A Minimal Irreversibility Memory Kernel in Selector-Time Theory: Path Dependence, CAL-like Stabilization, and Post-IRREV Contextual Typification

This article presents a minimal computational model for investigating how irreversible transitions may leave observable operational signatures in systems with path dependence, residual memory, and loss of invertibility. The proposal is developed within the context of Selector-Time Theory, or STT, an ontological framework under construction that distinguishes between a pre-operational domain of admissibility, selection, and hardening — the Second Degree — and a physical-operational domain of records, measurements, and observables — the Third Degree. The central objective is not to physically prove STT, but to transform an ontological hypothesis into a criticizable, reproducible, and adjustable computational protocol. To this end, Listing 01 introduces a minimal irreversibility memory kernel in which forward and reverse trajectories do not return to the same operational state, leaving residuals, hysteresis, and asymmetry. The model implements a discrete selection dynamics among candidate coherences, a CAL-like stabilization process, and a hardening criterion triggered when the selective difference, or tau gap, crosses critical thresholds associated with the CRIT and IRREV regimes. Listing 02 then adds a fundamental layer: post-IRREV contextual typification. In this stage, the event hardened by IRREV is not automatically treated as a final physical observable. It is represented as E_IRREV at the conceptual level and, in the computational protocol, by the operational analog E_IRREV_comp. This computational event passes through a contextual operation Pi_3, conditioned by a signature Sigma_3 and by a compatibility function K(E_IRREV_comp, Sigma_3). Only when this compatibility exceeds the threshold epsilon_3 does the event receive an observable form O3_type. The article therefore proposes the formal chain: Omega_adm -> REV -> CRIT -> tau(Omega_adm) = chi_sel -> CAL(chi_sel) -> gamma_sel -> IRREV(gamma_sel) -> E_IRREV -> E_IRREV_comp -> Pi_3(E_IRREV_comp, Sigma_3) -> O3_type. The computational results indicate that the model is able to produce operational marks compatible with residual memory, path dependence, transition boundaries, and contextual typification. The scientific value of the model lies in moving part of STT from a purely conceptual formulation toward a regime of mathematical and computational risk, where its hypotheses can be refined, compared, criticized, and eventually falsified.