PRIORITY FILING - INITIAL DRAFT
A fully revised and formatted version will follow.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 19, 2026
From the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics by Gregory Caldwell Beier (forthcoming 2026).
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
The companion paper on incentive compatibility (Beier, 2026r) micro-founds Assumption J4 - the independence of the judgment market's return from price efficiency - by partitioning the constraint environment into a price-reflected component theta_P and a forward-looking component theta_C that the price does not and cannot reflect. The willingness to pay for calibrated judgment depends on theta_C, which is structurally independent of the price's informativeness about theta_P (Proposition 9). A natural objection arises: as the Disclosure Incentive succeeds and constraint information enters prices, theta_C could shrink toward zero, undermining J4 and collapsing the mechanism's own foundation. This paper proves that theta_C > 0 is a permanent feature of any constraint environment that evolves over time, not a contingent property of the current information architecture. Three independent arguments establish the result. Proposition IC-1 uses the non-stationarity of physical constraint environments: because the parameters governing constraint dynamics change over time (climate shifts recharge patterns, technology alters extraction rates, regulation modifies interaction structures), any price that incorporates the current constraint state is immediately rendered incomplete by the next parameter change, regenerating theta_C as fast as disclosure fills it. Proposition IC-2 uses the combinatorial growth of interaction configurations: as new agents enter and existing agents change behavior, the number of distinct interaction configurations in a constraint domain grows combinatorially, and the price system's finite channel capacity cannot carry the new configurations as fast as they arise. Proposition IC-3 invokes computational irreducibility: for constraint environments governed by nonlinear dynamics with sensitive dependence on initial conditions, no finite computation - whether performed by the price system, an AI system, or the Disclosure Incentive itself - can reduce the forward-looking uncertainty to zero, because the system's future trajectory is computationally irreducible from its current state. The three arguments are complementary: non-stationarity regenerates theta_C through parameter change, combinatorial growth regenerates it through configuration change, and computational irreducibility guarantees that even perfect knowledge of the current state leaves theta_C > 0 about the future trajectory.
Keywords: theta_C, forward-looking constraints, inexhaustibility, non-stationarity, combinatorial growth, computational irreducibility, J4, judgment market
JEL Codes: D82, D83, C65, Q54
The companion paper on incentive compatibility (Beier, 2026r, Proposition 9) establishes that the principal's willingness to pay for calibrated judgment depends on the forward-looking constraint component theta_C rather than the price-reflected component theta_P. The argument partitions the information about the constraint environment theta into two components: theta_P (what the current price reflects) and theta_C (what concerns future constraint states that the price does not and cannot reflect because those states have not yet been realized). Since the principal's delegation problem concerns future decisions under constraint uncertainty, the demand for calibrated judgment is independent of how informative the price is about the current state.
The partition is correct at any moment in time. The question this paper addresses is whether the partition is stable over time. If the Disclosure Incentive is successful, constraint information that was previously in theta_C (invisible to the price) migrates to theta_P (incorporated into the price through disclosure). Does theta_C shrink toward zero as the information architecture improves?
The answer is no, for three independent reasons that this paper formalizes. Each reason establishes a mechanism through which theta_C is regenerated as fast as - or faster than - disclosure fills it.
Proposition IC-1 (Non-stationarity regeneration). Let theta(t) denote the state of a constraint environment at time t, governed by parameters phi(t) that evolve over time. Let theta_P(t) denote the constraint information reflected in the price at time t, and theta_C(t) = theta(t) - theta_P(t) denote the forward-looking component. Suppose:
(NS1) The parameters phi(t) are non-stationary: there exists no finite time T after which phi(t) = phi(T) for all t > T.
(NS2) The price at time t reflects information about theta conditional on the current and past parameter values phi(s) for s <= t but does not reflect information about future parameter values phi(s) for s > t.
(NS3) The constraint environment's future trajectory depends on future parameter values: for any t, E[theta(t+h) | phi(s), s <= t] differs from E[theta(t+h) | phi(s), s <= t+h] for some h > 0 with positive probability.
Then theta_C(t) > 0 for all t, regardless of how much constraint information has been incorporated into the price at time t.
Proof. At any time t, the price reflects theta_P(t), which incorporates all available information about the constraint state given current and past parameters. By (NS3), the future constraint trajectory depends on future parameter values that are not yet known. By (NS2), the price does not reflect information about these future values. Therefore the component of theta(t+h) that depends on phi(s) for s > t is in theta_C(t) for all h > 0.
By (NS1), there is no time T after which the parameters stop changing. Therefore at every time t, there exist future parameter changes that the current price does not reflect, and these changes affect the future constraint trajectory. The information about these future changes constitutes theta_C(t) > 0.
