A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics

In a major advance and simplification of this field, we show that A Local Resolution of the Problem of Time - also viewable as A Local Theory of Background Independence - can at the classical level be described solely by of Lie's Mathematics. This comprises i) Lie derivatives to encode Relationalism, including via solving the generalized Killing equation. ii) Lie brackets to formulate Closure, via Lie's Algorithm suitably extended to accommodate Dirac and topological insights, producing generator Lie algebraic structures: Lie algebras or algebroids. iii) Observables defined by Lie brackets relations, recastable as explicit PDE systems to be solved using the Flow Method, and constitute observables Lie algebras. iv) The `passing families of theories through the Dirac Algorithm' approach to Spacetime Construction from Space, and to obtaining more structure from less internally to each of space and spacetime separately, are identified as deformations that work selectively when Lie Algebraic Rigidity is encountered. v) Reallocation of Intermediary-Object (RIO) Invariance: the general Lie Theory's commuting-pentagon analogue of posing Refoliation Invariance for GR. i) to v) cover respectively the Relationalism, Closure, Observables, Deformations and RIO super-aspects of Background Independence, Lie Theory moreover already collates i) to iii) and the internal case of iv) as multiple interacting aspects. The Problem of Time's multiple interacting facets are then explained as, firstly, having 2 copies of this Lie collation, 1 for each of spacetime and space primalities. Secondly, a Wheelerian two-way route between these two primalities, comprising v) and iv)'s `spacetime from space' version. We further develop the Comparative Theory of Background Independence thus. We can even give a `pillars of the Foundations of Geometry' parallel of our Background Independence super-aspects.