DEPRECATED Mapping the Time: Time as a Projective Shadow Sufficient Geometric Conditions for the Emergence of Temporal Structure

This paper presents the first installment of the Mapping the Time research program and investigates the emergence of temporal structure as a purely geometric phenomenon. We show that a structure behaving as time can arise from non-invertible projection between smooth manifolds, without assuming time as a fundamental entity. Given an extended manifold S and a surjective, non-injective projection onto an observed space O, we derive sufficient geometric conditions under which one coordinate of O acquires the structural role of time. The analysis is deliberately pre-dynamical and kinematic: No equations of motion: No dynamical laws are introduced. No evolution parameter: No temporal flow is postulated. No arrow assumptions: No thermodynamic, entropic, or causal arrow assumptions are made. No ontological priority: No preeminence of the extended space is assumed. Core Thesis: Temporal ordering, causal stratification, and Lorentzian signature emerge as consequences of projection asymmetry and information loss, rather than from dynamics or physical postulates. Time is treated strictly as a structural role induced by projection, not as an intrinsic coordinate. This work establishes a minimal geometric foundation for emergent time and serves as the kinematic basis for subsequent papers addressing multi-level structure, compatibility with dynamics, and explicit realizations within physical frameworks. 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