Operational Liquidity State Beyond Price: A Formal Non-Identifiability Result for Constant-Product AMMs

WLRT Validation Series — Note No. 1 (VR-001), version 1.1. This note addresses Q-001: Does there exist an operationally defined market liquidity state that is not reducible to price alone? For a two-asset constant-product automated market maker, it defines liquidity state as the positive reserve pair and reserve-ratio price as the ratio of those reserves. It proves that price does not uniquely identify the reserve state across admissible pool depths. It further shows that states with the same reserve-ratio price can produce different outputs for the same finite input. A separate integer-arithmetic example illustrates the distinction under the published Uniswap v2 exact-input rule. The result is a formal, mechanism-specific answer to Q-001 within the validation program associated with Wave Liquidity Redistribution Theory (WLRT). It does not establish market prediction, causality relative to external prices, or applicability to all market structures. “PROVED” denotes the status of the stated mathematical results within the author’s validation register, not peer review or institutional endorsement.