The Syntonic Capacity Theorem and Extensions: Gaussian, Correlated, and Non-Gaussian Adaptive-Tracking Bounds

Version 1.2 of the SCT, V2.0 of the GCE and NGSCT (following the July 2026 adversarial review) Combined deposit: Syntonic Capacity Theorem V1.2, Correlated Gaussian Extension V2.0, Non-Gaussian Correlated Extension V2.0. This version implements the corrections of the July 2026 adversarial review campaign (Corpus Status Register and Errata, DOI 10.5281/zenodo.21627116); the withdrawal it records is printed inside the corrected documents themselves. The Syntonic Capacity Theorem 1.2 gives a sharp collapse bound for first-order adaptive tracking. For a scalar system hit by i.i.d. Gaussian innovations \(\eta_k \sim \mathcal N(0,\sigma^2)\) at intervals \(T_{\rm env}\), coherent tracking is defined by \(V^\star < \sigma^2\), where \(V^\star\) is the stationary pre-innovation tracking-error variance. Direct theorem. Under linear relaxation \(\dot e = -\lambda e\), the stationary variance is \(V^\star_{\rm lin} = \sigma^2/(e^{2\lambda T_{\rm env}}-1)\), and coherence holds if and only if \(\lambda > \lambda_c = \ln 2/(2T_{\rm env})\). Converse theorem. By the Entropy Power Inequality, no deterministic strategy in the scalar Markov flow class \(\dot e=f(e,t)\), under (R1)--(R4) and the per-cycle entropy-contraction budget \(\int_0^{T_{\rm env}}\mathbb E[-f']\,dt \le \lambda T_{\rm env}\), beats \(V^\star_{\rm lin}\); the budget equals \(\lambda T_{\rm env}\) exactly for affine flows, and the bound limits every member's variance through \(V^\star \ge N_e\). Among autonomous flows the equality case selects exponential forgetting \(f(e)=-\lambda e\) (classically anticipated by Muth 1960 on the forecasting side). Version 1.2 states the criterion family: for \(V^\star < c\,\sigma^2\) the threshold is \(\lambda_c(c)=\ln(1+1/c)/(2T_{\rm env})\), and the constant is criterion- and phase-dependent, not structural: the time-averaged criterion \(\bar V