Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 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 covariance matrices on which modern financial economics depends are composed predominantly of human-decision-determined quantities. Their eigendecompositions are temporary. This paper formalizes the consequences.
A stability condition S = LD/(sigma_e + sigma_b(LD)) is derived from a network model of heterogeneous agents, where L is aggregate learning capacity, D is aggregate disclosure rate, sigma_e is the rate of physical structural change, and sigma_b is the rate of human-generated structural change. The central result is that sigma_b is itself a strictly decreasing function of LD: the agents whose learning and disclosure determine whether structural change is detected are the same agents whose decisions generate the structural change. This reflexive feedback produces a phase transition at S = 1 with an unstable fixed point, self-stabilizing dynamics above the threshold, and self-destabilizing dynamics below it.
The corrected dynamics with bounded re-entry exhibit a saddle-node bifurcation at a computable critical parameter mu_crit. Below mu_crit, the system has two stable equilibria: a collapse trap and a well-functioning state. An informational impossibility result follows: a system in the collapse trap cannot determine from its own data whether its collapse is permanent or recoverable, because the parameters distinguishing these regimes are not identifiable from crisis outcomes. These results hold for any continuous, strictly decreasing Beier drift function, not only the specific parameterization used for computation.
The paper further shows that cognitive suppleness - the capacity to absorb large model revisions without degraded learning - unconditionally amplifies stability and conditionally expands the usable dimensionality of the eigenspace when drift rates fall within the attention budget. The suppleness multiplier is derived from established results in cognitive psychology on belief updating under disconfirmation.
Keywords: Beierspace, eigenvalue instability, phase transition, bistability, informational impossibility, covariance non-stationarity, cognitive suppleness, reflexivity
The eigendecomposition of economic covariance matrices is a foundational tool of financial economics. Portfolio theory (Markowitz, 1952), factor models (Ross, 1976; Fama and French, 1993), risk management (Longerstaey and Spencer, 1996), and macroeconomic forecasting all depend on decomposing observed covariance structures into principal components and treating those components as stable features of the economy. The documented instability of these decompositions during crises - correlation breakdowns, regime switches, factor rotation - is typically modeled as an exogenous process: the regime changes, and the model tracks it (Hamilton, 1989; Ang and Bekaert, 2002; Engle, 2002).
This paper proposes a different interpretation. The entries of economic covariance matrices are generated predominantly by human decisions - trading, lending, pricing, hiring, consuming - rather than by physical processes. Quantities whose statistical properties are constituted by the beliefs and actions of participants, rather than determined independently of those beliefs, are here termed Beier-space quantities to distinguish them from eigen-space quantities whose statistical properties arise from physical law. The instability of the eigendecomposition during crises is then not an exogenous regime switch but a direct consequence of the ontological status of the quantities being decomposed: human decisions change under stress, and the covariance structure they constitute changes with them.
The paper formalizes this observation through a stability condition
S = LD / (sigma_e + sigma_b(LD))
where L and D are the aggregate learning capacity and disclosure rate of market participants, sigma_e is the rate of physical structural change, and sigma_b is the rate of human-generated structural change. The central analytical result is that sigma_b is endogenous: it is a strictly decreasing function of LD, derived from a network model in which the same agents whose learning and disclosure suppress Beier drift are the agents whose decisions generate it. This reflexive coupling produces a phase transition at S = 1, self-stabilizing dynamics above the threshold, and self-destabilizing dynamics below it.
The corrected dynamics with bounded re-entry exhibit a saddle-node bifurcation at a computable critical re-entry intensity mu_crit. Below this threshold, the system possesses two stable equilibria - a collapse trap and a well-functioning state - separated by an unstable fixed point. The identification analysis reveals that while the composite stability ratio S is identifiable from crisis outcomes, the structural parameters governing the Beier drift mechanism are not separately identifiable from cross-sectional data. Combining these results yields an informational impossibility theorem: a system in the collapse trap cannot determine from its own data whether its collapse is permanent or recoverable, because the parameters distinguishing these regimes are precisely those the collapsed system's degraded information production cannot estimate.
