A Local Resolution of the Problem of Time. III. The other classical facets piecemeal

Abstract: We introduce nine further local facets of the classical Problem of Time, and underlying Background Independence aspects, the previous two Articles having covered one further facet-and-aspect each for a total of eleven. I.e. 1) Constraint Closure, 2) Assignment of Observables, 3) Space Construction from Less Structured Space, A) Spacetime Construction from Space, 0') Spacetime Relationalism, 1') Spacetime Generator Closure, 2') Assignment of Spacetime Observables, 3') Spacetime Construction from Less Structured Spacetime, and B) Foliation Independence. These are classically implemented piecemeal by the Dirac Algorithm for 1) and the Lie Algorithm for 1'). Taking Function Spaces Thereover: over phase space for 2) and over the space of spacetimes for 2'). Feeding Deformed Families into the Dirac Algorithm for A) and into the Lie Algorithm for 3) and 3'), working out whenever Rigidity is encountered. Spacetime's diffeomorphism-invariance and perturbative Lie derivative for 0'). Finally, Refoliation Invariance following from the Dirac algebroid of the GR constraints for B). Each of these approaches is moreover grounded in Lie's Mathematics in accord with this Series' claim. The generalizations required to extend these resolutions to solve combined rather than just piecemeal facets is the main subject of Articles V to XIII. We include novel work on, firstly, Closure, including a blockwise breakdown into more specific Closure Problems. Secondly, on a coordinate-free Multi-tensor Calculus suitable for joint treatment of Background Independence aspects, presenting in particular the Dirac Algorithm, and ensuing theory of observables and of deformations. Finally, aspect 3') is new to the current Article.