The one-period Gaussian Kyle model has exactly one equilibrium

In the one-period Gaussian Kyle~(1985) model, a single informed trader observes a Gaussian asset value, while independent Gaussian noise demand is submitted to competitive market makers. The market makers observe aggregate order flow and set the price equal to the inverse regression of value on order flow, while the insider chooses demand to maximise expected profit pointwise in the observed asset value. We prove that every equilibrium is Kyle's affine equilibrium: there is no other Borel-measurable equilibrium strategy, and the equilibrium pricing rule is unique up to almost-sure equality. Unlike earlier uniqueness results, ours leaves the original Gaussian model unchanged and imposes no restriction on admissible strategies beyond Borel measurability, thereby resolving a long-standing open question. The proof is probabilistic and convex-analytic and uses no complex analysis. Its central economic insight is that, under competitive pricing, the noise traders' expected loss is the covariance between noise demand and the price. The insider's profit maximisation forces this covariance to attain its sharp upper bound; in the normalised model, the maximal loss is exactly one half. We establish the bound for every strategy and prove its saturation by combining a posterior-resampling balance identity with a one-sided Stein inequality applied to the signed root of the insider's value function.