Three further results are established. First, all qualitative findings - the phase transition, the bistability, the bifurcation - hold for any continuous, strictly decreasing Beier drift function satisfying four conditions (class B'), not only the specific power-law parameterization used for computation. Second, cognitive suppleness - formalized through a sigmoid multiplier derived from established results in belief-updating psychology (Festinger, 1957; Lord et al., 1979; Sherif and Hovland, 1961) - unconditionally amplifies stability and conditionally expands the usable dimensionality of the eigenspace when drift rates fall within the attention budget. Third, the optimal allocation of attention across dimensions follows the same water-filling structure as the rate-distortion theory of price formation developed in a companion paper (Beier, 2026c), unifying the two results at the mathematical level.
The paper's scope is information economics with connections to cognitive science and dynamical systems theory. The broader ambitions that motivated the research - a unification of economics, computer science, mathematics, and physics through the Beier-space concept - exceed what the current formal results support and are noted as future directions rather than established claims.
The core ideas in this paper have antecedents. The specific formalization and the results it produces are new.
Soros (1987) articulated the reflexivity principle: market participants' biased perceptions affect fundamentals, and the changed fundamentals affect perceptions, producing boom-bust dynamics. The present paper formalizes this as sigma_b(LD) decreasing in LD and derives the phase transition, bifurcation structure, and impossibility result that Soros's verbal framework could not reach.
Taleb (2012) identified the fragile-robust-antifragile trichotomy and argued that systems should be designed to benefit from volatility. The present paper provides the mechanism (sigma_b endogenous to LD), the threshold (S = 1), and the boundary conditions (the LD ceiling on dimensionality expansion). Taleb's "antifragile" means gaining from disorder; "supple" means maintaining coherence through continuous updating - related but distinct mechanisms.
The ecological resilience literature (Holling, 1973; Gunderson and Holling, 2002) has argued for five decades that adaptive capacity constrains system survival and that rigidity leads to catastrophic collapse. The key difference: resilience theory applies to ecological systems where the eigenstructure is physically determined. The present paper extends the analysis to systems where the eigenstructure is human-decision-determined, introducing the reflexive amplification that ecological systems lack.
Chapter 12 of Keynes (1936) describes investors pricing assets based on beliefs about others' beliefs - the beauty contest. This is the Beier-space observation stated without the formal apparatus of information theory, which did not exist in 1936.
MacKenzie (2006) demonstrated sociologically that financial models reshape the markets they describe (performativity). The present paper provides the information-theoretic formalization and derives the stability consequences.
The DCC-GARCH literature (Engle, 2002; Tse and Tsui, 2002) tracks time-varying correlations empirically without explaining why correlations change. Regime-switching models (Hamilton, 1989; Ang and Bekaert, 2002) treat the switch as exogenous. The present paper endogenizes correlation instability through the information architecture of participants.
The rational inattention literature (Sims, 2003) provides the microeconomic foundation for the attention budget constraint in the eigenspace extension theorem. The present paper extends this to the population level, connecting channel capacity to the dimensionality of the tracked eigenspace.
What is new. The specific combination: (1) derivation of sigma_b as decreasing in LD from a network model; (2) phase transition at S = 1 with computable eigenvalue; (3) saddle-node bifurcation at mu_crit; (4) informational impossibility connecting bistability to identification failure; (5) functional form robustness across class B'; (6) cognitive derivation of the suppleness multiplier. Each component has antecedents. The assembly is new.
Consider a K-dimensional vector of economic quantities whose covariance matrix Sigma(t) is estimated from observed data. The eigendecomposition Sigma = V Lambda V' produces eigenvalues and eigenvectors treated as features of the system to be discovered.
This treatment is appropriate when the quantities composing Sigma are physically determined - when their statistical properties arise from processes independent of human decisions. Temperature, orbital mechanics, atmospheric chemistry: these are eigen-space quantities with covariance structures reflecting physical law.
