Certifiable State Under Information Loss

Data systems routinely transform state without preserving everything that made the state analytically trustworthy. This paper introduces a governed-state object, K=(K,Γ,E), and a proof-relevant graded derivability judgment, K⊩gc, to distinguish operational information from what a state remains entitled to certify. Governed transformations induce contravariant claim transport and witness transport; evidence-neutral transformations may preserve or weaken warrant but cannot manufacture stronger warrant. State composition and certificate composition can therefore diverge. Approximate state adds a second separation: Theta-style sketches can preserve valid certificate semantics while the informativeness of the resulting statistical claim weakens. The central result is that information loss, evidential loss, and loss of informativeness are distinct phenomena. A transformation can preserve computation, warrant, and informativeness independently.