Prefix-Exposure Spectra and Statistical Horizons in Saturated Autoregressive Continuation

For a known source prompt law, a target prompt law, and a saturated class of context-specific continuation distributions, we characterize the consistency boundary for target-weighted continuation estimation when prompt depth grows logarithmically with sample size. Under Poissonized sampling, we prove the exact minimax identity (\mathcal R_{t,H}^{\star,\mathrm{Pois},Q\mid P}=\mathbb E_{U\sim QH}r\ell(tPH(U))), so the governing object is the full target-weighted prefix-exposure spectrum rather than one effective sample size. If (Hn/\log n\to c), the source surprisal under (QH) satisfies a good large-deviation principle with rate (J), and the one-prefix risk has high-exposure exponent (\beta), then the fixed-sample risk exponent is (\infx{cJ(x)+\beta(1-cx)_+}). This yields regular high-exposure, rare-prefix frontier, and invisible-prompt phases, and identifies the information shells carrying the risk. For finite-state Markov prompt laws, Perron--Frobenius pressure makes the thresholds and Gaussian or endpoint-modulated critical windows explicit. Supplementary benchmark results describe how parameter sharing, deterministic decoding, rollout, and adaptive selection alter the relevant exposure geometry.