The treatment is inappropriate when the quantities are human-decision-determined. Prices, interest rates, credit ratings, exchange rates, and fiat currency: these are Beier-space quantities existing because humans collectively sustain them. Their covariance structure reflects the current pattern of human decisions, which can change whenever the decisions change.
At any single time t, the eigendecomposition of Sigma(t) cannot distinguish between eigen-space and Beier-space components. The distinction matters at t + dt. The eigen-space component changes only when the physical world changes. The Beier-space component changes whenever the decisions that constitute it change - without any physical trigger, because it is constituted by beliefs about beliefs.
The stability of Sigma(t) is therefore not a physical property of the economic system. It is a property of the information architecture that maintains the coordination of human decisions.
Let x = LD denote aggregate information capacity. The total structural drift is sigma(x) = sigma_e + sigma_b(x), and the stability ratio is
S(x) = x / sigma(x) = x / (sigma_e + sigma_b(x))
Proposition 4.1 (Reflexive Beier Drift). In a market of N CARA agents with heterogeneous beliefs M_i(t) about the K-dimensional return covariance structure, updating through a disclosure channel at rates determined by learning capacities l_i and disclosure rates d_j, the Beier drift rate sigma_b is strictly decreasing in x = LD.
The proof proceeds through four steps. (i) The disclosure channel contracts model disagreement V(t) = (1/N) sum_i ||M_i - M_bar||^2_F exponentially at rate 2LD, yielding steady-state V_ss proportional to sigma^2_v / LD. (ii) Model disagreement produces action dispersion W proportional to V through the CARA-normal allocation rule a_i = (1/gamma) M^{-1}_i mu. (iii) Action dispersion produces Beier-space covariance change sigma_b proportional to V through heterogeneous order flow. (iv) Therefore sigma_b is proportional to V_ss, which is inversely proportional to LD, giving d(sigma_b)/d(LD) < 0.
Remark. The main result (sigma_b decreasing in LD) is confirmed numerically across the full range tested. The exponential contraction rate 2LD is accurate to within 6% of the simulated rate. The intermediate proportionalities are monotone in the correct direction but approximate rather than tight.
Definition 4.2 (Class B'). A function sigma_b : [0, x_max] -> R+ is in class B' if:
(B1) sigma_b is continuous on [0, x_max] and C^1 on (0, x_max).
(B2) sigma_b is strictly decreasing: sigma'_b(x) < 0 for all x in (0, x_max).
(B3) sigma_b(0) > 0 (possibly infinite).
(B4) sigma_e + sigma_b(x_max) < x_max.
Condition (B4) defines the framework's domain of applicability. When physical drift plus irreducible Beier drift exceeds maximum information capacity, no information architecture can stabilize the system.
Proposition 5.1 (Critical Threshold). For any sigma_b in B', there exists a unique x* in (0, x_max) such that S(x*) = 1, equivalently x* = sigma_e + sigma_b(x*).
Proof. Define phi(x) = x - sigma_e - sigma_b(x). Then phi(0) = -sigma_e - sigma_b(0) < 0 and phi(x_max) = x_max - sigma_e - sigma_b(x_max) > 0 by (B4). Since phi'(x) = 1 + |sigma'_b(x)| > 1, phi is strictly increasing. A unique zero exists by the intermediate value theorem.
Theorem 5.2 (Phase Transition). x* is an unstable fixed point of the uncorrected dynamics x_dot = eta(S(x) - 1)x with eigenvalue
lambda = eta [1 + |sigma'_b(x*)|] > eta > 0
The instability is a property of the class B', not the parameterization.
Proof. Linearizing: dF/dx|_{x*} = eta[S'(x*)x* + (S(x*) - 1)] = eta S'(x*)x* since S(x*) = 1. Computing S'(x*) = [sigma(x*) + x*|sigma'_b(x*)|] / sigma(x*)^2, and using sigma(x*) = x*, gives lambda = eta[1 + |sigma'_b(x*)|].
Trajectories starting above x* converge to x_max (self-stabilization). Trajectories starting below x* converge to 0 (self-destabilization).
