Correction, 30 October 2026. Theorem 10.1 (corrected). Let each agent i observe a set P_i of pi >= 1 dimensions no other agent observes, and let at least a fraction rho_2 of the dimension pairs (a, b) with a in P_i, b in P_j, i != j, carry constraint interactions.
Theorem 12.1(iv) (corrected). Withdrawn for fixed capacity.
Theorem 13.1(iii) (corrected). Let dLD/dt = G(LD) - sigma(LD) on LD >= 0 with 0 absorbing, G continuous and concave with G(0) = 0 and G <= 0 beyond some K_c (for instance G(LD) = r LD (1 - LD/K_c)), and sigma_b convex and strictly decreasing. A summary of this correction is on the papers page at https://www.susarb.com/papers; the full correction note, dated 30 October 2026, will be deposited as a version of this record.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 2026
version 3.4
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.
This paper identifies five axioms from which the core theorems of the Prolegomena follow. Axiom 1 (Observation-Maintenance Identity): observation of a coordination-dependent structure and maintenance of that structure are the same act. Axiom 2 (Bounded Capacity): the total interpretively coherent observation bandwidth of any finite population is bounded above. Axiom 3 (Compression Inevitability): communication between finite agents necessarily compresses multi-dimensional sources. Axiom 4 (Irreducible Heterogeneity): cross-dimensional constraint interactions are identifiable at actionable resolution only through observation that spans both interacting dimensions. Axiom 5 (Reflexivity): for coordination-dependent sources, observation and the source are coupled through the agents who both observe and constitute the source. From these axioms together with Shannon's rate-distortion theory, the paper derives five theorems: Universal Excess Distortion (any scalar codec on a multi-dimensional Gaussian source produces strictly positive excess distortion above a source-dependent rate threshold), the Beier Factorial (combinatorial growth in discoverable constraint interactions through heterogeneous disclosure, with a proved quadratic lower bound under pairwise coupling and a combinatorial ceiling of O(2^N) under deep higher-order coupling), Universal Dissipation (coordination-dependent dimensions receiving insufficient observation bandwidth lose coherence at a rate bounded below by the rate-distortion function), Universal Goodhart (scalar targets cause agents to reallocate observation bandwidth toward the target, accelerating constraint-dimension dissipation), and the Reflexive Phase Transition (a stability threshold at S = 1 with self-stabilizing dynamics above and self-destabilizing dynamics below). All proofs are contained in this paper. The Grossman-Stiglitz Resolution through track record capital is presented as the first application of the axiomatic framework to a specific institutional design, demonstrating how the axioms operate in the market setting. Three conjectures are stated: Dimensionality Growth, the Encoding Theorem, and the Asymmetry Principle. The five axioms are shown to be independent. The Beierspace fraction b parameterizes the framework's applicability, producing differential predictions that scale with the coordination-dependence of the source. One empirical calibration is presented.
Keywords: axioms, coordination, compression, observation-maintenance identity, bounded capacity, compression inevitability, irreducible heterogeneity, reflexivity, rate-distortion theory, dissipation, Goodhart's Law, Beier Factorial, phase transition, Beierspace, eigenspace, formal foundations
The axioms operate within Shannon's rate-distortion theory (Shannon, 1948, 1959; Berger, 1971; Cover and Thomas, 2006). The framework provides: a source V = (V_1, ..., V_K)' with K dimensions and positive definite covariance Sigma with eigenvalues lambda_1 >= ... >= lambda_K > 0; a codec mapping V to a lower-dimensional signal; a rate-distortion function D*(R) characterizing the minimum achievable distortion at compression rate R; and a distortion measure (mean squared error throughout; robustness to alternatives is addressed in the Boundaries section).
Definition 2.1 (Eigenspace quantity). A quantity whose statistical properties arise from physical law and persist independently of whether any agent observes or acts on them. The flow rate of an unmanaged river. The decay rate of a radioactive isotope. The thermal conductivity of copper.
Definition 2.2 (Beierspace quantity). A quantity whose statistical properties are partially or wholly constituted by the beliefs, decisions, and coordinated actions of the agents who observe and act on them. The market capitalization of a firm. The trust in a political institution. The adoption rate of a network platform.
Definition 2.3 (Beierspace fraction). For any source V, the Beierspace fraction b(V) in [0,1] is the share of V's total variance attributable to coordination-dependent rather than physically determined drivers (Beier, 2026k). When b = 0, V is purely eigenspace. When b = 1, V is purely Beierspace. Most real-world sources occupy the interior.
Theorem 2.4 (Mixing Theorem; Beier, 2026i, Theorem 8.1). The eigenspace-Beierspace distinction is not a partition but a continuous spectrum parameterized by maintenance bandwidth. Any dimension's position on the spectrum is a function of the observation bandwidth directed at it: dimensions with high maintenance bandwidth behave as eigenspace (stable, persistent); dimensions with low maintenance bandwidth and high coordination content behave as Beierspace (unstable, observation-dependent). The same dimension can move along the spectrum as maintenance bandwidth changes.
Proof. See Beier (2026i), Theorem 8.1. The proof derives from Axiom 1 applied to dimensions with varying Beierspace content and the rate-distortion characterization of the maintenance-coherence relationship.
Why b matters for this paper. The Beierspace fraction parameterizes every prediction the framework makes. At b near 0, only the Excess Distortion Theorem applies (compression wastes bandwidth but does not degrade the source). At b near 1, all five axioms operate at full strength and the full consequence set follows. The framework produces differential predictions that scale with b: more dissipation, stronger Goodhart effects, and sharper phase transitions in higher-b domains. Section 15 provides an empirical calibration.
Three measurement proxies for b are developed in Beier (2026k): analyst disagreement decomposition, Tobin's Q decomposition, and options-implied variance attribution. Each has limitations identified in that paper.
