Time-Scalar Field Theory Derivation of π via Transdimensional Identity

We extend Time-Scalar Field Theory (TSFT), as introduced in the seminal work Zebra Poker: The Ultimate Unification of Physics [1], to derive the mathematical constant π through the Transdimensional Identity. Building on Ramanujan’s modular forms and hypergeometric series [2, 3], we show that TSFT’s scalar-time phase-closure across Θ-layers enforces an elliptic integral identity, yielding ultra-fast algorithms for π computation, including the Gauss–Legendre quadratic iteration, Borwein quartic iteration, and Chudnovsky series. This unifies geometric constants with TSFT’s conserved energy-flux framework, achieving record convergence rates without additional parameters.