No Free Parameters --- The Fixed Point Law

We take the exact self-reproduction equation T_K(Σ) = K as primitive. Nothing analytically or geometrically precedes this requirement. The point of the theory is that admissibility, overlap structure, induced geometry, and normalization are selected by the existence of a kernel that closes on itself. A kernel satisfying this exact self-consistency requirement is called a Distinguished Reproducing Kernel (DRK).

We study the realized hyperbolic class in which the closure obstruction can be computed explicitly. In that class, the weighted closure fails exact reproduction by a purely scalar obstruction. We compute that obstruction exactly, identify its residue with boundary contact volume, and reformulate the same asymptotics intrinsically using contrast-ball truncations. Any uncancelled scalar residue yields T_K = cK with c ≠ 1 and therefore violates the primitive fixed-point equation. Exact non-projective closure consequently forces all normalization channels to collapse into one unified exact normalization package.

This unified exact normalization forces complex dimension 2 and canonical holomorphic sectional curvature −1. On the exact complex-hyperbolic branch in complex dimension 2, the raw kernel diagonal is K(z,z) = c_K(1−||z||²)^−3, while the canonically normalized metric representative is g_* = 2 ∂∂̄(−log(1−||z||²)). It follows that the realized geometry selected by the equation is the canonically normalized complex hyperbolic plane CH², hence real dimension 4.