Operator-Gram Sensing at Vanishing Visibility: Hölder Minimax Laws and Coherent Heterodyne Seclusion

We study active coherent experiments in which a hidden vector- or operator-valued response is observed only through scalar powers. The local response has the form $$D_x(t) = t^r A(t) x,$$ where the field-amplitude visibility vanishes to order $r$, and the observable information factors through the hidden-codomain gauge-invariant right-Gram geometry. For a $C^m$ operator germ we obtain the sharp fixed-dimensional minimax exponent: $$p_{m,r} = 2 + \frac{2r}{m}.$$ For noninteger Hölder smoothness $\beta = k+\alpha$, the exponent becomes: $$p_{k,\alpha,r} = 2 + \frac{2r}{\beta}.$$ The lower bound is not stable under an arbitrary coherent query algebra: a remote nonzero same-mode field can act as a heterodyne reference and restore linear sensitivity. We therefore introduce the observable Gram-tangent kernel: $$\dot{K}_0(s,t) = F(s)^* G(t) + G(s)^* F(t),$$ and prove a continuous query-operator criterion separating three regimes: Exact quadratic seclusion The sharp quadratic regime Heterodyne-dominated shortcutting We then realize the Hölder law in an actual finite-difference coherent dark-port experiment, including a genuinely $C^{k,\alpha}$ but not $C^{k+1}$ hard pair, and give a structurally independent spherical self-calibration realization with $$(r, m) = (2, k-2).$$ Polarization, phase retrieval, classical quadratic-functional minimax theory, nulling interferometry, and curvature reconstruction are treated as prior-art inputs. The contribution claimed here is the coupling of: vanishing visibility, right-Gram quotient sensing, the sharp Hölder minimax law, and the global Gram-tangent heterodyne-seclusion criterion, rather than novelty of any one imported component.