Not framed as a globally invertible map, sPNP structure is a quotient plus a kernel-smoothed pushforward: configurations that differ only inside a fiber are identified by the projection and the observed spacetime quantity is the kernel-weighted average over that fiber. That makes emergence explicitly many-to-one without pretending the microscopic map can be reversed pointwise.
This also sharpens Gaussoherence. It is not just “smearing” in a vague sense; it is the heat-kernel coarse-graining that removes unresolved structure while preserving the physically relevant coarse data. If the underlying object is a density or a Fisher-geometric state, then the right monotonicity statement is the familiar one: coarse-graining cannot increase the distinguishable information. So the important theorem-level question is not whether kernels integrate — they do — but whether the physical observables cared about are stable under that projection in the appropriate sense.
The Jacobian lesson is a useful reminder that local regularity does not buy global invertibility. In sPNP, that is not a problem to hide from; it is the expected geometry of emergence. The theory lives on regular strata and local sections, while the projection itself is a covariant pushforward, not a globally reversible change of variables.
