Discrete Expansion and Emergent Time — The Sayır Formulation

This repository presents the foundational formulation and observational assessment of the Discrete Expansion and Emergent Time (DEET) framework. What DEET Is DEET is a theoretical framework in which physical time is not fundamental, but emerges dynamically from the gradient of a scalar phase field, denoted Θ. The Θ field is not a bookkeeping device or an auxiliary variable; it represents the minimal physical structure required for time to exist locally. Where the gradient of Θ vanishes, no physical time is produced and no dynamical evolution can occur. The local flow of physical time is defined covariantly as d\tau \equiv \nabla_\mu \Theta \, dx^\mu , and all physical rates are measured with respect to this emergent proper time τ. In regions where the Θ-gradient is smooth and timelike, standard kinematics and dynamics emerge naturally. Dynamical Consequence In the Newtonian limit relevant for galactic dynamics, spatial variations in the Θ-gradient lead to an effective acceleration of the form \mathbf{a} = -c^2 \nabla \ln u_0(\mathbf{r}), where u_0 is the local time-production rate. This acceleration generates a logarithmic effective potential without introducing dark matter halos, modifying gravitational laws, or postulating a universal acceleration threshold. Observational Test The present work applies DEET to a sample of 175 disk galaxies with well-measured rotation curves, focusing on their outer, approximately flat regions. Using a model-independent fitting procedure, we find that DEET reproduces the observed outer rotation behavior with very high statistical significance across the vast majority of the sample. Deviations are rare and strongly correlated with known observational limitations or non-equilibrium dynamics, rather than with galaxy mass, size, or baryonic content. Relation to Existing Frameworks Unlike ΛCDM, DEET does not invoke non-luminous matter components to explain galactic dynamics. Unlike MOND, it introduces no universal acceleration scale or interpolating function. Instead, DEET reinterprets galactic rotation curves as probes of the spatial structure of physical time production. Scope and Purpose This document serves as a foundational reference for the DEET framework and its application to galactic rotation curves. It does not claim to replace ΛCDM cosmology or to model inner-galaxy dynamics, bars, or feedback-driven processes. Its scope is deliberately restricted to the outer-disk regime, where the assumptions of quasi-stationarity and weak baryonic gradients are well justified. Reproducibility and Outlook All analyses are based on publicly available rotation-curve data and deterministic fitting procedures. The DEET framework makes clear, falsifiable predictions and motivates future tests involving environmental dependence, weak lensing, and independent time-sensitive observables.