Math Series Paper 0: The Transdimensional Identity Mathematical Invariants of a Fractal Scalar Manifold (A Structural Corollary of Time-Scalar Field Theory)

This paper formalizes a canon-safe mathematical keystone for the π–φ–primes series: a single transdimensional identity that packages (i) phase-closure invariance (Euler → π), (ii) scale-eigenvalue invariance (self-similarity → φ), and (iii) irreducible multiplicative decomposition (discreteness → primes) as projections of one invariant structure. The purpose is not to introduce new physical axioms, nor to re-found Time-Scalar Field Theory (TSFT), but to extract the minimal mathematical invariants that TSFT already necessitates through its closed-manifold, fractal, transdimensional architecture. The central result is an explicit operator-level and productlevel construction whose distinct limits recover the canonical e, i, π closure, the golden-ratio scaling fixed point, and Euler-product encoding of prime atoms. A numerical program is specified for testing stability of the proposed invariant under truncation, contour deformation, and scale reparameterization.