Lemma 2: Derivation of the Invariant Photon Sea ($N_{\gamma} = 1,977,326,742$) and the 21-Element Anchor Matrix

We formalize Lemma 2 of the Fundamental Pocket Model (FPM) framework, deriving the exact integer invariant governing the cosmological background photon sea density, $N_{\gamma} = 1,977,326,742$. We show how this photon ocean arises as the invariant trace of a $21\times21$ dimensional discrete coupling tensor—the Anchor Matrix $A_{21}$. By integrating scale-damping dynamics across the 100-iteration fractal boundary, we demonstrate that the photon-to-baryon ratio $\eta^{-1}$ and the CMB energy density are exact structural consequences of recursive metric invariants rather than arbitrary initial conditions.