Constraint-Reduced Quantum Geometry in the Quaternionic Wavefront Framework: Exact Radial Inversion, Projective Continuum Dilation, and a Parameter-Free One-Cell Ultraviolet Completion

We construct the constraint-reduced quantum sector of the quaternionic wavefront framework. Hamiltonian-CMC reduction yields one Lichnerowicz-York solved-volume generator for transverse-traceless (TT) shape and root data. The H-phase selects X = R² and canonical inversion fixes the radial constraint. One-loop shells close through four derivatives modulo Euler and equation-of-motion directions; their a₆ remainder is O(M⁻²) on intrinsic double windows, whose projective exhaustion gives a minimal continuum dilation. The two-derivative family nevertheless inherits the essential two-loop invariant I₃. Harmonic localization identifies the shifted scalar Dirichlet-to-Neumann generator and |D_H| and fixes e^{-ℓ_P |D_H|}. Its trace has ν_H = 3π/2, contributes g_{6,P} = 1/(1290240π²), and has a TT Hessian that crosses zero at ℓ_P|p| = 6.514453... and is noncoercive in the ultraviolet. The same one-cell transfer uniquely fixes the coefficient-free H-energy defect 𝒫_TT (1 − e^{-2ℓ_P√𝒫_TT}). An exact Dawson bound makes the completed TT Hessian uniformly order-two coercive, and Schur elimination preserves the gap. The exact coarea block identity cancels the vertical metric against the Gaussian Hessian determinant; the induced Schur metric gives J_π = 1. The full SL(3,ℝ)/SO(3) affine Jacobian satisfies the sharp global bound 0 ≤ D² log J ≤ G₀/4; centered York decomposition and Kato transport have unit real Jacobian. Thus no independent measure or moving-frame exponent remains. The unresolved local quantity is the connected non-Gaussian Poisson-plus-storage shell Hessian; York caustics and finite-history boundaries remain global problems.