Cosmological Update Dynamics: Structural Future Tendency and the Origin of Selection Bias

This paper extends the selection-driven probability framework introduced in Cosmological Update Dynamics (CUD) v4 by providing a phenomenological definition of the selection bias term ω. In previous work, the realization probability of structurally accessible paths was expressed as P_i ∝ ω_i exp(-ds,i/ℓ_α), where d_s is the structural distance, ℓ_α is the structural coherence scale, and ω represents the relative preference among competing possibilities. While CUD v4 established the necessity of this selection bias, its physical interpretation remained unspecified. The present work introduces the concept of Structural Future Tendency and models the selection bias phenomenologically as ω_i ∝ exp(βχ_i + ηH_i), where χ denotes Critical Proximity (present structural pressure) and H denotes Structural Historical Weight (accumulated structural memory). This formulation establishes the temporal relationship H (past) + χ (present) → ω (future), suggesting that future realization emerges from the combined influence of structural history and present instability. The resulting normalized realization probability depends on three complementary factors:(1) structural similarity,(2) present criticality,and (3) accumulated structural history. CUD v4.2 provides a mathematical framework for describing the relative tendency of possible outcomes to be realized and offers a quantitative interpretation of future selectability within the broader CUD program.