TI06math — Foundations of TGG Time: Radon–Nikodym Geometry, Finsler Structures, and Diff⁺(ℝ) Reparametrization

This paper develops the strict mathematical foundations of generated timewithin the Trans–Integral (TI) program.Rather than treating time as a fixed background parameter, generated timeis modeled as an observer–dependent Radon–Nikodym flowdτθ = ρθ(t) dt,in which each observer produces its own proper time through a measurabledeformation of a reference measure. From this principle we construct a unified geometric framework calledTime–Generated Geometry (TGG).Generated time is shown to simultaneously define(i) a Radon–Nikodym measure flow,(ii) a canonical Finsler–type geometry on the time axis, and(iii) a natural action of the reparametrization group Diff⁺(ℝ).The nonlinear dynamics induced on this group admits affine maps as itsunique stable fixed points. These affine fixed points constitute the TGG–Lorentz regime.Lorentz–type transformations thus appear as linear equilibrium solutionsof the generated–time flow, not as fundamental spacetime postulates.Quantities such as TGG–velocity and TGG–energy arise here strictly asgeometric invariants of generated time, serving as geometric precursorsfor later physical interpretation rather than as physical observables orconserved quantities. The paper also introduces a correlation geometry for generated timesbetween observers, leading to a natural pseudo–metric on observer spaceand suggesting the possibility of intrinsic geometric bounds oncorrelations. TI06math deliberately adopts a restricted scope.It fixes the boundary within which generated time admits a coherentlinear representation, and renders explicit the stabilizations that aretypically left implicit.In doing so, it provides the rigorous mathematical groundwork upon whichthe physical reconstructions developed in TI06 are formulated, whilepreparing the structural perspective extended further in TI07 and itsCompanion volumes.---Version note (v1.1):Clarifies the methodological scope and interpretive position of TI06math.No mathematical results are changed; the revision fixes the boundary atwhich generated time admits linear stabilization and synchronizes thetext with subsequent TI developments. Related works: TI06 — Generative Time and Emergent Lorentz Geometryhttps://doi.org/10.5281/zenodo.17835394 Mathematical context and dependencies:TI00 — Generative Time, RN Flows, and the Geometric Foundations of Path Integralshttps://doi.org/10.5281/zenodo.17834833 TI05 — Measure Flows, Dynamic σ–Algebras, and the Calculus of Generated Timehttps://doi.org/10.5281/zenodo.17772983 TI06math isolates the purely mathematical TGG–Lorentz layer from the broader physical reconstruction presented in TI06.