Axiom 1. For any source with Beierspace fraction b > 0, observation through an information channel and maintenance of the source's dimensional coherence are the same act.
(1a) Every bit of observation bandwidth directed at dimension k updates the observing agents' models toward mutual consistency on dimension k, increasing dimensional coherence.
(1b) Every bit of maintenance bandwidth directed at dimension k constitutes an observation, because maintaining coordination requires generating and transmitting information about the current state.
(1c) Dimensions receiving zero observation bandwidth receive zero coherence maintenance. Coordination on those dimensions decays.
Scope. The axiom applies only when b > 0. For purely physical sources, observation is not maintenance.
Status. This axiom is an empirically motivated starting point for the deductive system, in the tradition of applied axiomatics from Newton's Axiomata through Kolmogorov's probability axioms. It is chosen because it matches the observed relationship between institutional observation and institutional persistence, and it is productive of consequences that can be tested against the world. First stated in Beier (2026i).
Axiom 2. The total interpretively coherent observation bandwidth of any finite population of agents is bounded above at any moment. The bound depends on population size, individual cognitive resources, shared interpretive infrastructure, and available technology. It can change over time but is finite at every instant.
Clarification. The bound is on the rate at which agents can receive information, interpret it through shared models, and integrate it into coordinated action. Data received but not interpreted does not maintain coordination.
Axiom 3. Any agent with a finite internal state, transmitting their assessment of a multi-dimensional source V to another agent with a finite internal state, must compress. The compression is a necessary consequence of communication between finite agents.
Formal content. Agent i has representational capacity C_i bits. Agent j has capacity C_j bits. Source V has entropy H(V). The transmission carries at most min(C_i, C_j, C_channel) bits. For any multi-dimensional source with continuous components, H(V) exceeds any finite bound and the transmission is lossy.
Why this is not derivable from Shannon alone. Shannon's source coding theorem takes the channel rate R as given. Axiom 3 identifies the deeper source of the rate constraint: the agents themselves are finite. Even with an infinite-capacity channel, communication from a finite agent to another finite agent compresses, because neither can represent V without loss. The compression is in the endpoints, not only in the wire.
Axiom 4. Cross-dimensional constraint interactions are identifiable at actionable resolution only through observation that spans both interacting dimensions. No amount of observation confined to a single dimension can identify the mechanism, magnitude, and conditions of an interaction involving a dimension outside the observation set, at the resolution required for agents to intervene before the interaction produces consequences.
Formal content. Let constraint interaction I_{jk} exist between dimensions j and k of V. Let agent i observe subset O_i. If j and k are not both in O_i, agent i cannot identify I_{jk} at actionable resolution - the resolution at which the mechanism of the interaction (how j affects k), its magnitude (by how much), and its conditions (under what circumstances) are specified precisely enough for an agent to adjust their decision before the interaction's consequences materialize. The interaction I_{jk} is identifiable at actionable resolution only by an agency A_S whose combined observation set includes both j and k.
The latent variable objection. Factor analysis and latent variable models can infer the existence of unobserved dimensions from their effects on observed dimensions. The axiom does not deny this. What the axiom states is that inferring the existence of an interaction is not identifying it at actionable resolution. You can infer from return correlations that some unobserved factor is driving co-movement. You cannot infer from return correlations alone the specific mechanism by which aquifer depletion beneath a data center interacts with grid load in the surrounding region to produce a cascading infrastructure failure at a specific threshold, within a specific timeline. The identification of mechanism, magnitude, and conditions - at the resolution required for actionable intervention - demands observation that spans the interacting dimensions.
What "actionable resolution" means. The criterion is external to the Prolegomena's specific constructs. It references real-world decision-making: can an agent, given the identification, adjust their decision to avoid or mitigate the interaction's consequences before those consequences materialize? If the identification is "some unobserved factor exists" (the latent variable output), the agent cannot act because the mechanism, magnitude, and conditions are unknown. If the identification is "aquifer depletion at rate X interacts with grid load at threshold Y to produce failure mode Z within timeline T" (the heterogeneous agency output), the agent can act. The axiom states that the second type of identification requires observation spanning both interacting dimensions.
Axiom 5. For any source with Beierspace fraction b > 0, observation of the source and the constitution of the source are coupled: the agents who observe V are participants in the coordination that constitutes V, so every observation changes the source being observed.
Formal content. Let V(t) be a coordination-dependent source. Let O(t) denote the aggregate observation. The dynamics are coupled:
V(t + dt) = f(V(t), O(t), epsilon(t))
O(t) = g(V(t), M(t), eta(t))
V determines what is observed. The observation determines how V evolves. Neither is independent of the other.
Distinction from related concepts. This is not the Heisenberg uncertainty principle (physical disturbance by measurement). The disturbance here is informational: observation updates beliefs, beliefs change coordinated actions, actions change the source. This is not performativity in the MacKenzie (2006) sense alone. Performativity describes how observation reshapes the source. Axiom 5 includes this and adds the complement: how the absence of observation degrades the source (by Axiom 1), and how the degradation feeds back into subsequent observations.
The five theorems below are derived from the axioms plus rate-distortion theory. Each proof traces its dependencies explicitly.
Remark 8.1. Between any two finite agents communicating about a continuous multi-dimensional source V, information is lost. This follows directly from Axiom 3 and the definition of continuous source entropy. It is a consequence of finitude rather than a deep result, but it grounds the framework: the question is always about managing distortion, not eliminating it.
Theorem 9.1. Let V ~ N(mu, Sigma) with K >= 2 and eigenvalues lambda_1 >= lambda_2 >= ... >= lambda_K > 0. Let (a, w, p = a'V + w) be any scalar codec. The excess distortion Delta = D_codec - D*(R_codec) satisfies:
(i) Delta = 0 when R_codec <= R_0.
