  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.
