Mixed Poincar\'e and Fefferman--Phong inequalities for measure potentials on $2$-PI spaces

We develop a local-to-global method for comparing a positive Radon measure potential with a reciprocal-scale multiplier on an unbounded complete metric measure space supporting a doubling measure and a weak $(1,2)$-Poincaré inequality. The central local object is the genuinely mixed oscillation integral \[
\int_B\int_B |u(x)-\widetilde u(y)|^2\,d\omega(x)\,d\pi(y), \] which cannot be controlled by the ordinary $d\omega\,d\omega$ Poincaré inequality when $\pi$ is singular. We first prove a fixed-outer-domain content--capacity estimate at every positive codimensional gain. A ball-growth condition for $\pi$ then yields capacitary domination and a Maz'ya-type trace inequality, hence the desired mixed Poincaré estimate on normalized balls. A bounded-overlap normalized cover globalizes the local result and gives two-sided Fefferman--Phong inequalities and equivalence of the corresponding energy completions.
The abstract theory is verified for Euclidean $A_2$ weights with generalized Schrödinger measure potentials, for reverse-Hölder function potentials, on Carnot groups, and for lower-dimensional singular measures. We also show that the natural structural assumption is the quadratic PI property: $A_2$ is not necessary, whereas a bare $A_p$ assumption with $p>2$ is insufficient. The Euclidean application yields form-domain equivalence, smooth form cores, self-adjoint realization, resolvent energy estimates, and local critical-multiplier bounds. Finally, the method supplies the mixed-measure step missing from a previously published $A_2$ generalized Schrödinger argument and gives a fixed-dilate finite-scale $A_2$ extension, for the naturally augmented measure, of the generalized Poincaré mechanism used in a later $A_1$ theory.