22.9 Statistical and language-valued tangent sites

Most present experiments use deterministic perturbations, controlled synthetic data, or a fixed collection of language probes. Statistical LINCS needs obstruction objects that carry uncertainty before scalarization. Required ingredients include:

• support and overlap conditions for tangent probes;

• simultaneous inference across many paths, intersections, and blocks;

• uncertainty propagation through localization and quotienting;

• sequential validity under adaptive covers and repairs;

• robustness to biased or learned tangent estimators; and

• abstention under weak identification.

Infinitesimal causality makes this discipline especially visible: a smooth response field acquires causal meaning only under explicit intervention, support, and identification assumptions [ Mahadevan , 2026f ] . Similar care is needed when tangent directions are inferred from model activations rather than supplied by a physical protocol.

Language-valued systems add another layer. Argument roles, propositions, sources, and semantic equivalence classes are estimated by fallible models. A robust language tangent site should separate at least:

1. presentation variation such as paraphrase or reordering;

2. semantic variation that preserves the downstream decision;

3. structural variation that changes a role or restriction map; and

4. evidential variation that changes whether a claim is supported.

Inter-annotator variation, parser uncertainty, model-to-model variation, and context drift should remain visible through the repair decision. Otherwise a language model can certify a gluing repair using the same unsupported interpretation that generated it.