Formally: theta_C(t) >= I(theta(t+h); phi(s), s > t | phi(s), s <= t) for all h > 0, and by (NS1) and (NS3), this mutual information is strictly positive for all t. QED.
Empirical grounding. Every physical constraint environment satisfies (NS1)-(NS3). Climate change alters the parameters governing precipitation (affecting aquifer recharge), atmospheric chemistry (affecting carbon cycle dynamics), and extreme weather frequency (affecting infrastructure resilience). Technological change alters extraction rates, efficiency parameters, and the set of constraint-relevant activities. Regulatory change alters the institutional parameters governing agent behavior within constraint domains. None of these parameter processes is stationary, and all affect future constraint trajectories in ways the current price cannot reflect. The Ogallala Aquifer's depletion trajectory depends on future precipitation patterns under climate change, future irrigation technology adoption, and future agricultural policy - all parameter values that are unknown at time t and that no amount of current disclosure can determine.
Proposition IC-2 (Combinatorial regeneration). Let N(t) denote the number of agents operating in a constraint domain at time t, and let the constraint state depend on the interaction configuration C(t) - the specific pattern of which agents interact with which through the shared constraint. The number of distinct interaction configurations is |C(t)| = O(2^{N(t)^2}) for pairwise interactions. Let R_price denote the price system's information rate in bits (from the companion paper Beier, 2026c, Section 3.3). Then for sufficiently large N(t):
log_2 |C(t)| >> R_price
and the price system cannot carry the information required to distinguish among the possible interaction configurations, regardless of how efficiently it operates on the consensus value dimension.
Proof. The number of distinct pairwise interaction configurations among N agents is 2^{N choose 2} = 2^{N(N-1)/2}, since each pair either interacts or does not. The information required to specify the configuration is log_2 |C(t)| = N(N-1)/2 bits. The price system's information rate is R_price = (1/2) log_2(1 + SNR_1) bits (Beier, 2026c, Proposition A), which is bounded and does not grow with N.
For N > 2 * sqrt(2 * R_price + 1), the configuration entropy exceeds the price system's capacity: N(N-1)/2 > R_price. Since N(t) grows over time in most constraint domains (more agents, more activities, more interactions), the gap between the configuration entropy and the price system's capacity grows without bound.
The forward-looking theta_C includes the information about which new configurations will emerge as new agents enter the constraint domain and existing agents change their behavior. This information grows combinatorially with N while the price system's capacity to carry it remains bounded. Therefore theta_C(t) >= N(t)(N(t)-1)/2 - R_price, which grows without bound. QED.
Interpretation. The Disclosure Incentive's factorial coordination paper (Beier, 2026ag) established that the information value of N disclosures grows as Omega(N^2). This proposition establishes the complementary result: the information NEEDED to characterize the constraint environment also grows as Omega(N^2), because the interaction configurations grow combinatorially. The disclosure ecosystem can reduce theta_C - each disclosure covers some configurations - but the configuration space grows at least as fast as the disclosure volume, ensuring theta_C > 0 permanently.
Proposition IC-3 (Computational irreducibility bound). Let theta(t) evolve according to a dynamical system with sensitive dependence on initial conditions (positive maximal Lyapunov exponent lambda_max > 0). Let delta_0 denote the uncertainty in the current constraint state at time t. Then the uncertainty in the constraint state at time t+h grows as:
delta(h) >= delta_0 * e^{lambda_max * h}
for h in the regime where the linearization holds. No finite computation performed at time t - whether by the price system, an AI system, a disclosure ecosystem, or any combination - can reduce the uncertainty about theta(t+h) below this bound, because the prediction horizon for a system with positive Lyapunov exponent and initial uncertainty delta_0 is:
T_predict = (1 / lambda_max) * log(delta_threshold / delta_0)
where delta_threshold is the uncertainty level beyond which the prediction has no value. For h > T_predict, theta_C(t) about the state at t+h is bounded below by the system's natural variability, regardless of all information available at time t.
Proof. This follows from the definition of the maximal Lyapunov exponent and the theory of sensitivity to initial conditions in nonlinear dynamical systems (Ott, 1993). Two trajectories that begin delta_0 apart diverge exponentially at rate lambda_max. Any finite measurement at time t leaves residual uncertainty delta_0 > 0 about the current state (by the uncertainty principle for continuous systems, or by measurement precision limits for discrete measurements). The future trajectory is unknowable beyond the Lyapunov prediction horizon T_predict, because the exponential divergence amplifies the residual uncertainty beyond any useful threshold.
