Opponent Modeling in Incomplete-Information Continuous Colonel Blotto

We study which reduced opponent states preserve the interim decision problem of a realized type in incomplete-information continuous Colonel Blotto. For any continuous observable summary of the opponent posterior, we prove an exact ambiguity identity: the worst-case payoff gap over posterior laws with the same state equals twice the sup-norm distance of the payoff slice from the closed observable-additive span. Thus a state is payoff-exact if and only if every payoff slice lies in that span. Specializing to exact-budget Blotto, we characterize first-order battlefield marginals: they are exact precisely for coordinate-additive opponent slices. The two-battlefield case is degenerate, whereas from three battlefields onward first-order marginals can discard payoff-relevant dependence; more generally, exact-budget allocation games exhibit a strict $r$-order marginal hierarchy. The same characterization separates estimation from representation: exact states admit vanishing statistical error, while non-exact states impose a sample-independent ambiguity floor for state-restricted learners. Finally, exact and efficiently learnable first-order states do not eliminate equilibrium complexity: approximate Bayesian Nash equilibrium remains PPAD-hard in a strict diagonal regularized Blotto subclass.