What does this equation mean?

This is the object this article is actually about, so its bookkeeping has to be stated in full before any number is produced. $D_{\rm det}$ is a map from pairs of “treaty configurations” — worldline, field state, coupling type, and the gap/switching pair used to build the spectrum — to the real interval [0,1] ; it carries no physical units, because it is built entirely from normalized probability densities over $\Omega$ , whose own units of inverse time cancel between numerator and denominator of every term in the integral. It inherits, rather than assumes, the property of being a genuine metric — non-negative, symmetric, zero exactly when $p_x$=$p_y$ almost everywhere, and satisfying the triangle inequality — because total variation distance is a metric on probability measures as a matter of elementary measure theory, independent of anything physical being fed into it. That inheritance is itself a check worth stating plainly: an ad hoc “difference score” invented for this article would need its metric properties verified by hand; $D_{\rm det}$ gets them for free because it was built from an object mathematics already understood.