TabPFN is a transformer-based foundation model for tabular prediction that performs inference without task-specific training by conditioning on a support set and query inputs. Despite its strong empirical performance, its internal behavior on structurally difficult tabular geometries remains poorly understood. We study this behavior using zigzag persistent homology, treating TabPFN layer representations as evolving point clouds. We construct a controlled benchmark of synthetic tabular tasks with known true probabilities and varied intrinsic topology, including warped circles, tori, spheres, Hopf links, trefoil knots, and Swiss rolls. Across these tasks, we find that the topology of TabPFN's internal representation geometry is strongly associated with dataset-level reliability; for example, the zeroth homology group $H_0$ fragmentation count correlates positively with mean absolute residual across controlled tasks, and this association strengthens in a high-resolution warped circle case study at large sample size. Harder geometries induce a dual topological signature: increased $H_1$ loop activity and increased $H_0$ fragmentation, while the $H_1$ persistence becomes shorter-lived. These descriptors correlate with Bayes error, mean absolute residuals, and overconfidence. Our results suggest that zigzag persistence diagnoses the reliability of the inferred in-context task geometry and provides a context-level view of when TabPFN operates in topologically stressed regimes.
Teams measuring whether large language models (LLMs) recommend a brand face a reproducibility problem: ask the same question twice and the answer moves. Practice resamples each prompt a few times (commonly five) and averages, treating within-prompt resampling as the source of the noise. But a measured brand score moves for at least four separable reasons: within-prompt resampling, prompt paraphrase, model identity, and query language. We specify a crossed random-effects (generalizability-theory) decomposition that partitions the total variance of a response-level brand outcome into these four sources, and embed the components in a decision-study allocation that returns how many repeats, paraphrases, models, and languages to buy for a target reliability. We apply it to a fully crossed corpus of 12,933 LLM responses on 20 Central and Eastern European brands, 8 languages, and 3 models (GPT-5.2 and Gemini 3 Flash in parametric mode, Perplexity in grounded retrieval), with a stability subset of 1,435 cells resampled about five times. The outcome is per-response multilingual sentiment polarity. Query language is the largest systematic facet (26.5% of the variance of one response) against 1.5% for brand identity (ICC 0.0146), so a single AI answer carries almost no brand-discriminating signal. Once a cell term isolates pure resampling, resampling is 34.8% of variance and the brand-in-context interaction 29.6%; brand-by-language is 8.6% (a bilingual penalty) while brand-by-model and brand-by-prompt are near zero. Per unit of query budget, adding languages and models reduces relative-error variance far more than adding repeats: a repeat past the fifth reduces it by only 0.0003. Brand-ranking reliability stays low, near 0.01 for a single answer and about 0.36 at the full crossed design, so reliability is bought by spreading across languages and models, not by repeating one prompt.
High test accuracy and good aggregate calibration do not show whether an individual prediction is structurally supported by its evidence. In tabular decision systems, failures often occur when a feature family becomes unavailable, delayed, noisy, stale, or low-trust while the model remains highly confident. Existing calibration, uncertainty, selective-prediction, explanation, and perturbation methods provide scalar scores or attribution maps, but not a recomputable audit object answering: under a declared evidence-failure protocol, what trajectory makes this prediction lose support? We introduce Counterfactual Fragility Certificates (CFC), a model-agnostic protocol-level audit certificate-not a formal robustness certificate-that maps each prediction into an ordered evidence-failure trajectory summarized by greedy flip budget, normalized margin-collapse area, degradation thresholds, and fragility dominance score. Across seven tabular benchmarks and strong linear, tree-based, boosting, and neural baselines, CFC-FDS identifies independently brittle high-confidence cases with 0.915 AUROC, improving over the strongest non-certificate score by +0.405. The advantage persists across perturbation, permutation-importance, group-SHAP, baseline-choice, seed-variance, budgeted-review, and naturalistic field-unavailability checks. Under a 20% review budget, CFC-FDS captures 88.9% of brittle high-confidence cases, compared with 31.8-37.4% for confidence and energy scores. We also evaluate fragility-aware regularization and brittleness-aware temperature correction as secondary uses. CFC provides a concrete reliability framework for exposing high-confidence brittleness missed by ordinary score-centric evaluation.