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In a universal framework that expresses any market system in terms of statetransition rules, we prove that every DeFi market system has an invariantfunction and is thus by definition a CFMM; indeed, all automated market makers(AMMs) are CFMMs.  Invariants connect directly to arbitrage and to completeness, according totwo fundamental equivalences. First, a DeFi market system is, we prove,arbitrage-free if and only if it has a strictly increasing invariant, wherearbitrage-free means that no state can be transformed into a dominated state byany sequence of transactions. Second, the invariant is, we prove, unique if andonly if the market system is complete, meaning that it allows transitionsbetween all pairs of states in the state space, in at least one direction.  Thus a necessary and sufficient condition for no-arbitrage (respectively, forcompleteness) is the existence of the increasing (respectively, the uniquenessof the) invariant, which, therefore, fulfills in nonlinear DeFi theory thefoundational role parallel to the existence (respectively, uniqueness) of thepricing measure in the Fundamental Theorem of Asset Pricing for linear markets.  Moreover, a market system is recoverable by its invariant if and only if itis complete; and in all cases, complete or incomplete, every market system has,and is recoverable by, a multi-invariant. A market system is arbitrage-free ifand only if its multi-invariant is increasing.  Our examples illustrate (non)existence of various specific types of arbitragein the context of various specific types of market systems -- with or withoutfees, with or without liquidity operations, and with or without coordinationamong multiple pools -- but the fundamental theorems have full generality,applicable to any DeFi market system and to any notion of arbitrage expressibleas a strict partial order.

All AMMs are CFMMs. All DeFi markets have invariants. A DeFi market is arbitrage-free if and only if it has an increasing invariant