Mechanistic Localization Is Distribution-Relative: Receiver Sufficiency, Context Invariance, and the Support Shadow

Mechanistic localization is distribution-relative. A component set does not determine the optimal rule obtainable from the internal representation it exposes: even with the network, receiver, phenomenon, and literal support fixed, changing only probability weights can reverse the Bayes-optimal rule. We therefore include the realized probability state in the localization problem and distinguish a structural location $K$, a receiver-level presentation $(K,h)$ (the selected receiver set together with a decoder), and the induced action $[h{Q}_K{]}_{\mu }$. The receiver map determines what state is exposed; an admissible decoder class determines how that state may be used. We take all measurable receiver-level decoders as the maximal receiver-sufficiency class and recover architecture-specific continuation classes by restriction. This separation yields two different laws. Passive receiver refinement can only improve predictive adequacy, but it does not order context invariance. Under log loss, the value of revealing context is $I(Y_{\Phi };C\mid Q_K(X))$, and adding receiver $j$ changes this value by $I(Y_{\Phi };Q_j\mid Q_K,C)-I(Y_{\Phi };Q_j\mid Q_K)$; either sign occurs. Exact localization is strictly coarser. In finite $0$-$1$ settings, zero risk forgets probability weights and reduces to support-level factorization, functional dependency, and decision-reduct structure. Fixing the network does not fix this exact localization family: one Boolean OR network with fixed coordinate receivers realizes every nonempty upward-closed exact localization family by varying support alone. A circuit label is therefore not a complete mechanistic localization claim without the realized distribution, access, admissible downstream use, phenomenon, and adequacy criterion under which it is asserted.