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|<1/2. The 1/2 adversarial l-infinity boundary is sharp. Reusing the same frozen noise fields, the direct route recovers 1,000/1,000 cases at every n=6, 8, 10, 11 and 12. If the unprojected noisy response is transformed first, exact minimal-output recovery instead holds if and only if L(eta)≤tau<U(eta) for each fixed realization. This is an exact transformed-domain characterization, not an optimal binary support estimator or a data-computable universal threshold. The frozen n=6 calibration gives the pointwise Wilson-95%-qualified segment 0.32388115740199136≤tau<0.83601591378172. In the dimension scan, the transformed-domain fixed threshold at 1/2 falls from 1,000/1,000 exact recoveries at n=6 to 0/1,000 at n=12, while the raw route remains 1,000/1,000. The collapse is transform-domain noise amplification, not an intrinsic binary support-recovery limit. Möbius inversion, the monotone obstruction correspondence, blocker duality, raw midpoint separation and exact-oracle minimal-set enumeration are classical and explicitly attributed. The contribution is their domain-typed fixed-parent proof-to-intervention composition, task-sensitive routing, executable verification and finite physical proof of operation. Raw thresholding is selected for calibrated binary support-only recovery; Möbius coordinates for signed amplitudes, interaction order and cancellation structure; and Walsh coordinates for parity. The crisp chain is checked over all 7,773 antichain families for n≤5 with zero counterexamples. 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. Shared executable final seal: this package carries the complete public Cross-Field Spectral Capacity Probe. In its twelve-case deterministic ablation, capacity alone resolves 9/12 cases; capacity plus Silver spectral shape resolves 10/12; capacity plus Möbius carrier order resolves 11/12; and the complete diagnostic resolves 12/12. The engine, frozen outputs, hashes and replay instructions are included in SCP_TRIAD_FINAL_SEAL/. 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.