The Reflexive Path-Law Framework: Unpredictability, Identification, and Market Equilibrium

Abstract

Return unpredictability is an observational property relative to a filtration, horizon, target, loss, and strategy class; it does not by itself establish correct valuation, no-arbitrage, net non-exploitability, causal exogeneity, or dynamic stability. This paper develops the Reflexive Path-Law Framework (RPLF), which represents a market as a constrained feedback system on the joint law of prices, order flow, liquidity, beliefs, balance-sheet capacity, and institutional constraints. A law-to-policy map and a deployment-and-clearing map define a policy-dependent path-law operator, so equilibrium is the fixed point of the law agents learn from and alter through their actions. The paper proves three groups of results. First, observationally equivalent Gaussian price--flow models generate the same joint price--flow law while implying different interventional price responses. A bound on the common-news channel yields a sharp interval for structural impact; removing the bound gives complete non-identification, whereas excluded idiosyncratic flow contracts the interval. Second, a stationary exponential Hawkes process and a Cox process driven by a square-root diffusion can share the same mean intensity, Bartlett spectrum, and asymptotic dispersion despite opposite event-feedback mechanisms. Spectra below the Poisson floor reject the regular Cox class, but do not identify a unique alternative. Third, a Wasserstein contraction theorem gives existence, uniqueness, geometric convergence, and intervention sensitivity for reflexive equilibria. A local bound decomposes loop gain into persistence, impact, policy response, volatility targeting, and financing constraints, while a separate spectral criterion characterizes stability of a finite-dimensional expectational equilibrium map. The framework also separates martingality from equality to a declared valuation benchmark and shows that deterministic path representations cannot restore causal identification lost in the observed law. It therefore replaces ``market efficiency'' with five declared predicates: predictive, net non-exploitability, fundamental, causal, and equilibrium stability. The empirical implication is a disciplined protocol combining observational moments with instruments, interventions, liquidity variation, higher-order event statistics, and explicit transport assumptions. Return unpredictability can coexist with endogenous price formation and fragile equilibrium; stronger efficiency conclusions require a transparent ledger of information, valuation, identification, selection, and deployment assumptions. The claim ledger states the epistemic status and boundary conditions of every result, separating proved statements from empirical designs and maintained assumptions.