What Intervention Loss Certifies: Sharp Bounds for Factor Separation

An intervention loss can be small even when a frozen representation exhibits substantial factor mixing if the available actions fail to probe the relevant dependencies. We characterize precisely what separate upper bounds on such losses can certify about the representation. We show that the largest compatible off factor variation is given by a fractional cover linear program. An orthogonal variance decomposition reduces the exponential interaction structure to one variable per forbidden factor while an additive construction attains the resulting bound. To make the certificate data dependent we develop simultaneous confidence bounds that permit principled selection of certificate weights. We further reduce statistical conservatism through predictable variance adaptation and deterministic fiber envelopes without changing the certified quantity. On nonlinear neural encoders predictable concentration tightens a pair mask certificate from $0.00169$ to $0.00130$ compared with oracle leakage of $0.00085$; restricting to overlapping tests yields a nearly two fold reduction. On the public Pendulum generator three image only autoencoder families achieve raw output certificates within factors of $1.02$ to $1.04$ of exact finite reference errors using $32,768$ pairs per action with exhaustive range audit costs reported separately. These results establish dependence certification as a distinct objective from normalized disentanglement scores full identifiability and information retention. They provide a quantitative framework for determining what intervention based evidence can and cannot certify about a representation.