(ii) Delta > 0 when R_codec > R_0.
(iii) The result is invariant to the domain of V.
Step 1: Codec distortion. For V Gaussian and p = a'V + w, the conditional distribution V|p is Gaussian with conditional covariance Sigma_{V|p} = Sigma - (Sigma a a' Sigma)/(a'Sigma a + sigma_w^2). The codec distortion is:
D_codec = tr(Sigma_{V|p}) = tr(Sigma) - (a'Sigma^2 a)/(a'Sigma a + sigma_w^2)
For the eigenbasis-aligned codec (a = e_1):
D_codec = lambda_1/(1 + SNR_1) + sum_{k=2}^{K} lambda_k
where SNR_1 = lambda_1/sigma_w^2.
Step 2: Codec rate. R_codec = I(V;p) = (1/2)log_2(1 + a'Sigma a/sigma_w^2). For the eigenbasis-aligned codec: R_codec = (1/2)log_2(1 + SNR_1).
Step 3: Optimal distortion. The Gaussian rate-distortion function: D*(R) = sum_{k=1}^{K} min(lambda_k, theta*), where theta* satisfies R = sum_{k=1}^{K} max(0, (1/2)log_2(lambda_k/theta*)).
Step 4: Threshold. At R_0 = (1/2)log_2(lambda_1/lambda_2), the water level theta* = lambda_2, only dimension 1 is active, and D*(R_0) = D_codec(R_0). Delta = 0.
Step 5: Strict positivity. For R_codec > R_0, theta* < lambda_2, the optimal allocation distributes positive bits to at least two dimensions. The scalar codec concentrates all bits on one direction. The Gaussian rate-distortion problem is strictly convex (Cover and Thomas, 2006, Section 10.3.3). The unique minimizer differs from the scalar codec. Therefore D_codec > D*(R_codec) and Delta > 0.
Step 6: Partial loading. The eigenbasis-aligned codec minimizes D_codec among unit-norm observation directions at fixed sigma_w^2 by a Rayleigh quotient argument. The eigenbasis-aligned Delta is a lower bound. Partially loaded codecs produce larger Delta.
Step 7: Domain invariance. Every step references only eigenvalues, observation direction, noise variance, and rate. No step references what V represents.
Step 8: Non-Gaussian robustness. Among all sources with covariance Sigma, the Gaussian has the highest rate-distortion function (Cover and Thomas, 2006, Chapter 10, Theorem 10.3.3). For any non-Gaussian source V' with the same covariance, D*_{V'}(R) <= D*_{Gaussian}(R). Since D_codec depends on the covariance (not the distributional form), D_codec is identical for V' and the Gaussian. Therefore Delta_{V'} >= Delta_{Gaussian}. The Gaussian excess distortion is a conservative lower bound. For heavy-tailed distributions - which characterize financial returns, income distributions, and many of the domains this paper addresses - the excess distortion is at least as large. QED.
The contribution is not the mathematics of scalar compression, which is standard. The contribution is the identification of institutional scalar signals - prices, votes, grades, diagnoses, headlines, KPIs, zoning codes, premiums, relevance scores, model weights - as codecs operating on coordination-dependent sources, to which the standard result applies.
Theorem 10.1. Let N agents observe a source V with observation sets {O_i}_{i=1}^{N}. Define the heterogeneity parameter delta = min_{i != j} max(|O_i \ O_j|, |O_j \ O_i|), the minimum pairwise non-overlapping observation size. Assume delta >= 1 (every pair of agents observes at least one dimension the other does not). Assume sufficient spread: for distinct pairs (i,j) and (i',j'), the cross-boundary dimension sets (O_i \ O_j) x (O_j \ O_i) and (O_{i'} \ O_{j'}) x (O_{j'} \ O_{i'}) have overlap bounded by a constant eta < 1 independent of N. Under Axiom 4, if the pairwise constraint interaction density rho_2 > 0:
K*(N) >= (1 - eta) * rho_2 * delta^2 * N(N-1)/2
where K*(N) is the number of constraint interactions discoverable by at least one agency formed from the N agents. The growth is quadratic in N. The marginal contribution of the (N+1)th agent is at least (1 - eta) * rho_2 * delta^2 * N.
Under higher-order coupling (rho_m > 0 for m >= 3), additional terms contribute: m-way interactions discoverable only by m-agent agencies produce terms scaling as C(N,m) * rho_m * delta^m. The theoretical ceiling on discoverable interactions is O(2^N), the number of possible non-empty subsets of agents. The proved lower bound is quadratic under pairwise coupling.
Step 1: Define heterogeneous agencies. For each pair of distinct agents (i,j) with i < j, the non-overlapping observation sets are O_i \ O_j and O_j \ O_i. By the heterogeneity condition delta >= 1, both are non-empty. The pair can form an agency whose combined observation set O_i union O_j spans dimensions that neither agent covers alone.
Step 2: Count discoverable pairwise interactions. By Axiom 4, a constraint interaction I_{ab} between dimensions a and b is identifiable at actionable resolution only when both a and b are in the agency's combined observation set. For pair (i,j), the interactions discoverable by this agency but not by either agent alone are those involving one dimension in O_i \ O_j and one in O_j \ O_i. The number of such dimension pairs is |O_i \ O_j| * |O_j \ O_i| >= delta^2. The fraction with non-trivial interactions is bounded below by rho_2.
