PROJECTION TOPOLOGY OF THE INFLATIONARY VACUUM

We establish a rigorous mathematical foundation unifying the quasicrystal spacetime proposed by Fang et al. (2016) with the Representation Elasticity Principle. By formally defining the vacuum via the “cut-and-project” method from a higher-dimensional unconstrained state, we identify the geometric representation strain with the quasicrystalline phason field.
Through an exact variational derivation, we demonstrate that the non-minimal coupling between macroscopic spacetime curvature (R) and this hidden phason strain (S) generates an effective action mathematically identical to Starobinsky inflation (R+αR2). The coefficient α is derived deterministically from the phason rigidity modulus. To prevent unbounded phason drift and stabilize the vacuum, the projection manifold must possess absolute arithmetic rigidity. We identify the Deterministic Spectral Manifold (DSM-861), anchored at the class-number-one Heegner discriminant D = −163, as the exact arithmetic topology required to bound the inflationary epoch.