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The estimate $T \\geq \\frac{mn}{m+n} -1$ has been conjectured for some time but could only be proved under additional hypotheses for $\\theta$ and $\\kappa$. This paper proves a weaker estimate for $T$ while also reducing the strong estimate to a prominent conjecture on intersections of subvarieties of split "},"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":"2506.01123","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-06-01T19:00:18Z","cross_cats_sorted":[],"title_canon_sha256":"f90b20cefe09518c2e911081901598a5b06f2105e59c93ae264781cfc7d0fee3","abstract_canon_sha256":"330cab54a6692fad2d083634bdc2842da3513cc6feb0b61c4920d1f79ad990ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:13:46.909839Z","signature_b64":"pRObc+Ff/8v8k47kKC/KhkunkFdb2eP1xfCXdsyv2B3Pd94Igw7vmJFza6H1V+V+IOzZyBCHpz/NZTYoXn2oCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b88e95efe8083647bb67fa22198e7060b4a0ceb2cc5e038f54824f71bfea0df6","last_reissued_at":"2026-07-05T11:13:46.909431Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:13:46.909431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Transcendence degrees of fields generated by exponentials of products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Heinrich Massold","submitted_at":"2025-06-01T19:00:18Z","abstract_excerpt":"Let $\\theta=(\\theta_1,\\ldots,\\theta_m) \\in \\R^m, \\kappa=(\\kappa_1,\\ldots,\\kappa_n) \\in \\R^n$ be two tuples of real numbers each linearly independent over $\\Q$, and $T$ the transcendence degree of the field generated by $\\{\\exp(\\theta_i \\kappa_j) | i=1,\\ldots,m, \\; j=1,\\ldots,n \\}$ over $\\Q$. The estimate $T \\geq \\frac{mn}{m+n} -1$ has been conjectured for some time but could only be proved under additional hypotheses for $\\theta$ and $\\kappa$. This paper proves a weaker estimate for $T$ while also reducing the strong estimate to a prominent conjecture on intersections of subvarieties of split "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.01123","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.01123/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.01123","created_at":"2026-07-05T11:13:46.909486+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.01123v1","created_at":"2026-07-05T11:13:46.909486+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.01123","created_at":"2026-07-05T11:13:46.909486+00:00"},{"alias_kind":"pith_short_12","alias_value":"XCHJL37IBA3E","created_at":"2026-07-05T11:13:46.909486+00:00"},{"alias_kind":"pith_short_16","alias_value":"XCHJL37IBA3EPO3H","created_at":"2026-07-05T11:13:46.909486+00:00"},{"alias_kind":"pith_short_8","alias_value":"XCHJL37I","created_at":"2026-07-05T11:13:46.909486+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC","json":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC.json","graph_json":"https://pith.science/api/pith-number/XCHJL37IBA3EPO3H7IRBTDTQMC/graph.json","events_json":"https://pith.science/api/pith-number/XCHJL37IBA3EPO3H7IRBTDTQMC/events.json","paper":"https://pith.science/paper/XCHJL37I"},"agent_actions":{"view_html":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC","download_json":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC.json","view_paper":"https://pith.science/paper/XCHJL37I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.01123&json=true","fetch_graph":"https://pith.science/api/pith-number/XCHJL37IBA3EPO3H7IRBTDTQMC/graph.json","fetch_events":"https://pith.science/api/pith-number/XCHJL37IBA3EPO3H7IRBTDTQMC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC/action/storage_attestation","attest_author":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC/action/author_attestation","sign_citation":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC/action/citation_signature","submit_replication":"https://pith.science/pith/XCHJL37IBA3EPO3H7IRBTDTQMC/action/replication_record"}},"created_at":"2026-07-05T11:13:46.909486+00:00","updated_at":"2026-07-05T11:13:46.909486+00:00"}