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We introduce nine further local facets of the classical Problem of Time, andunderlying Background Independence aspects, the previous two Articles havingcovered one further facet-and-aspect each for a total of eleven. I.e. 1)Constraint Closure, 2) Assignment of Observables, 3) Space Construction fromLess Structured Space, A) Spacetime Construction from Space, 0') SpacetimeRelationalism, 1') Spacetime Generator Closure, 2') Assignment of SpacetimeObservables, 3') Spacetime Construction from Less Structured Spacetime, and B)Foliation Independence. These are classically implemented piecemeal by theDirac Algorithm for 1) and the Lie Algorithm for 1'). Taking Function SpacesThereover: 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 LieAlgorithm 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 GRconstraints for B). Each of these approaches is moreover grounded in Lie'sMathematics in accord with this Series' claim. The generalizations required toextend these resolutions to solve combined rather than just piecemeal facets isthe 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 jointtreatment of Background Independence aspects, presenting in particular theDirac Algorithm, and ensuing theory of observables and of deformations.Finally, aspect 3') is new to the current Article.

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