The Heegner–johnson Modularity Theorem

The Heegner-Johnson Modularity Theorem resolves the open algebraic, geometric, and analytic gaps of the discrete vacuum manifold. It proves that an ideal of norm eight hundred sixty-one cannot exist in the maximal class-number-one imaginary quadratic field because the primes three and seven are inert, ruling out naive ideal quotient constructions. Instead, the eight hundred sixty-one lattice nodes arise as the unordered pairs of points on the projective line over the residue field of the split prime forty-one, rigorously establishing the underlying geometry as an eighty-regular distance-transitive Johnson graph and resolving the six-regularity dilemma. The combinatorial Laplacian spectrum decomposes under permutation symmetry into explicit boundary and bulk representations, including a tripartite division of bulk modes into three symmetric sectors. A two-tier multiscale architecture resolves the finite-quotient modularity paradox by separating the finite graph partition function—completed as a quantum modular form—from the continuous transverse zero-mode theta series of level one hundred sixty-three. Finally, an order-seventy-two discrete symmetry operator cancels the modular framing anomaly in differential cohomology, while the singular modulus connects the lattice analytic torsion directly to Monstrous Moonshine representations and exact integer defect dissipation sums