Discrete Jackson $q$-Linearization of the Kaprekar Kernel: $3$-Adic Factorial Isomorphism, Causal Continuum Limits, and Spectral Models for Quadratic Irrationals

Motivated by Borel's open problem on the normal distribution of digits of quadratic irrationals on the real line $\mathbb{R}$, we develop an operator-theoretic framework connecting non-Archimedean Kaprekar dynamics with discrete quantum $q$-calculus ($q = 1/b = 1/10$). To model the suppression of non-local Archimedean digit carries along the periodic continued fraction carrier of $\sqrt{2} - 1 = [0; (2)]$, we construct a $q$-deformed dynamic kernel regulated by an infinite-state $S_3$-valued digit transducer. Our primary algebraic result establishes an exact $3$-adic valuation isomorphism for Jackson $q$-factorials, $\nu_3([n]_q!) = \nu_3(n!)$, proving that discrete $q$-quantization introduces no small denominator obstructions or hidden Galois phase shifts. Using this invariance, we prove the existence and uniqueness of a regular Hensel root $S_q^*(\Delta) \in 1 + 9\mathbb{Z}_3$ via a convergent quantum Bernoulli--Taylor series. In the continuum limit $q \to 1^{-}$, we establish the weak Prokhorov convergence of the resulting causal model measures to the Lebesgue measure under uniform Weyl remainder assumptions, verify boundary non-atomicity via Liouville's Diophantine inequality, and demonstrate structural compatibility with the positive spectral gap of the Gauss--Kuzmin--Wirsing transfer operator. Finally, we formulate open hypotheses regarding full Borel normality on $\mathbb{R}$ and higher-degree radical correspondences for sparse generating functions $F_\alpha(t)$ ($\alpha \ge 3$).