Non-Geometric Derivation of Pi from Dimensional Frequency Scaling

This paper presents a novel derivation of the mathematical constant pi from recursive frequency scaling, without invoking any geometric assumptions. Within the Holon Theory framework, dimensional transitions are governed by recursive coherence processes that determine the dominant oscillator frequency per spatial degree of freedom. A log-log regression of theoretical frequency values across dimensions D = 1 to 4 yields a slope of approximately -0.464, matching the predicted scale contraction derived from the recursion law r = pi / (1 + pi). This relationship closes at a structural fixed point, pi = r / (1 - r), without requiring arcs, areas, or coordinate metrics. The derivation reframes pi as a coherence-based invariant that emerges from recursive balancing of dual emission states, rather than as a purely geometric constant. Empirical testability is addressed via suggested observations of log-log scaling relationships in physical constants such as the fine-structure constant alpha(nu), or entropy thresholds in cosmological data. The broader implication is that spacetime structure may be emergent from coherence dynamics, suggesting a new pathway for unification beyond geometry-based models