Hazard Conservation Under Translation: A Stricter Invariance for Honest Compilers

Mutual information is the polite invariant; hazard (rare-event rate) is the serious one.An honest compiler preserves the rate function I(θ) of target events across idioms up to gap-damped error; a fake one does not. We formalize conditions under which hazard commuteswith translation–compression and scale contraction, prove upper bounds on rate distortion,and give a falsifier that fits in a page. The law is simple: if your translation changes exponentsoutside the gap window, you lost sufficiency or you lied.1 SetupLet Πα→β be a translation–compression from idiom α to β. For an observable with cumulantgenerating function Λ(θ), define the rate I(θ) = supη{θη − Λ(η)}. Let γ be the mixing gap ofthe underlying runtime dynamics.Definition 1 (Hazard conservation). A compiler Π conserves hazard for class C if for allobservables in C,Iβ (θ) = Iα(θ) ± O(γ−1). (1)2 Sufficient conditionsTheorem 1 (Conservation under sufficiency). If (i) Π preserves a sufficient statistic for theobservable (likelihood ratios commute), (ii) compression is refinement-monotone, and (iii) G, Rresiduals are gap-damped, then hazard is conserved:supθIβ (θ) − Iα(θ) ≤ C γ−1. (2)Proposition 1 (Tightness). If (i)–(iii) fail, there exist observables for which |Iβ − Iα| isbounded away from zero independent of γ. In particular, style-first translations that project ontonon-invariant subspaces change exponents.3 Scale consistencyLemma 1 (RG commuting). If hazard is conserved across idioms at one scale and R is Markovand contractive, then hazard is conserved along the RG tower up to the same gap constant.4 FalsifierPick a task and a tilt θ. Estimate Iα(θ) and Iβ (θ) from streaming outputs before and aftertranslation. If the difference exceeds a gap window, translation is either insufficient (lost kernel)or dishonest (wrapper intrusion). The test fits on a page; no metaphysics required.1