26-Mar-2026 Verion2: Gamma-DG Core I: Time-Core and the Minimal Temporal Arrow (Version 2). This revised version strengthens the original Core I manuscript while preserving its overall architectural role as the minimal axiomatic nucleus of Gamma-DG. The principal revision concerns the proof of the Minimal Temporal Arrow: instead of relying on a merely sequential restatement of the hypotheses, the argument is reformulated through witnessed stabilized tails. More precisely, witness data for stabilized tails are isolated, the corresponding family T(tau) of stabilized indices is shown to be non-empty, and the least stabilized tail is selected by the well-ordering principle. The contradiction argument is then carried out inside that least stabilized tail, separating a monotonicity case from a persistence case. In addition, the three temporal roles are sharpened structurally: absolute time is equipped with a preorder morphism condition, relative time with an acyclicity requirement, and undefined time with a partial resolution function. The manuscript also includes a fully worked prototype domain certificate for a number-theoretic descent setting, where the primitive data are given explicitly and the core axioms A0--A6 are verified concretely. Overall, Version 2 preserves the original purpose of Core I while improving mathematical clarity, proof substance, and readiness for transfer to concrete domains. Verion1: Gamma-DG Core I is the core axiomatic manuscript of the Gamma-DG framework, organized around structure, time roles, monotonicity, stabilization, the lift--descent interface, barriers or obstructions, and the transfer principle. It presents Core I as a reusable minimal nucleus extracted and axiomatized from existing concrete constructions. A supplementary note provides additional clarification of its mathematical hierarchy and structural position.
Yoshihito Kawanishi (Thu,) studied this question.