Step 3: Count unique discoveries. The number of distinct agent pairs is C(N,2) = N(N-1)/2. Each pair discovers at least rho_2 * delta^2 interactions. By the sufficient spread condition, the fraction of pair (i,j)'s discoveries that duplicate those of any other pair (i',j') is bounded by eta. Each interaction discovered by pair (i,j) is duplicated by at most eta * C(N,2) other pairs. The number of unique discoveries is therefore at least:
K*(N) >= rho_2 * delta^2 * C(N,2) * (1 - eta) = (1 - eta) * rho_2 * delta^2 * N(N-1)/2
Since eta < 1, the factor (1 - eta) is bounded away from zero, and K*(N) grows as Theta(N^2). The sufficient spread condition is the formal requirement: it guarantees that the observation sets are diverse enough that distinct pairs of agents discover substantially distinct interactions, rather than all pairs discovering the same interactions from different angles.
Step 4: Marginal contribution. When agent N+1 joins with observation set O_{N+1} satisfying the heterogeneity and sufficient spread conditions, the new agent forms N new pairs, one with each existing agent. Each new pair discovers at least rho_2 * delta^2 interactions involving dimensions in O_{N+1} \ O_i that were not previously spanned by any agency (by Axiom 4, these dimensions were not identifiable without observation spanning them). By the sufficient spread condition, at least a fraction (1 - eta) of these are unique to the new pair. The marginal contribution K*(N+1) - K*(N) >= (1 - eta) * rho_2 * delta^2 * N, which is increasing in N.
Step 5: Higher-order terms. Under m-way coupling (rho_m > 0 for m >= 3), m-agent agencies discover interactions among m dimensions that no proper subset can identify (by Axiom 4 extended to m-way interactions). The number of m-agent agencies is C(N,m). Each discovers at least rho_m * delta^m novel m-way interactions. The total contribution from m-way agencies is C(N,m) * rho_m * delta^m, which grows as N^m/m!. The sum over all orders m from 2 to N is bounded above by sum_{m=2}^{N} C(N,m) * rho_m * delta^m <= 2^N * max_m(rho_m * delta^m). The theoretical ceiling is O(2^N). The proved baseline is the quadratic term (m=2). QED.
Remark on the name. "The Beier Factorial" refers to the combinatorial process of agency formation, in which the number of possible agencies is 2^N - 1. The proved lower bound on discoverable interactions under pairwise coupling is quadratic. Under deep higher-order coupling, the growth can approach the combinatorial ceiling. The name references the process and its ceiling. The proved baseline is the quadratic floor.
Theorem 11.1. Let V be a source with b > 0, coordination complexity K >= 2, and total observation bandwidth M. Under Axioms 1 and 2:
(i) If M < sum m_k* (bandwidth insufficient for full maintenance), there exist dimensions with m_k < m_k*, and coordination coherence on those dimensions decays.
(ii) The minimum aggregate dissipation at bandwidth M is D_min(M) = sum_{k=1}^{K} min(lambda_k, theta_M*), the rate-distortion function applied to the maintenance problem.
(iii) The magnitude of dissipation on dimension k is proportional to the Beierspace content b_k of that dimension.
(i) By Axiom 2, M is finite. M < sum m_k* by assumption. By the pigeonhole principle, at least one k has m_k < m_k*. By Axiom 1(c), coordination on dimensions receiving insufficient bandwidth decays. The decay rate on dimension k is proportional to b_k: dimensions with b_k = 0 do not decay from lack of observation.
(ii) The bandwidth allocation problem (minimize dissipation subject to sum m_k = M) is formally identical to the Gaussian rate-distortion problem. The reverse water-filling solution applies. Dimensions with lambda_k > theta_M* receive positive bandwidth and contribute theta_M* to dissipation. Dimensions with lambda_k <= theta_M* receive zero bandwidth and contribute lambda_k. D_min(M) > 0 when M < sum m_k*. QED.
Theorem 12.1. Let V be a source with b > 0, observed by a population of agents indexed i = 1, ..., N, each with observation capacity C_i. Let t = b'V be a scalar target. Let agent i's reward be R_i = h(E[(t - t_hat_i)^2]) where t_hat_i is agent i's estimate of t and h is strictly decreasing (lower estimation error yields higher reward). Assume observation noise is independent across dimensions. Under Axioms 1 and 2:
(i) Each agent's optimal bandwidth allocation concentrates on the direction b.
(ii) The aggregate bandwidth on dimensions orthogonal to b decreases relative to the pre-targeting allocation.
(iii) Dissipation on dimensions orthogonal to b increases.
(iv) The magnitude of the reallocation is increasing in the reward sensitivity |h'|.
Note on independent noise. The proof below assumes observation noise is independent across dimensions. Under correlated noise (which arises in any real observation system - observing revenue and costs through the same financial report introduces correlated measurement error), the separable decomposition in Step 2 does not hold. The optimal allocation under correlated noise may not concentrate on b as cleanly. The qualitative result - that scalar targets distort bandwidth allocation away from constraint dimensions - holds under correlated noise because the agent's objective still weights dimensions by their contribution to t, but the closed-form first-order conditions require the independence assumption.
Step 1: Agent optimization problem. Agent i solves:
max_{m_i} h(E[(t - E[t | observation_i(m_i)])^2])
subject to sum_k m_{i,k} <= C_i, m_{i,k} >= 0
where m_i = (m_{i,1}, ..., m_{i,K}) is the agent's bandwidth allocation across dimensions and observation_i(m_i) is the signal the agent receives under allocation m_i.
Since h is strictly decreasing, maximizing h is equivalent to minimizing E[(t - E[t | observation_i])^2].
