When Can We Trust the Matching Principle? Robust Deployment Geometry Under Finite-Sample and Model Uncertainty

Reliable semi-dense matching is essential for modern geometric vision systems. Designed under a coarse-to-fine paradigm, it achieves an optimal balance between performance and computational cost. However, existing methods often struggle to provide well-quantified uncertainties, where catastrophic coarse-assignment failures are ignored, leading to truncated error distributions and severely misjudged geometric estimations. In this paper, we propose a lightweight, post-hoc overall uncertainty estimation framework that introduces a two-component calibrated Laplace mixture model with only 9 learnable parameters. The objective is to explicitly capture both the sharp local refinement noise and the broader tail of coarse-assignment failures. We introduce the Coarse-success posterior Refit (CoRe) method, a geometric refitting module that utilizes the posterior probability of coarse-assignment success as soft correspondence weights. Extensive experiments show that our method consistently improves downstream geometric accuracy across various pretrained-only matchers and robust estimators with minimal computational overhead. Our code is available at https://github.com/khoavpt/Probabilistic-matching.

Robustness, domain adaptation, photometric/occlusion invariance, sensor drift, and alignment style are treated as separate literatures with separate method families. Under label-preserving deployment shift they share one geometric object: the covariance Sigma_task = Cov_{Q_n}(n) of ways inputs can change without changing the label. CORAL, adversarial training, augmentation, metric learning, Jacobian penalties, and alignment constraints are not independent tricks--they are estimators of Sigma_task. Fix that object and the Jacobian penalty is pinned by a matrix Sigma' whose range must cover range(Sigma_task)--the matching principle. We prove optimality in a linear-Gaussian model (Thm. A), necessity of range coverage for any quadratic penalty that zeros deployment drift (Thm. G), and the same dichotomy at global minima (Thm. A_global). Wrong-direction/signal-aligned controls (Lemma C; Cor. E/E) and seven estimators (Lemmas D1--D7), plus label-free TDI, yield a falsifiable recipe when Sigma_task must be learned. Thirteen blocks (ML through Qwen2.5-7B) test matched vs isotropic vs wrong-direction penalties on geometry and deployment drift. Twelve match theory where identifiability holds; Office-31 is a named eigengap failure. Partial passes: geometry can improve without every headline task metric moving. A pilot 7B DPO run (one epoch, 240 pairs): matched style-PMH preserves Style TDI where standard DPO degrades it. We do not claim standard training reaches global minima (assumption (O) is open), that estimated Sigma_task is always identifiable, or dominance on every leaderboard. We claim a falsifiable design recipe: estimate Sigma_task, match Sigma', run the controls, report task and geometry separately.

Open-ended Theory-of-Mind (ToM) trackers emit valid beliefs absent from finite references. A finite-reference-plus-matcher pipeline marks unmatched outputs false, creating proxy labels that can reverse proper-score model selection on fixed outputs. Holding 259 beliefs and paired scores fixed, reference recoding lowers weighted prevalence from 0.783 to 0.295 and reverses strictly proper Brier risk: a frozen source-prior rule leads native confidence by 0.227 under reference labels and trails by 0.152 under blinded adjudication, in all six authored scenarios. A reference-only Platt recalibrator reverses further. An ICE-specific reversal appears in a released 301-question NQ-open DPR-BERT pipeline: its average-confidence baseline improves instance-level calibration error by 0.045 under exact match but worsens it by 0.074 under human correctness, with both intervals excluding zero. On independently authored OpenToM narratives, 90-96% of audited unmatched beliefs are literally true and the paired direction again reverses. An exact decomposition attributes the distortion to omitted truths, and a closed-form criterion correctly classifies comparisons from twelve released systems. Frozen-audit retrospective replay shows 50 attempted annotations recover ranking direction with probability at least 0.996. TriSource-Restore anchors full-frame reference labels and frozen automatic judgments to a probability-sampled human pilot, maintains at least nominal coverage, narrows intervals, and repairs confidence subject to a base-rate deployment gate.