Numerical Provenance and Decision Architecture: A General Mathematical Framework for Hidden Selectors, Architecture Sensitivity, and Claim Calibration

Modern scientific, medical, mathematical, engineering, computational, financial, regulatory, and automated decision systems frequently depend on numerical or procedural selectors that are not treated as primary objects of review. These selectors include thresholds, timing intervals, window lengths, weighting rules, cutoffs, regularization strengths, fitted lags, branch choices, normalizations, priors, stopping conditions, and data-processing conventions. This paper develops a general mathematical framework for Numerical Provenance and Decision Architecture. A decision architecture is represented as a composition that maps data and prior knowledge, through a vector of selectors and a sequence of transformations, to a calculated result and final claim or decision. The framework defines material and hidden selectors, provenance classes, source-attribution labels, provenance graphs, earliest free choices, admissible selector domains, local and global architecture sensitivity, decision-boundary fragility, uncertainty decomposition, architecture equivalence, architecture closure, and claim calibration. Seven formal propositions establish that materially controlling selectors carry an equal provenance burden whether they are explicit or hidden; numerical convergence cannot substitute for selector provenance; one-at-a-time sensitivity tests may miss selector interactions; threshold decisions may be unstable despite apparently precise values; and operational reproducibility does not imply architecture independence. Applications are developed for experimental science, medicine, artificial intelligence, engineering safety, computer-assisted mathematics, finance, law and public policy, manufacturing, cybersecurity, education, and other systems that convert data into consequential decisions. The framework is constructive rather than automatically adversarial. It preserves valid observations, calculations, and operational results while preventing the certainty of a final claim from exceeding the strength of the complete provenance chain supporting it.