One noisy inventory observation: an exact finite recourse reduction and two information thresholds

For two fixed soil models with interval initial stock and one bounded-error inventory observation, this theoretical note proves an exact finite recourse reduction for every successful fixed control prefix. At most three fixed tails suffice; the outer prefix optimization remains. A fully solved three-period construction separates the soil-label identification threshold 19/2 from the budget-value threshold 27/2, including the first-contact jump and a lower bound covering every prefix. Technical note. AI tools assisted with mathematical derivation, drafting, and internal proof review.