When Does Sequential Detection Collapse to a Scalar? A Necessary and Sufficient Characterisation

For $K=2$ regimes, scalar thresholding of the Bayesian posterior is Bayes-optimal. For $K \geq 3$, the posterior evolves on a $(K-1)$-dimensional simplex $\Delta^{K-1}$, yet virtually all deployed detection systems reduce it to a scalar score without theoretical justification. We resolve this question completely. We introduce **Extended Decision Sufficiency (EDS)**—three linear constraints on emission densities and transition dynamics (Rank-One Emissions, Markov Factorisation, Normal Factorisation), verifiable in $O(K^2)$ operations from model parameters alone—and prove the following exact characterisation: **Main result.** EDS holds if and only if the Bayes-optimal stopping value function is constant on every level set of a scalar statistic $\phi : \Delta^{K-1} \to \mathbb{R}$. Sufficiency follows from algebraic closure of the Bellman operator $\mathcal{B}$ under EDS. Necessity is proved by a coupling-based fixed-point contradiction: for each failure mode we exhibit an explicit belief pair with equal $\phi$-values but a strictly positive total-variation gap on predictive densities, which propagates to a value-function gap via the Lipschitz bound on $\mathcal{B}$ and the $\rho$-contraction of $\mathcal{T}$, without assuming any structure on $V$ beyond continuity. Three consequences are sharp and quantitative: * **(i)** When EDS fails, *every* scalar rule—regardless of architecture, capacity, or training data—incurs a strictly positive, computable, algorithm-independent lead-time loss. * **(ii)** When EDS holds, optimal expected lead time is given in closed form by a hitting-time functional of a scalar Markov chain. * **(iii)** Under approximate EDS with deviation $\eta$, performance degrades at rate $O(\eta/(1-\rho))$, making $\eta$ a practical model-diagnostic with direct performance implications. Across 72 synthetic configurations and the CICIDS2017 benchmark, predicted and observed lead times agree within 0.4 steps (MAE). A controlled experiment with matched parameter counts confirms that the lead-time advantage is structural, not a regularisation artefact, thereby validating the theory as the sole causal mechanism.