DEPRECATED VERSION Mapping the Time: Explicit Toy Models of Projective Time Concrete Instantiations of an Emergent Temporal Framework

This paper constitutes the fourth installment of the Mapping the Time research program and provides explicit toy models that concretely instantiate the abstract framework developed in the preceding papers. Building on the kinematic emergence of time established in Paper 1, the tri-level projective architecture introduced in Paper 2, and the dynamical constraints derived in Paper 3, this work constructs low-dimensional, fully explicit examples that realize all structural requirements simultaneously. The goal is not to introduce new theoretical claims, but to demonstrate that the framework is operationally consistent and mathematically realizable. The paper presents two hierarchical toy models that explicitly implement: A tri-level structure consisting of an Extended Volumetric Substrate (EVS), Observed Spacetime (OS), and a Reduced Informational Manifold (RIM); Two strictly non-invertible projections linking these levels; A volumetric spatial coordinate in the EVS that projects to the temporal coordinate in observed spacetime; Projection-compatible dynamics defined on the extended level; Emergent irreversibility at the observed and informational levels despite reversibility at the extended level; A clear distinction between the abstract evolution parameter and observed time. The central results demonstrate that: All constraints derived in Papers 1–3 can be satisfied simultaneously within simple, explicit models; Structural irreversibility arises purely from projective constraints, without invoking thermodynamics, entropy, or fundamental time asymmetry; The induced dynamics at the observed and informational levels naturally form semigroups, even when the underlying dynamics remain reversible; The mechanism is robust under changes in coordinate choices and fiber geometry. Importantly, the analysis remains strictly structural and pedagogical: No physical interactions or forces are modeled; No empirical claims are made; No new postulates are introduced beyond those already established in the trilogy; The models are illustrative rather than physically realistic. By providing explicit realizations, this paper closes the foundational phase of the Mapping the Time program. It validates the internal coherence of the framework and clarifies how the abstract results of the previous papers can be instantiated concretely, preparing the ground for future work on physically motivated models or applications. Keywords: emergent time, toy models, projective time, non-invertible projections, tri-level structure, projection-compatible dynamics, structural irreversibility, pre-dynamical physics, spacetime emergence, information loss, mathematical physics, differential geometry, foundations of time, theoretical physics.