Step 2: Optimal allocation. t = b'V = sum_k b_k V_k. Under Gaussian observation with independent noise on each dimension, the estimation error on t decomposes:
E[(t - E[t|obs])^2] = sum_k b_k^2 * Var(V_k | obs_k) = sum_k b_k^2 * lambda_k / (1 + m_{i,k} * SNR_k)
where SNR_k is the signal-to-noise ratio per unit of bandwidth on dimension k. The agent minimizes this subject to the capacity constraint. The Lagrangian first-order condition:
b_k^2 * lambda_k * SNR_k / (1 + m_{i,k} * SNR_k)^2 = mu for all k with m_{i,k} > 0
where mu is the Lagrange multiplier on the capacity constraint. Dimensions with larger b_k^2 * lambda_k * SNR_k receive more bandwidth (the marginal return to bandwidth on dimension k is higher when the target loads more heavily on k and k has higher variance). Dimensions with b_k = 0 receive zero bandwidth under the target-optimizing allocation, regardless of lambda_k.
Step 3: Reallocation. Compare the target-optimizing allocation m_i^target to the pre-targeting allocation m_i^pre (which, absent the scalar target, allocates bandwidth to minimize total MSE on V, distributing bandwidth more evenly across dimensions). Under targeting, bandwidth shifts toward dimensions in the support of b and away from dimensions orthogonal to b. The shift is bounded by C_i and is larger when the reward sensitivity |h'| is larger (because the reward gradient amplifies the return to accuracy on t relative to other dimensions).
Step 4: Aggregate reallocation. Summing across agents, the aggregate bandwidth M_k^target = sum_i m_{i,k}^target on each dimension k changes: M_k^target > M_k^pre for k in the support of b, M_k^target < M_k^pre for k orthogonal to b. The aggregate capacity sum_k M_k = sum_i C_i is conserved.
Step 5: Increased dissipation. By Theorem 11.1, dissipation on dimension k is a decreasing function of M_k. The reduction in M_k on constraint dimensions (those orthogonal to b) increases dissipation on those dimensions. The increase is bounded below by D_min(M^target) - D_min(M^pre) > 0 on the constraint subspace.
Step 6: Proportionality. The magnitude of each agent's reallocation is increasing in |h'| (the reward sensitivity). Stronger scalar incentives produce larger bandwidth shifts. The aggregate reallocation, and therefore the increase in constraint-dimension dissipation, is increasing in the population-average reward sensitivity. QED.
Domain invariance. The proof references the source covariance, the target direction b, the agents' capacity constraints, and the reward function h. It does not reference the domain. A KPI driving corporate behavior, a GPA shaping student effort, a benchmark score directing AI development, and a p-value structuring researcher incentives all produce the same bandwidth reallocation through the same mechanism.
Clarification on the nature of the Goodhart effect. Goodhart's Law is traditionally stated as "when a measure becomes a target, it ceases to be a good measure." The theorem proves something more specific and more insidious. The measure remains perfectly accurate on its own narrow dimension - the KPI still measures what it measures, the GPA still reflects assessed performance, the benchmark still scores what it scores. What degrades is the system the measure was meant to represent. The constraint dimensions that the scalar target does not load on lose observation bandwidth and, by the Dissipation Law, lose coherence. The organization hits its revenue target while institutional knowledge erodes. The student achieves a high GPA while deep understanding atrophies. The AI model tops the benchmark while real-world robustness degrades. The scalar signal looks better while the multi-dimensional source it was supposed to summarize deteriorates beneath it. The traditional formulation captures the symptom. The theorem identifies the mechanism.
Theorem 13.1. Let V be a source with b > 0. Let the total observation bandwidth be LD(t). Let the structural drift be sigma(LD) = sigma_e + sigma_b(LD), where sigma_e > 0 is exogenous drift and sigma_b(LD) is endogenous drift, strictly decreasing in LD with sigma_b(0) = sigma_b^max > 0 and lim_{LD->inf} sigma_b(LD) = 0. Under Axioms 1 and 5:
(i) The dynamics of LD are governed by dLD/dt = F(LD) where F(LD) = LD - sigma(LD) at equilibrium. The fixed point LD* satisfying F(LD*) = 0 is unstable.
(ii) For LD > LD*, LD increases over time (self-stabilization). For LD < LD*, LD decreases over time (self-destabilization).
(iii) Under bounded re-entry (sigma_b^max < infinity), the system exhibits bistability: a collapse trap (low LD) and a well-functioning state (high LD), separated by a saddle-node bifurcation.
(iv) In the collapse trap, the parameters distinguishing recoverable from permanent collapse are not identifiable from the system's own data.
All dynamics are stated in LD-space. Define the stability ratio S(LD) = LD / sigma(LD) as a derived quantity.
Step 1: Fixed point. The fixed point LD* satisfies LD* = sigma(LD*) = sigma_e + sigma_b(LD*). This is S(LD*) = 1.
Step 2: Instability. Compute F'(LD) = 1 - sigma'(LD) = 1 - sigma_b'(LD). Since sigma_b is strictly decreasing, sigma_b'(LD) < 0, so F'(LD) > 1 > 0 at every LD, including LD*. In the one-dimensional ODE dLD/dt = F(LD), F'(LD*) > 0 means F crosses zero from below at LD*. For LD slightly above LD*, F(LD) > 0, so dLD/dt > 0 and LD moves upward. For LD slightly below LD*, F(LD) < 0, so dLD/dt < 0 and LD moves downward. The fixed point is unstable.
Step 3: Self-stabilization above LD.* When LD > LD*, F(LD) > 0. By Axiom 5, the increase in LD reduces sigma_b (because more observation reduces the rate at which agents' decisions generate structural change). The reduced sigma_b reduces sigma(LD), which makes F(LD) larger, which accelerates the increase. The positive feedback drives LD upward toward a high-LD steady state where the observation bandwidth is large relative to the drift.
Step 4: Self-destabilization below LD.* When LD < LD*, F(LD) < 0. By Axiom 5, the decrease in LD increases sigma_b (agents with degraded observation make more erratic decisions, generating more structural change). The increased sigma_b increases sigma(LD), which makes F(LD) more negative, which accelerates the decrease. The negative feedback drives LD downward.
