Contextual Observability and Grammar Singularity for Compositional Task Families

We study Bayesian meta-learning when tasks are sampled from a latent compositional family rather than treated independently. A task family is modeled as a stochastic grammar over reusable linear modules, and grouped datasets are observed across many tasks. The target is the latent program law together with the shared module library, modulo explicit language symmetries. We prove four results. First, an exact sufficient-statistic reduction turns grouped compositional meta-learning into a finite mixture of matrix-normal laws. Second, local quotient identifiability holds under positive linearized observability and component separation, while in the single-occurrence regime zero contextual observability yields exact non-identifiability. Third, the posterior for the grouped predictive law contracts at the explicit non-i.i.d. rate $$ \delta_{m,n}^2=\frac{K_{\mathrm{eff}}\log m + rd^2\log(mn)}{m}, $$ and structural contraction occurs at the inverse exponent $\kappa$: $\kappa=1$ in regular families but $\kappa=2$ in an explicit duplicated-module singular family. Fourth, matching minimax lower bounds identify the same hardness parameters. These quantities yield a theory-native benchmark with provable phase transitions in observability, singularity, and anchor exposure.