Finite-Horizon Freedom Maximization in Directed Acyclic Growth

We formalize a finite-horizon version of the Principle of Maximal Freedom (PMF) for growth processes on directed acyclic graphs under explicit finite-branching assumptions. Structural freedom at depth His defined as the number of admissible extension sequences of length H. We prove finiteness and existence of maximizing extensions at every finite state, yielding well-defined deterministic and probabilistic growth operators. A logarithmic formulation reveals a bounded log-sum-exp recursion analogous to finite-depth value functions. Within a bounded-memory, bounded-branching constraint family, simulations and combinatorial analysis demonstrate systematic divergence from random admissible growth. A bottleneck mechanism based on eligibility saturation is identified, and structural lemmas clarify the relation between boundary capacity and future multiplicity and illustrate the non-myopic character of finite-horizon PMF. No claims are made regarding infinite-horizon limits or continuum behavior.