Reconstructing Reserve State from Price and Exact Finite-Input Response in Constant-Product AMMs

WLRT Validation Series — Note No. 2 (VR-002), version 1.0. This note addresses Q-002: Which price and finite-input response quantities are determined by the reserve state of a two-asset constant-product AMM, and can those observations identify the state? In a specified continuous model, it proves that an exact pre-trade reserve-ratio price and an exact, model-consistent finite-input response, both referring to the same pre-trade state, uniquely identify the two reserves when the positive input and effective-input factor are known. The note gives the inverse formulas, the admissible observation domain, and a sensitivity result. Under a separately specified transition rule, it also derives the post-trade reserve-ratio price. The result complements VR-001 (https://doi.org/10.5281/zenodo.22920393), which establishes that price alone does not identify reserve state. “PROVED” refers to the stated mathematical results within the specified continuous model. The note does not claim prediction of external market prices, exact reconstruction from rounded blockchain outputs, peer review, or institutional endorsement.