The price system, the disclosure ecosystem, and any AI system all operate on information available at time t. None can reduce delta_0 to zero (any measurement has finite precision). None can override the exponential divergence. Therefore theta_C(t) about the constraint state at time t+h for h > T_predict is bounded below by the system's attractor variance, which is strictly positive for any non-trivial dynamical system. QED.
Empirical grounding. Aquifer dynamics under coupled climate-extraction forcing have positive Lyapunov exponents when the system operates near threshold transitions (Scheffer, 2009). Atmospheric carbon dynamics have positive Lyapunov exponents through the ice-albedo, permafrost methane, and ocean circulation feedback mechanisms. Electrical grid dynamics under intermittent renewable penetration have positive Lyapunov exponents through the frequency stability mechanism. In each case, the prediction horizon is finite and the forward-looking theta_C about the post-horizon constraint state is bounded below regardless of current information quality.
The three propositions are complementary and operate on different timescales.
Proposition IC-1 (non-stationarity) guarantees theta_C > 0 because the parameters keep changing: even if you knew the current constraint state perfectly, you would not know what the parameters will be tomorrow, and the future state depends on the future parameters.
Proposition IC-2 (combinatorial growth) guarantees theta_C > 0 because the configuration space grows faster than the information system's capacity to map it: even if you knew all current parameters and interactions, new agents and new interactions are always emerging faster than any finite channel can carry.
Proposition IC-3 (computational irreducibility) guarantees theta_C > 0 because the dynamics themselves are unpredictable beyond a finite horizon: even if you knew the current state and all parameters exactly, the nonlinear dynamics amplify residual uncertainty exponentially, making the future trajectory unknowable beyond the Lyapunov time.
Together they establish that theta_C > 0 is a property of the constraint environment itself, independent of the information architecture's current limitations. No conceivable information architecture - no matter how rich the disclosure, how efficient the price, or how powerful the AI - can reduce theta_C to zero. The demand for calibrated forward-looking judgment is permanent, and J4 is secure because the constraint environment's intrinsic structure regenerates uncertainty faster than any architecture can resolve it.
Proposition 9 of the companion paper (Beier, 2026r) partitions theta into theta_P and theta_C and derives J4 from the fact that the principal's delegation problem depends on theta_C. The present paper's three propositions establish that this partition is permanent. The principal will always need calibrated judgment about forward-looking constraint environments because:
The parameters keep changing (IC-1), so current knowledge becomes incomplete as soon as the parameters shift. The interaction configurations keep growing (IC-2), so the constraint space the principal must navigate always exceeds what the price system can map. The dynamics are computationally irreducible (IC-3), so even perfect current knowledge leaves the future unknowable beyond the Lyapunov horizon.
J4 does not depend on the Disclosure Incentive failing. It holds even if the Disclosure Incentive succeeds completely - even if every constraint analyst in the world discloses every assessment with perfect calibration and the entire current constraint state is reflected in the price. theta_C is regenerated by the physics, the combinatorics, and the dynamics of the constraint environment itself. The mechanism cannot consume its own fuel because the fuel is produced by the world, not by the information architecture.
Proposition IC-1's assumptions (NS1)-(NS3) are empirically rather than logically necessary. A constraint environment with stationary parameters (if one existed) would not regenerate theta_C through the non-stationarity channel. The claim is that no economically relevant constraint environment has stationary parameters, which is an empirical generalization about Earth's physical, biological, and institutional systems rather than a theorem.
Proposition IC-2's combinatorial bound is a worst case. If interaction configurations are sparse (each agent interacts with a bounded number of neighbors rather than with all other agents), the configuration entropy grows as O(N * log N) rather than O(N^2), which still exceeds the price system's bounded capacity but at a slower rate. The bound is tightest for constraint domains with dense interaction networks (aquifers where every well's cone of depression can overlap with every other's) and loosest for constraint domains with sparse interactions (geographically isolated activities with minimal coupling).
Proposition IC-3 requires positive Lyapunov exponents, which not all constraint dynamics exhibit. Linear constraint environments (constant depletion rates, no feedback effects) have zero Lyapunov exponents and are predictable indefinitely. For such environments, IC-3 does not apply and theta_C about the trajectory is zero given perfect knowledge of the current state and parameters. IC-1 and IC-2 still apply (the parameters can change and new agents can enter), but the computational irreducibility argument specifically requires nonlinearity. The paper's claim is that the constraint environments most consequential for economic coordination - those involving thresholds, feedbacks, and interaction effects - are the ones with positive Lyapunov exponents, which is empirically well-supported (Scheffer, 2009) but not universal.