The micro-founded derivation (Appendix A) yields:
x_dot = eta(S(x) - 1)x + mu * h(R) * (x_max - x)
R_dot = sigma(x) - x
where R >= 0 is accumulated reality debt, mu > 0 is re-entry intensity, and h(R) = R/(1 + R) is the saturating re-entry response.
Definition 6.1. The critical re-entry intensity is
mu_crit = max_{x in (0,x*)} phi(x) / (x_max - x), where phi(x) = eta(1 - S(x))x
Remark. An earlier version of this analysis used mu_crit = D_max/x_max, which underestimates by 20-70% because it ignores the geometry of the re-entry force mu(x_max - x). The error was discovered by constructing a counterexample per step (e) of the proof discovery protocol (Section 11).
Theorem 6.2 (Bistability). For any sigma_b in B':
(A) For mu >= mu_crit: all trajectories starting below x* eventually recover and converge to x_max.
(B) For mu < mu_crit: the system has two stable equilibria - a collapse trap at x_low > 0 and the full-information state at x_max - separated by an unstable fixed point x_sep.
(C) At mu = mu_crit, the trap and its separatrix merge in a saddle-node bifurcation.
Proof. In the saturated limit (h(R) -> 1), the effective 1D dynamics are g(x) = eta(S(x) - 1)x + mu(x_max - x). At x = 0: g = mu * x_max > 0. At x = x*: g = mu(x_max - x*) > 0. The decay force phi(x) = eta(1 - S(x))x is zero at both endpoints and positive in the interior. For mu < mu_crit, the re-entry force mu(x_max - x) falls below phi(x) on an interval, creating at least two zeros by the intermediate value theorem. The lowest zero x_low is stable (the vector field transitions from positive to negative). For mu >= mu_crit, g > 0 throughout (0, x*).
Verified across five functional forms (power law, exponential, logistic, piecewise linear, concave square root). All exhibit the same bifurcation structure.
Remark. The bistability result does not depend on the decay force phi(x) being unimodal. If phi has multiple interior maxima, additional stable-unstable pairs may appear, but the outermost structure (a lowest trap x_low, the upper attractor x_max, separated by at least one unstable point) is preserved. The analytical proof of unimodality for the power-law class contains an algebraic error in the change-of-variables step; numerical verification confirms unimodality for all beta in {0.5, 1.0, 1.5, 2.0, 3.0} tested, but the written proof should be treated as incomplete. See Section 13.
The composite S is identifiable from crisis outcomes: the reduced-form probit P(crisis) = Phi(-(a_0 + a_1 log S)) recovers a_0 and a_1 tightly in simulation (100/100 replications converging).
The structural parameters theta = (sigma_0, alpha, beta) are not separately identifiable: for any target S_0,
sigma_0 = (x/S_0 - sigma_e)(1 + alpha * x)^beta
defines a two-dimensional surface in theta-space. The Fisher information matrix of the full structural probit has condition number exceeding 3,000.
Theorem 7.1 (Informational Impossibility). Consider a system governed by the corrected dynamics.
(i) If mu < mu_crit, the system has two stable equilibria separated by x_sep. Recovery from x_low requires a perturbation exceeding x_sep - x_low.
(ii) The composite S is the identified predictor of crisis outcomes, but the mechanism parameters determining mu_crit are not identifiable from crisis data alone.
(iii) Therefore: a system in the collapse trap cannot determine from its own crisis experience whether mu < mu_crit (permanent trap) or mu > mu_crit (recoverable collapse). The data that would distinguish these regimes is precisely the data the system cannot produce because it is trapped.
The collapsed system is informationally blind to its own prognosis. External diagnosis is required even to assess whether reform is feasible.
From the zone of tolerance framework (Sherif and Hovland, 1961), the biased assimilation literature (Lord et al., 1979), and the deliberate practice literature (Ericsson et al., 1993; Tetlock and Gardner, 2015):
g(s, sigma_k) = 1 / (1 + exp(kappa * (sigma_k - Z_0(1 + gamma * s))))
where kappa = 2.82 (calibrated from Lord et al.'s 40% fidelity reduction finding), Z_0 is the baseline zone of tolerance, and s is aggregate suppleness.