Step 5: Bistability. With sigma_b bounded above by sigma_b^max, the downward trajectory reaches a low-LD steady state where LD is near zero and sigma is near sigma_e + sigma_b^max - the collapse trap. The system has two stable states (high-LD and collapse) separated by the unstable fixed point LD*. The saddle-node bifurcation occurs at the parameter value where LD* merges with the high-LD equilibrium, below which only the collapse trap remains.
Step 6: Informational impossibility. In the collapse trap, the system's data are generated by the low-LD dynamics. The parameters of sigma_b(LD) in the high-LD region (which determine whether an exogenous boost to LD would push the system past LD* and into the self-stabilizing regime) are not identifiable from low-LD data, by standard identification arguments (the data do not contain variation in the high-LD region). The system cannot determine from its own observations whether the collapse is recoverable or permanent. QED.
Note on S. The stability ratio S(LD) = LD/sigma(LD) is a monotone function of LD (since sigma is decreasing in LD, S is increasing in LD). S > 1 corresponds to LD > LD* (self-stabilizing). S < 1 corresponds to LD < LD* (self-destabilizing). All dynamics are stated and proved in LD-space. S is a convenient summary statistic, not the state variable.
Note on functional form. The dynamics dLD/dt = LD - sigma(LD) encode a specific modeling choice: observation bandwidth grows at a rate proportional to itself in the absence of drift. The axioms guarantee the existence of coupled positive and negative feedbacks (Axiom 5) but do not pin down the functional form. The qualitative results - unstable fixed point, self-stabilizing above, self-destabilizing below, bistability under bounded re-entry - are robust to any dynamics dLD/dt = G(LD) - sigma(LD) where G is continuous, increasing, and satisfies G(0) = 0. The specific linear form F(LD) = LD - sigma(LD) is chosen for tractability.
Domain invariance. The proof requires Axioms 1 and 5 and the drift structure sigma = sigma_e + sigma_b(LD). It applies to any coordination-dependent source where agents both observe and constitute the source and where endogenous drift decreases with observation bandwidth. Civic observation stabilizes political institutions; when it declines, institutional erosion generates the instability that drives further disengagement. Peer review in scientific communities operates through the same feedback, as does community maintenance of open-source codebases where declining contribution accelerates technical debt beyond any individual's capacity to address. The parameters vary across these domains. The qualitative dynamics do not.
The five theorems above derive strictly from the axioms plus rate-distortion theory. The result below applies the axiomatic framework to a specific institutional setting - the market for information production - and relies on additional institutional conditions beyond the five axioms. It is presented as an application, not a theorem of the axiomatic system.
Application 14.1. In any domain where (a) agents can publish structured assessments of V attributed to persistent identity, (b) assessments are scored against outcomes using a strictly proper scoring rule, and (c) a market exists for calibration quality, the return to information production is decoupled from information monopoly and the Grossman-Stiglitz paradox (Grossman and Stiglitz, 1980) dissolves.
Argument. The GS paradox rests on an assumption embedded in the model's structure:
Assumption GS: The return to information production is captured exclusively through monopoly on the information produced.
Under Assumption GS, public information yields zero return to the producer. If prices are efficient, the trading return vanishes. No rational agent produces information. The paradox follows.
The contribution is threefold.
Identifying the assumption. Assumption GS was built into the Grossman-Stiglitz model's structure (agents choose between paying cost c and trading on a private signal, or not paying c and observing the price; there is no third option) and was not extracted for explicit examination in the forty-six years of subsequent literature. Every proposed resolution preserved it: noise trading (Black, 1986), strategic trading (Kyle, 1985), and rational inattention (Sims, 2003) each modified other features of the model while leaving the single-channel return structure intact. The idea that reputation can compensate information production is present in related work (Diamond, 1989, on monitoring; Holmstrom, 1999, on managerial reputation; Hong and Kubik, 2003, on analyst career outcomes). What is new is not the idea that reputation matters, but the specific mechanism design and the structural result below.
Designing the mechanism. Under the Brier scoring rule with coverage weight alpha < 2/3 (Beier, 2026d, Proposition 14, proved within the Hurwicz-Myerson framework), honest comprehensive disclosure is the strictly dominant strategy for every agent regardless of calibration quality. This is a mechanism design result, not an assumption about agent behavior.
Proving the return does not collapse. The principal's willingness to pay for calibration quality decomposes into a consensus premium (which vanishes as the price becomes efficient on V_1, as GS predicts for the trading channel) and a constraint premium W_constraint(q) that remains permanently bounded above zero and independent of price efficiency (Beier, 2026d, Proposition 7). The constraint premium exists because the price system is never efficient on constraint dimensions - Theorem 11.1 guarantees this through the Dissipation Law. The return to disclosure is bounded below by W_constraint. The paradox cannot reach the disclosure channel.
The connection to the axioms: Axiom 1 (observation-as-maintenance) explains why constraint-dimension information production has permanent value (the information maintains coordination that would otherwise decay). Axiom 5 (reflexivity) explains why the disclosure feeds back to improve the source (the observed constraint dimensions stabilize). The GS Resolution is an application of these axioms to the market's information production problem, augmented by conditions (a)-(c) that specify the institutional environment.
(See Beier, 2026d, for the formal CARA-normal general equilibrium model, the Hurwicz-Myerson derivation of incentive compatibility, and the proofs of Propositions 7 and 14.)
The Beierspace fraction b parameterizes the framework's predictions. This section provides one empirical calibration to demonstrate that the predictions are differential and testable.
The prediction. Domains with higher b should exhibit: (1) higher rates of Goodhart degradation when scalar targets are imposed, (2) faster coherence loss when observation bandwidth declines, and (3) sharper phase transitions between well-functioning and dysfunctional regimes.
