The Möbius–Obstruction Instrument: From Response Surfaces to Irreducible Supports and Minimal Interventions

Sharp binary-noise recovery, an exact transformed-domain support characterization, and a finite physical proof of operation. For one fixed finite parent and a declared subset-response surface, this work defines A_tau(Y)=min{S nonempty: |Delta Y(S)|>tau} and I_tau(Y)=(A_tau(Y), blocker(A_tau(Y))). For arbitrary real-valued responses, the outputs are inclusion-minimal retained supports and their minimal hitting sets; they become obstruction supports and minimal interventions only under the declared monotone-closure and fixed-deletion semantics. Newly explicit sharp boundary: for a calibrated binary closure Y=F_C+eta, the raw-response threshold at 1/2 recovers the complete truth table and its minimal obstruction family whenever max |eta|