When Does a Pathway Score Preserve a Mediated Effect? Aggregation Geometry, Minimax Adequacy, and Gap Inference for High-Dimensional Mediation
High-dimensional mediators are routinely collapsed into low-dimensional scores before mediation analysis, but aggregation can change the mediated estimand rather than merely reduce noise. We characterize when a linear score preserves the full mediated effect. Exact preservation for every outcome model occurs precisely when the score contains the covariance-adjusted exposure-predictive mediator direction, the classical sufficient reduction for exposure under a Gaussian inverse-regression model. We introduce a distribution-free minimax adequacy loss equal to the fraction of exposure-relevant mediator information discarded by the score and exactly linked to overlap recovered by aggregation. A split-sample inverse-Wishart correction yields unbiased estimators of retained and full exposure information in moderate dimensions. Under spherically symmetric outcome-effect benchmarks, relative aggregation error has a Cauchy location-scale profile, and paired influence-function inference targets the realized gap for an observed outcome. In simulations, paired interval coverage ranged from 93.3% to 97.5%. In an outcome-free TCGA lung adenocarcinoma audit of 1,314 smoking-responsive genes, the pooled gene-mean loss was 0.811 (descriptive 95% gene-bootstrap interval 0.797 to 0.823), versus a dimension-only random-subspace benchmark of 0.913; retaining promoter and non-promoter means separately reduced pooled loss to 0.625. Independent half-sample screening was unstable, but top-100 and top-250 cross-screen pooled gene-mean losses ranged from 0.827 to 0.932. The framework separates structural adequacy, benchmark fragility, and outcome-specific discrepancy.