Entropic Flow and Prime-Domain Computation: A Structural Study of the Collatz Dynamics

We propose a convergence conjecture identifying the minimal structural anchor C(n) → 5·2^t, obtained by partitioning odd integers into prime intervals that assemble a coherent Prime-Train system. This organization explains the hierarchical distribution and tail confluence of primes within the Collatz flow, showing that apparent randomness follows a modular and convergent order. From the exponential relation e^π, the stable prime anchor p_* = 23 arises, generating the arithmetic progression p = 23 + 6k that acts as the operational backbone of a compensated Collatz field. On this foundation, an entropy-guided formulation re-examines the Gate-5 structure, the role of prime generators, and the reduction of mathematical entropy under singular boundary constraints. Within this framework, logarithmic space serves as an entropy coordinate in which both Collatz odd–even transitions and Goldbach pairings contract into an integrable, computable flow. The structural relation T ≈ E/Δf links complexity to scale resolution, enabling the analytical derivation of the convergence depth t^* through the Lambert-W function. The resulting logarithmic prime-generator provides a constructive bridge between Collatz dynamics and additive number theory, transforming Goldbach’s infinite search into a finite, entropy-bounded convergence mechanism. The study suggests that structural compensation and logarithmic balance may constitute a hidden organizing principle for prime equilibrium within arithmetic dynamical systems.