THE MATHEMATICAL HEART OF GEOMETRIC ESSENTIALISM Complete longhand derivations of: I. The Present Moment Soliton II. The Lock, the Pi Quad, and the Koenig-Ramanujan Identity III. The Scale Uncertainty Principle

This paper provides complete longhand derivations of the three central mathematicalresults of the Koenig framework. Part I derives the present moment soliton: the uniquesolution d_eff(ζ) = 3 + tanh(γζ) connecting d=2 (UV) to d=4 (IR) through d=3 at ζ=0. Weprove it satisfies the logistic ODE, that ζ=0 is the point of maximum dimensionalvelocity, and that the arrow of time is the positivity of dd/dζ. Part II derives the Lockequation α·3π/2 = φ−7 from the Rogers-Ramanujan continued fraction, proves