DEPRECATED VERSION Mapping the Time: Time as a Context-Dependent Projection - A Tri-Level Geometric Framework Without Ontological Hierarchy

This paper constitutes the second installment of the Mapping the Time research program and develops the structural architecture underlying the projective emergence of temporal ordering. Building on the kinematic result established in Paper 1—namely, that time in observed spacetime can arise as a non-invertible geometric projection—this work introduces a tri-level geometric framework in which temporal ordering emerges at multiple descriptive levels without assuming fundamental time, dynamics, or ontological hierarchy. The framework is organized around three mathematical structures connected by two non-invertible projections: Extended Volumetric Substrate (EVS) Observed Spacetime (OS) Reduced Informational Manifold (RIM) The central results demonstrate that: A spatial volumetric coordinate in the EVS projects to the temporal coordinate in observed spacetime. A spatial volume functional in observed spacetime projects to an ordering parameter in the reduced informational manifold. These projections are structurally dual and strictly unidirectional, avoiding causal circularity. The roles of “time” and “space” are context-dependent, determined by the projection level rather than intrinsic labels. Crucially, the paper remains pre-dynamical and purely geometric: No equations of motion are introduced. No evolution parameter is postulated. No thermodynamic, entropic, or causal arrow is assumed. No space is granted ontological priority over another. Temporal ordering is treated as a structural role induced by irreversible information loss under projection, not as a primitive feature of reality. The tri-level architecture shows how multiple notions of ordering—temporal and informational—can coexist consistently across descriptive levels. This work answers the structural question left open by Paper 1: how projective emergence of time can be organized coherently across multiple levels, setting the stage for the analysis of dynamical constraints in Paper 3 and explicit realizations in Paper 4. Keywords: emergent time, projective geometry, non-invertible projections, temporal emergence, pre-dynamical physics, geometric foundations of time, causal structure, Lorentzian signature, information loss, kinematic time, projection-induced order, foundations of spacetime, mathematical physics, differential geometry, structural time, time without dynamics, coarse-graining geometry, irreversible projection, spacetime emergence, theoretical physics.