Full attention exposes every token pair, whereas kernel attention compresses a sequence into a fixed-dimensional sketch. We show that this distinction becomes exponential at the first context length containing two competing candidates. On Min-IP over Boolean inputs, rank-one normalized kernel attention solves every sequence of length at most two exactly. In contrast, any single normalized nonnegative kernel-attention head that succeeds on all three-token sequences with error strictly below $1/2$ requires $2^{Ω(m)}$ features, even with arbitrary finite-dimensional tokenwise values and an arbitrary query-dependent affine readout. Dense softmax solves the same task with $m$-dimensional scores and constant temperature. The conclusion survives position-dependent token maps and a causal final query. As context length grows, the lower bound approaches the exact $2^m$-feature realization. Separately, for deterministic multihead, multilayer sketch models whose cross-token channels have finite alphabets, we prove a transcript lower bound linear in the number of independent answers and logarithmic in their alphabet size.
We present a complete formal proof that transformer architectures, when their internal update mechanisms satisfy a Bayes joint-distribution condition, implement exact Bayesian posterior inference. Working within the measure-theoretic kernel framework, we define a hierarchy of abstractions -- from the core Bayesian transformer, through semantic transformers with explicit update kernels, to full transformer blocks with QKV/attention/residual/MLP pipelines, and finally multilayer stacks -- and prove at each level that the Bayes joint semantics implies the update kernel equals the posterior almost everywhere. For the block-level architecture, we derive the explicit Bayes formula through Radon-Nikodym differentiation and prove its normalization. We additionally prove that the softmax attention mechanism induces a valid probability distribution over keys, establishing the bridge between the abstract kernel framework and concrete attention implementations. The framework makes no architectural assumptions beyond the Markov kernel structure and exposes explicit conditions under which a transformer block is provably Bayesian. In essence, when this joint distribution condition is satisfied, the forward computation of a Transformer is formally equivalent to a rigorous Bayesian posterior update.