsingular real solution, and that the forms $(1,...,1) \\cdot \\nabla f_k$ are linearly independent. Let $\\boldsymbol{\\tau} \\in \\mathbb{R}^R$, let $\\mu$ be an irrational real number, and let $\\eta$ be a positive real number. We consider the values taken by $\\mathbf{f}(x_1 + \\mu, ..., x_n + \\mu)$ for integers $x_1, ..., x_n$. We ","authors_text":"Sam Chow","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-28T20:20:27Z","title":"Birch's theorem with shifts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7789","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:98a65c689f78ab72f01b1de0799fa361036c3ee7c7b237430ed6fe7cdbe51747","target":"record","created_at":"2026-05-18T00:22:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"34faf86ba97e887d6cf45dadc204767057a1b3a21fd6047908502d0e50e1d94a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-28T20:20:27Z","title_canon_sha256":"f7c488375613c157fd702e3cb8533e002bdea3a29e5a1e69fc6b575ad1b6f1d3"},"schema_version":"1.0","source":{"id":"1410.7789","kind":"arxiv","version":2}},"canonical_sha256":"8d834c89c60e26476a639958e42f0a098fc276a00653d64169997a714b95c632","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8d834c89c60e26476a639958e42f0a098fc276a00653d64169997a714b95c632","first_computed_at":"2026-05-18T00:22:40.647920Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:40.647920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A5wefzuuG9SYEbKBAmj/L9J7DbshaM9FzE76bYkq9xoDycZ8kHJBONY/6fJNpAovk44lSsKWaBqw60O4V+UYBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:40.648555Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.7789","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:98a65c689f78ab72f01b1de0799fa361036c3ee7c7b237430ed6fe7cdbe51747","sha256:3f38113fc89869ac7ced251034ded1536a96473a2d7d4ed8eab8214c2bdd671a"],"state_sha256":"d2b942c6b4674adf4b5f52af9fcc789e43abc16b8bc45d6f0e15594d7706e16c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T5TIr1/TRZqF6tl4B7x7g7aHdKc1bXKQfpNJBchMDaCXUQOTaXdXCr8sINXQlGStoMPzMNScaPLOrABHcvTWCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T15:05:10.079744Z","bundle_sha256":"b81608fed8906f2360a181560c9404afc7f9bfa1333f708153c0fc521f3f3e3e"}}