Exact Deterministic Chaotic Quantization and Cauchy Scale Transport: Locality, Global Synchronization, and Dissipative Exponential Expansion

Generalized Boole transformations are known to connect three measure-theoretic regimes: a finite- measure mixing phase with an invariant Cauchy probability law, an infinite-ergodic critical point with Lebesgue measure, and a dissipative phase. We separate this invariant-measure classification from the exact finite-time transport of an initially Cauchy density. For the transported width γt+1 = Aγt + β/γt, β > 0, we prove sharp global time laws: for 0 < A < 1, the invariant Cauchy width γ∗ = pβ/(1 − A) attracts every positive initial width; at A = 1, where no finite invariant Cauchy probability exists, γt2 = 2βt + (β/2) ln t + Cc + o(1); and for A > 1, in the dissipative phase, γt = CAt − [βA/(C(A2 − 1))]A−t + o(A−t). We then introduce a positive spatial scale-field model and prove subcritical homogeneous-scale stability and supercritical Perron-mode homogenization on finite connected regular graphs. For the microscopic coupled map lattice, synchronization is shown to depend qualitatively on network topology. Nearest-neighbor lattices lose uniformly stable complete synchronization in the thermodynamic limit because their transverse spectral gap closes. In contrast, the complete graph retains a finite transverse separation and admits complete chaotic synchronization even as N → ∞; in the finite-measure phase its thermodynamic threshold is εc = 2√α(1 − √α), matching the functional form obtained independently for randomly coupled infinite-dimensional generalized-Boole networks. In the dissipative expanding phase, complete-graph transverse exponents approach ln α, whereas nearest-neighbor long-wave exponents approach ln A > 0. Thus invariant statistics, scale transport, and trajectory synchronization are distinct structures. We reserve the term deterministic Cauchy quantization for the finite-measure mixing phase, and discuss the supercritical law only as a deterministic dissipative scale-expansion mechanism, not as an invariant Cauchy vacuum or a complete inflationary cosmology.