Grammatical Structure of Prime Gaps as Cross-Domain Validation of Sub-Limit Dynamics

We analyse the sequence of consecutive prime gaps as an instance of a constrained generative system (CGS) with trajectory-consumptive dynamics (T3) and branched attractor regime (R2). The gap sequence exhibits a four-level grammatical structure: (i) 19 forbidden bigrams in the 11×11 transition matrix; (ii) 54% suppression of consecutive equal gaps; (iii) statistically overwhelming second-order memory (G²/df = 113, p ≈ 0, 47/49 state pairs significant after Bonferroni correction) ; (iv) a generation wall at k=4 identical consecutive gaps, penetrated only by the primorial gap 30. The 49 conditional distributions P (g₍+₂ | g₍+₁, gₙ) lie close to a low-dimensional manifold: a single SVD component captures 94. 4% of the variance, three components capture 99. 2%. The same four-level structure is derived independently by Hardy and Littlewood from analytic number theory. The convergence of two frameworks sharing no common ingredients constitutes cross-domain validation of the sub-limit dynamics framework on its eighth independent domain. This is the final paper in the core CGS series (C1–C9). Companion reproducibility script included. Reproducibility: The companion script cgsₚrimesₛuiteᵥ2. py (open access) reproduces all numerical results reported in the paper. Requires only Python 3 and NumPy. Usage: python cgsₚrimesₛuiteᵥ2. py limit --figures