The replication crisis in science provides a natural experiment. Different scientific fields have different Beierspace content. Experimental psychology studies human coordination-dependent phenomena (beliefs, decisions, social behavior) - high b. The phenomena under study are partially constituted by the beliefs and coordinated actions of the people being studied. Particle physics studies fundamental physical law - b near 0 on the relevant dimensions. The phenomena persist regardless of whether anyone observes them.
The framework predicts that fields with higher b should exhibit lower replication rates, because: (1) the phenomena under study are more sensitive to observation conditions (by Axiom 5, the researcher is part of the coordination that constitutes the phenomenon), (2) the Goodhart effect of publication incentives (scalar target: p < 0.05) degrades unmeasured dimensions more severely when the phenomena are coordination-dependent (by Theorem 12.1, the dissipation from targeting scales with b), and (3) the stability of the findings is more dependent on continued observation under consistent conditions (by Theorem 11.1).
The observed data are consistent. The Open Science Collaboration (2015) found that 36% of psychology studies replicated (high b, low replication). Particle physics experiments replicate at rates approaching 100% within their stated uncertainty (low b, high replication; the field's 5-sigma discovery threshold and the independent confirmation of results across competing detector collaborations, as in the ATLAS and CMS Higgs confirmations, reflect an observation architecture matched to a low-b source). The ordering matches the prediction. The assignment of b to these fields is by qualitative argument from the definition (coordination-dependence of the phenomena under study) rather than by quantitative measurement using the proxies in Beier (2026k), which are financial-market constructs not directly applicable to scientific fields. Developing domain-appropriate measurement proxies for b outside financial markets is deferred to future work.
The framework's specific contribution is a distinguishing prediction that separates it from conventional explanations (measurement imprecision, small samples, methodological heterogeneity). The prediction: a field studying coordination-dependent phenomena with precise measurement and large samples should still replicate less than a field studying physical phenomena with imprecise measurement and small samples, if b is the operative parameter. Conventional explanations predict that precise measurement and large samples improve replication regardless of domain. The framework predicts that b dominates, because the sensitivity of the source to the observation process (Axiom 5) and the Goodhart degradation from scalar targets (Theorem 12.1) operate on the coordination-dependence of the source, not on measurement quality. This prediction is testable and would distinguish the framework from alternatives if confirmed.
Conjecture 16.1. As coordination structures become more complex, the coordination complexity K grows faster than observation bandwidth M can grow.
Evidence for. The Beierspace fraction of the S&P 500 rose from approximately 17% in 1975 to approximately 90% in 2020 (Ocean Tomo, 2020). Interaction terms grow combinatorially with base dimensions.
Evidence against. AI observation may grow fast enough to close the gap.
What a proof requires. A formal model of the co-evolution of K and M.
Conjecture 17.1. Physical construction converts Beierspace maintenance burden into eigenspace persistence. Civilization advances when cumulative encoding outpaces cumulative dissipation at every institutional layer.
Grounding. Physical infrastructure (bridges, power grids, water systems), automated software protocols (TCP/IP, DNS, TLS), and codified legal frameworks (constitutions, property registries, standardized contracts) each replace the need for active human monitoring on specific coordination dimensions. A bridge converts one specific dimension of a coordination problem - the crossing location - from Beierspace (requires ongoing coordinated belief) to eigenspace (the bridge persists physically). The encoding is not free of new maintenance: the bridge creates new Beierspace dimensions (maintenance funding, traffic management, inspection scheduling, seismic assessment). The conjecture claims that the aggregate maintenance cost of the physical infrastructure is lower than the aggregate maintenance cost of the coordination it replaced, on average, across civilizational time - that the net effect of encoding is positive. The convergence condition (Beier, 2026i, Theorem 10.2) provides the formal criterion for specific institutional hierarchies.
Conjecture 18.1. Agents producing negative externalities have a structural incentive toward opacity. Agents producing positive externalities have a structural incentive toward transparency. Negative capital hides. Positive capital is transparent.
Formal sketch. If E < 0, disclosing the externality assessment reduces the activity's value. The agent's pre-scoring incentive is concealment. If E > 0, disclosing builds track record. The agent's pre-scoring incentive is disclosure. The scoring mechanism (Application 14.1) makes honest disclosure dominant for all agents, but its effectiveness in overcoming the concealment incentive depends on the scoring lag and externality magnitude.
Five axioms:
Five theorems:
One application:
Independence is demonstrated by constructing, for each axiom, a toy model in which the other four axioms hold and the target axiom fails. In each case, at least one theorem that depends on the missing axiom does not hold.
Model 1: Axioms 2,3,4,5 hold; Axiom 1 fails. Consider a coordination-dependent source (b > 0) maintained entirely by blind algorithmic execution - a decentralized smart contract enforcing rigid parameters, or a deeply embedded social taboo sustained by unreflective conformity rather than interpretive observation. Agents are finite (Axiom 3), have bounded capacity (Axiom 2), observe different aspects of the system (Axiom 4), and their actions reflexively constitute the coordination (Axiom 5 - the smart contract executes because people transact through it, the taboo persists because people conform to it). But observation does not maintain the source: the coordination is sustained mechanically or dogmatically without interpretive modeling. In many cases, observing and interpreting a taboo dissolves rather than maintains it - the coordination depended on unreflective compliance, and reflective observation undermines it. Axiom 1 fails because observation and maintenance are decoupled. Consequence: Theorem 11.1 (Dissipation) does not apply in its standard form - the coordination does not decay from lack of observation but from execution failure or from the erosion of the mechanical or dogmatic substrate that sustains it.
