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

Exact support recovery, a necessary-and-sufficient noisy-output gate, 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))). The outputs are irreducible retained interaction supports and their minimal interventions. Möbius inversion, the monotone obstruction correspondence, blocker duality and exact-oracle minimal-set enumeration are classical and explicitly attributed. The contribution is their domain-typed fixed-parent proof-to-intervention composition with signed response amplitudes, an exact noisy-output gate, a transform-selection boundary, executable verification and a finite physical proof of operation. Shared numerical coordinates across applications do not identify their observables as one physical variable. The crisp chain is checked over all 7,773 antichain families for n≤5 with two independent transforms, Dedekind-pinned enumeration, independent semantic-intervention tests and zero counterexamples. Five additional 200-trial randomized implementation stresses cover declared regimes at n=6, 7 and 8. For Y=F_C+eta, exact minimal-output recovery holds if and only if L(eta)≤tau<U(eta) for each fixed realization. A separate frozen n=6 benchmark with two disjoint triples and 1,000 iid bounded-noise fields gives the pointwise Wilson-95%-qualified segment 0.32388115740199136≤tau<0.83601591378172. It is neither a simultaneous confidence region nor a universal tolerance interval. Exact controls select Möbius coordinates for monotone closure and Walsh coordinates for parity, while a counterexample rejects universal Möbius sparsity. In the finite moment-constrained mobility demonstration, conserved moments through orders 1, 2 and 3 require minimal non-cancelling unit-charge supports 4, 6 and 8; fixed two-event seeds require cardinality-minimal additions 2, 4 and 6. Co-released triad: The Möbius–Obstruction Instrument: From Response Surfaces to Irreducible Supports and Minimal Interventions (doi:10.5281/zenodo.21745631); The Finite-Binary Möbius Conjugate-Response Law (doi:10.5281/zenodo.21745774); and The Silver Web: A Cross-Field Carrier–Capacity Correspondence (doi:10.5281/zenodo.21746428). Each release is standalone; neither companion is a premise here, and the finite verification batteries are not pooled as evidence for one physical law. This work supplies the response-to-support-to-intervention role, the conjugate-response paper the finite-binary response role, and the Silver Web the spectral-capacity role. Publication materials and recorded results are CC BY 4.0; first-party executable code is Apache-2.0. Mixed files are licensed by portion, and bundled fonts retain their upstream license.