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Temporal Ratio Model (TRM): Extended Framework v2 Author: Matthew Dominik (Hollis Black) Affiliation: Dominik Research Institute, Cleveland, OH License: CC-BY 4.0 Abstract: The Temporal Ratio Model (TRM) proposes that the observed arrow of time arises from the normalized ratio between forward-directed (T■) and backward-directed (T■) temporal propensities. Unlike entropy-driven or thermal-time explanations, TRM defines the time direction as an emergent, continuous variable determined by asymmetry in underlying temporal contributions. This expanded version formalizes the mathematics, explores cosmological and quantum applications, integrates TRM into the broader Dominik Convexity Framework, and outlines empirical implications. 1. Definition of the Temporal Ratio: Let T■ represent forward-propagating temporal influence and T■ represent reverse-propagating temporal influence. The observed arrow of time is: T_obs = (T■ - T■) / (T■ + T■) T_obs ranges from -1 to +1, giving a continuous scale of temporal direction. 2. Boundary Regimes: Big Bang Epoch: T■ >> T■ implies T_obs ≈ +1, maximal forward arrow. Heat Death or Bounce: T■ ≈ T■ implies T_obs ≈ 0, temporal neutrality. Black Hole Horizons:
Both T■ and T■ diverge symmetrically, slowing local temporal flow. 3. Relation to Entropy: Entropy does not *cause* the arrow of time. Instead: dS/dt ∝ T_obs Entropy gradients reflect underlying temporal asymmetry but are not fundamental. 4. TRM and CPT Symmetry: CPT symmetry requires that time-reversed solutions exist. TRM accommodates this by allowing T■ > 0 without forcing macroscopic backward causation. 5. TRM in Quantum Systems: Quantum revivals, weak value anomalies, and two-state vector formalisms imply bidirectional temporal structure. TRM formalizes this by providing the ratio governing their macroscopic suppression. 6. TRM and Planetary Convexity: Within the Convexity Framework: - Life stabilizes increasing T■ dominance through recursive complexity. - Biospheric regulation increases temporal asymmetry coherence. - Evolutionary directionality becomes a secondary expression of TRM. 7. TRM as a Scaling Function: Define a temporal potential: Φ_t = T_obs * R where R is the local curvature radius or decoherence rate. Φ_t predicts: - time dilation gradients, - collapse asymmetry in quantum measurement, - directional structure in cosmological inflation.
8. Observational Predictions: - Regions of low entropy but high curvature may show TRM temporal thinning. - Early universe anisotropies correlate with temporal asymmetry injection. - Black hole information patterns reflect T■/T■ domain boundaries. Conclusion: TRM offers a simple, symmetric, normalized framework for explaining the direction of time, embedding entropy, CPT structure, decoherence, and cosmology within a single ratio-based law. It integrates cleanly into the Dominik Convexity Series and stands as a mathematically coherent speculative model suitable for further development.
