{"paper":{"title":"Transversality and Geometric Regularisation in Distributional Statistical Models","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Generic kernels in rich families place distributional statistical models in transversal position to degeneracy loci.","cross_cats":["math.DG","stat.ME","stat.TH"],"primary_cat":"math.ST","authors_text":"R. Labouriau","submitted_at":"2026-05-06T06:24:41Z","abstract_excerpt":"The distributional statistical framework replaces classical probability densities by distribution-kernel pairs $(T, \\varphi)$, where $T$ is a tempered distribution and $\\varphi$ is a rapidly decaying kernel. We develop the thesis that the kernel acts as a geometric regulariser, placing parametric statistical models in generic (transversal) position relative to degeneracy loci encoding non-identifiability, singular information, moment indeterminacy, and representation failure.\n  Using the transversality theorems of Whitney, Thom, and Mather, we prove a finite-dimensional weak transversality the"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for a generic kernel in any sufficiently rich family, the kernel-induced feature map avoids degeneracy strata of sufficiently high codimension. We establish verifiable conditions -- formulated as rank conditions on the Jacobian of the joint feature map -- under which the transversality hypothesis can be checked.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the family of kernels is sufficiently rich for the transversality theorems to apply, and that the rank conditions on the Jacobian hold for the specific models (location families, log-normal, etc.) without post-hoc adjustments.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Kernels serve as geometric regularizers ensuring generic transversality of kernel-induced feature maps to high-codimension degeneracy strata in parametric distributional models, via a weak transversality theorem from Whitney-Thom-Mather.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generic kernels in rich families place distributional statistical models in transversal position to degeneracy loci.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3e4c46121a101fa514bd3edec16d3286d004272f047f899657fcde9c17a37604"},"source":{"id":"2605.04536","kind":"arxiv","version":2},"verdict":{"id":"87e42739-ae26-4826-b9b2-6e325b506a11","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T17:22:54.862743Z","strongest_claim":"for a generic kernel in any sufficiently rich family, the kernel-induced feature map avoids degeneracy strata of sufficiently high codimension. We establish verifiable conditions -- formulated as rank conditions on the Jacobian of the joint feature map -- under which the transversality hypothesis can be checked.","one_line_summary":"Kernels serve as geometric regularizers ensuring generic transversality of kernel-induced feature maps to high-codimension degeneracy strata in parametric distributional models, via a weak transversality theorem from Whitney-Thom-Mather.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the family of kernels is sufficiently rich for the transversality theorems to apply, and that the rank conditions on the Jacobian hold for the specific models (location families, log-normal, etc.) without post-hoc adjustments.","pith_extraction_headline":"Generic kernels in rich families place distributional statistical models in transversal position to degeneracy loci."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.04536/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T11:40:31.895527Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.726459Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:21:08.134493Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"6aa0552e74ce36a4c5576842c7d7682884337da4f4cf2390dac954febaceb681"},"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"}