Model 2: Axioms 1,2,4,5 hold; Axiom 3 fails. Consider a hypothetical two-agent system where each agent has unbounded internal state capacity (infinite memory, infinite processing). Both agents observe a coordination-dependent source V with b > 0 (Axiom 1 holds), reflexive coupling (Axiom 5 holds), cross-dimensional interactions (Axiom 4 holds), and finite aggregate bandwidth at any instant (Axiom 2 holds - even infinite agents face time constraints on interpretation). But each agent can represent V without loss internally (Axiom 3 fails). Consequence: Remark 8.1 (Inevitable Distortion) does not apply - lossless representation is possible for these agents. The aggregate bandwidth constraint (Axiom 2) still produces dissipation (Theorem 11.1 holds), but the compression is a bandwidth choice rather than a finitude constraint.
Model 3: Axioms 1,2,3,5 hold; Axiom 4 fails. Consider a two-dimensional coordination-dependent source V = (V_1, V_2) with b > 0, where V_2 = f(V_1) for a known, deterministic, invertible function f. Agents are finite (Axiom 3), have bounded capacity (Axiom 2), observe and maintain V (Axiom 1), and their observations affect V reflexively (Axiom 5). A cross-dimensional interaction exists between V_1 and V_2 (they are deterministically coupled). But an agent observing only V_1 can deduce V_2 = f(V_1) perfectly, identifying the mechanism, magnitude, and conditions of the interaction at actionable resolution without ever observing V_2 directly. Axiom 4 fails: heterogeneous observation spanning both dimensions is not required because the deterministic invertibility allows complete inference from a single dimension. Consequence: Theorem 10.1 (Beier Factorial) degenerates - a single agent can discover all interactions, and the combinatorial requirement for heterogeneous agencies collapses. The Factorial's growth in N vanishes because N = 1 suffices.
Model 4: Axioms 1,2,3,4 hold; Axiom 5 fails. Consider a coordination-dependent source V maintained by a population of observers (Axiom 1 holds) who are finite (Axiom 3), capacity-bounded (Axiom 2), and heterogeneous (Axiom 4 holds). But the observers' decisions do not affect V: the source is maintained by observation but the observers are external to the coordination they observe (perhaps they are regulators monitoring a market they do not trade in). Axiom 5 fails: V is maintained by observation but not reflexively coupled through the observers' participation. Consequence: Theorem 13.1 (Phase Transition) does not hold - without reflexive coupling, there is no endogenous drift sigma_b(LD) and therefore no self-destabilizing feedback below the threshold. Dissipation still occurs (Theorem 11.1 holds) but it is monotone rather than bistable.
Model 5: Axioms 1,3,4,5 hold; Axiom 2 fails. Consider a hypothetical system with unbounded aggregate observation bandwidth (an AI-augmented population with no effective capacity ceiling). The source is coordination-dependent (Axiom 1), the agents are individually finite (Axiom 3), the observation sets are heterogeneous (Axiom 4), and the source is reflexively coupled (Axiom 5). But M = infinity: the population can observe every dimension simultaneously with unlimited bandwidth. Consequence: Theorem 11.1 (Dissipation) does not apply - with unbounded bandwidth, every dimension receives its full-maintenance allocation, the water level theta_M* drops to zero, and no dimension is undermaintained. Theorem 12.1 (Goodhart) does not apply in its strong form - with unbounded capacity, agents can allocate bandwidth to the scalar target without reducing bandwidth on constraint dimensions, because the capacity constraint that forces the tradeoff is absent.
Each model demonstrates that the missing axiom cannot be derived from the remaining four and that at least one theorem fails without it.
Axiom 1 fails for purely physical sources (b = 0). Excess distortion exists but the source does not degrade.
Axiom 2 relaxes as AI augments observation. The bound moves with technology.
Axiom 3 holds for any finite agent. Would fail for hypothetical agents with unbounded internal states.
Axiom 4 fails for sources with zero cross-dimensional interactions (uncoupled dimensions).
Axiom 5 fails for sources whose evolution is independent of observation.
The framework applies in full when all five hold: the source is coordination-dependent (b > 0), the agents are finite, the source has cross-dimensional interactions, and the observation is reflexively coupled. It applies partially when some hold and others do not. The Beierspace fraction b parameterizes the transition from partial to full applicability.
Five axioms, five theorems, one application, three conjectures. The base is smaller than any of the individual papers it unifies. The reach is larger than any of them claimed.
The first two axioms - observation-as-maintenance and bounded capacity - produce the Dissipation Law and the Goodhart derivation directly. Compression inevitability (Axiom 3) grounds the universality by showing that the distortion cannot be designed away: it is a property of communication between finite agents, not a flaw in any particular institution. The irreducibility of heterogeneous observation (Axiom 4) produces the Beier Factorial's quadratic floor and combinatorial ceiling, grounding the case for diverse disclosure networks in the structure of the discovery problem rather than in a social argument. Reflexivity (Axiom 5), coupled with Axiom 1, generates the phase transition that separates self-stabilizing from self-destabilizing coordination regimes.
The Beierspace fraction b parameterizes every prediction, and the empirical calibration against replication rates across scientific fields is consistent with the framework's ordering. The Gaussian source model yields conservative estimates: for non-Gaussian sources with the same covariance, the excess distortion is at least as large.
The axioms are empirically motivated starting points for a deductive system. They define the scope. When the world falls outside the scope (b = 0, uncoupled dimensions, infinite agents), the system does not apply. Within the scope, the consequences follow with the full force of deductive inference.
The Prolegomena's title names four fields. The axioms hold wherever finite agents coordinate through compressed signals, which extends well beyond those four. The title understates the claim. The axioms do not.
This paper states the axiomatic foundations of the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics. Five axioms and one mathematical framework produce five theorems, one application, and three conjectures. The axioms are falsifiable. The theorems are proved. The boundaries are identified. All referenced papers are publicly available on Zenodo with persistent DOIs under CC BY-NC-ND 4.0.