Symbolic Field Interaction Theory (SFIT): Bounded Complexity, Topological Bifurcations, and Predictive Transformation Pathways

This paper develops the bifurcation and classification layer of Symbolic Field Interaction Theory (SFIT), building on earlier work on finite‑capacity reservoirs and nonlocal memory. Using a capacity‑closed SFIT toy model, we define a topological order parameter based on persistent homology of the coherence field and analyze its behavior under changes in a control parameter g. Finite‑size scaling of the order parameter, susceptibility, and Binder cumulant demonstrates that topological complexity is bounded: the first cohomology dimension H^1 grows only up to a finite maximum ⁡Cₘₐₓ and then declines, even as chaos and broad exploration persist. We identify a sharp topological transition at a critical coupling g_c, where the memory kernel activation creates a burst of new topological features. In contrast, thermodynamic observables derived from the reservoir energy density (average ρχρ_χρχ, variance, and fluctuation magnitude) remain smooth across g_c, showing that SFIT can generate a genuine topological threshold without a conventional thermodynamic singularity. The scaling of the finite‑size susceptibility peak yields a preliminary estimate of the universality exponent ννν, consistent with the Class I (Hamiltonian‑like) band defined in the periodic table of dynamical matter. We formulate a predictive bifurcation theory in which admissible transformation pathways are constrained by (Cₘₐₓ,ν) and capacity saturation. This connects SFIT’s microscopic feedback loop to macroscopic phenomena such as topological bursts, post‑peak simplification, and constrained evolution in the space of dynamical matter. Symbolic Field Interaction Theory (SFIT) is developed across three companion papers. Foundations, Regime, and Effective Dynamics introduces SFIT as a finite‑capacity slow–fast field framework in which nonlocal memory and noise emerge from a dynamical reservoir. Explicit Derivation of Nonlocal Memory and the Capacity‑Closed Reservoir provides the microscopic derivation of the SFIT feedback loop, showing how integrating out a finite‑relaxation reservoir generates a capacity‑closed memory kernel. The present paper, Bounded Complexity, Topological Bifurcations, and Predictive Transformation Pathways, applies SFIT to concrete models, demonstrating bounded topological complexity, sharp topological thresholds with smooth thermodynamic fields, and a universality‑based periodic table of dynamical matter. Together, the three works form a coherent SFIT stack from first principles to measurable predictions.