{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MFRXYMU7H5CTA2MCV2UKHXERJF","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":"b3a6b97133cf25cbd3256e3828ccb10cb7f19d10d032dbc8949e1d8f84e923c9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T17:59:30Z","title_canon_sha256":"af2b0c0e513247cf44c27e5d60aa57ad7a9c95ffaacf9567b414cba161c4eecd"},"schema_version":"1.0","source":{"id":"2605.13844","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13844","created_at":"2026-05-18T02:44:09Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13844v1","created_at":"2026-05-18T02:44:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13844","created_at":"2026-05-18T02:44:09Z"},{"alias_kind":"pith_short_12","alias_value":"MFRXYMU7H5CT","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"MFRXYMU7H5CTA2MC","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"MFRXYMU7","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:473bc0dce5cf5126bd64cc086e981e0451ba9af7d46380052e4932402a5d2770","target":"graph","created_at":"2026-05-18T02:44:09Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum of 2-dimensional torsion forms."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The base field is hereditarily Pythagorean and the extension has transcendence degree exactly 1 while remaining real; if the hereditarily Pythagorean condition fails or the degree is higher, the decomposition may not hold."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Torsion quadratic forms over real fields of transcendence degree 1 over hereditarily Pythagorean bases decompose into orthogonal sums of 2-dimensional torsion forms."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Over real fields that are transcendence degree one extensions of hereditarily Pythagorean bases, every torsion quadratic form decomposes into an orthogonal sum of 2-dimensional torsion forms."}],"snapshot_sha256":"ddfbfdf911a541d1c6f67c0b5c0a37d31a10ed6088be5a9e7ea264bb5d97c932"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d76953a0c89391b37b22c6931bd538cf458b64e07b1b5bcc5f2f2f4ef9e42962"},"paper":{"abstract_excerpt":"Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum of 2-dimensional torsion forms. This is obtained from a more general study of weakly isotropic forms over henselian valued fields and over function fields in one variable.","authors_text":"Karim Johannes Becher, M. Archita","cross_cats":[],"headline":"Over real fields that are transcendence degree one extensions of hereditarily Pythagorean bases, every torsion quadratic form decomposes into an orthogonal sum of 2-dimensional torsion forms.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T17:59:30Z","title":"Fields where torsion forms decompose"},"references":{"count":12,"internal_anchors":0,"resolved_work":12,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"J. K. Arason, A. Pfister. Zur Theorie der quadratischen Formen ¨ uber formalreellen K¨ orpern.Math. Z. 153 (1977), 289–296","work_id":"a665ef4f-19ef-4b8c-8589-a3c2a1ca42ac","year":1977},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"K.J. Becher. Minimal weakly isotropic forms. Math. Z., 252 (2006), 91–102","work_id":"4ab774fc-62e9-4731-a0c2-ed21688052e0","year":2006},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"K.J. Becher, N. Daans, Ph. Dittmann. Uniform existential definitions of valuations in function fields in one variable. Preprint (2025), https://arxiv.org/abs/2311.06044","work_id":"64db4179-d2d7-4d3f-8175-f89fa59e931f","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"K.J. Becher, N. Daans, D. Grimm, G. Manzano-Flores, M. Zaninelli. The Pythagoras number of function fields. Preprint (2024), https://arxiv.org/abs/2302.11425","work_id":"389884d5-cd19-4aec-94d3-55004a66f4a6","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"K.J. Becher, N. Daans, V. Mehmeti. The u-invariant of function fields in one variable. Preprint (2025), https://arxiv.org/abs/2502.13086","work_id":"8490e6fd-da8a-4cad-a2fd-200b058992bd","year":2025}],"snapshot_sha256":"d8616bd677d15aa095dbdb8feb874b69949d74e2a6b7dbbbc2f66a65ead9367d"},"source":{"id":"2605.13844","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T17:34:06.492939Z","id":"61050a93-b3a1-4257-9607-71f17e163f97","model_set":{"reader":"grok-4.3"},"one_line_summary":"Torsion quadratic forms over real fields of transcendence degree 1 over hereditarily Pythagorean bases decompose into orthogonal sums of 2-dimensional torsion forms.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Over real fields that are transcendence degree one extensions of hereditarily Pythagorean bases, every torsion quadratic form decomposes into an orthogonal sum of 2-dimensional torsion forms.","strongest_claim":"Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum of 2-dimensional torsion forms.","weakest_assumption":"The base field is hereditarily Pythagorean and the extension has transcendence degree exactly 1 while remaining real; if the hereditarily Pythagorean condition fails or the degree is higher, the decomposition may not hold."}},"verdict_id":"61050a93-b3a1-4257-9607-71f17e163f97"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:698e85b7777ae31f015227b523b887a8814b9c90350455ed9d6f72eb5625be13","target":"record","created_at":"2026-05-18T02:44:09Z","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":"b3a6b97133cf25cbd3256e3828ccb10cb7f19d10d032dbc8949e1d8f84e923c9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T17:59:30Z","title_canon_sha256":"af2b0c0e513247cf44c27e5d60aa57ad7a9c95ffaacf9567b414cba161c4eecd"},"schema_version":"1.0","source":{"id":"2605.13844","kind":"arxiv","version":1}},"canonical_sha256":"61637c329f3f45306982aea8a3dc91496e91248d22e4292f6821709ea5b37c22","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"61637c329f3f45306982aea8a3dc91496e91248d22e4292f6821709ea5b37c22","first_computed_at":"2026-05-18T02:44:09.381769Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:09.381769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I8+NkTnucHUvlk2P30zCP4Zq93CJ0j60gTk1sgCV16qEjWl0tBgXBu3PZLhz6D2RzeySImIAIchKJta/7ESjBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:09.382242Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13844","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:698e85b7777ae31f015227b523b887a8814b9c90350455ed9d6f72eb5625be13","sha256:473bc0dce5cf5126bd64cc086e981e0451ba9af7d46380052e4932402a5d2770"],"state_sha256":"e02eaed7a207db7f298d592bc3091795742361b7c30ca37061d406a72afd5f02"}