Theorem 8.1 (Eigenspace Extension). Let K dimensions have drift rates sigma_1 < ... < sigma_K.
(A) K*(s) = |{k : LD * g(s, sigma_k) > sigma_k}| is non-decreasing in s, with jumps at
s_k = (sigma_k - (1/kappa) ln(LD/sigma_k - 1) - Z_0) / (Z_0 * gamma)
(B) K* has a ceiling: K*_max = |{k : sigma_k < LD}|. Dimensions with drift rates exceeding LD are never usable regardless of suppleness.
(C) The stability amplification dS/ds > 0 holds unconditionally for all drift regimes.
Part (C) follows from the chain rule: dS/ds = (dS/dx_eff)(dx_eff/ds), where both factors are strictly positive.
The lossy compression paper (Beier, 2026c) shows that the price system introduces excess distortion Delta = D_price - D* > 0 by concentrating all information rate on the consensus dimension. Within the stability framework, this distortion is non-stationary: as S declines, the eigendecomposition rotates and the price system compresses along an axis that no longer corresponds to maximum variance.
The GS resolution paper (Beier, 2026d) shows that track record capital provides a return to information production independent of price efficiency, with constraint premium W_constraint bounded above zero. Within the stability framework, this mechanism prevents LD from collapsing to zero, which is the condition under which sigma_b grows without bound.
The following cases are consistency checks, not calibration. The stability condition has sufficient degrees of freedom to classify known outcomes ex post.
| Case | L | D | sigma_e | S | Outcome |
|---|---|---|---|---|---|
| Netherlands (post-1953) | 0.81 | 0.93 | 0.08 | 4.28 | 0 storm-surge deaths in 73 years |
| Taiwan COVID (as of Dec 2020) | 0.88 | 0.85 | 0.40 | 1.505 | 0.30 deaths/million |
| US COVID (as of Feb 2021) | 0.30 | 0.20 | 0.40 | 0.056 | 1,027 deaths/million |
| 2008 Crisis | 0.12 | 0.06 | 0.01 | 0.008 | Lehman, systemic failure |
| SVB (2023) | 0.08 | 0.06 | 0.01 | 0.005 | $42B withdrawn, 48hr failure |
Table 1: Historical illustrations. Minimum stable S (Taiwan, 1.505) is 27x the maximum collapsed S (US COVID, 0.056). COVID mortality figures are snapshot values at the dates indicated; cumulative totals rose substantially in subsequent waves. Netherlands figure covers storm-surge flooding (the Delta Works' domain); the 2021 Limburg river floods caused zero fatalities. Sources: WEF GCI, WGI, KNMI, Taiwan CDC, US CDC, FDIC Material Loss Review.
Several proofs in this paper contained errors discovered only through systematic verification. The incorrect mu_crit formula was found by constructing a counterexample. The false separatrix claim was found by asking what else the analysis revealed. The recovery transient was found by extending the simulation tenfold. Each error had a common cause: analysis guided by a desired conclusion rather than by the structure of the equations.
The protocol: (a) check dimensional reduction; (b) enumerate all fixed points; (c) draw the phase portrait; (d) sweep all parameters through bifurcation values; (e) for each claim, state precise conditions and construct a counterexample outside them; (f) report unrequested findings. Verify numerically only after (a)-(f), at >= 10x the apparent settlement timescale.
Prediction 1. Countries with higher pre-pandemic LD exhibit smaller Frobenius norm changes in DCC-GARCH covariance estimates during COVID, controlling for shock magnitude, GDP, and demographics.
Prediction 2. After Dodd-Frank stress testing (2011-2012), eigenvalue volatility decreases for financial sector ETFs relative to control sectors, with 6-18 month lag.
Prediction 3. Sectors with higher constraint-dimension disclosure volume exhibit smaller correlation spikes during stress events.
A preliminary mortality test (VA scores vs. COVID deaths/million, n = 94) produced r = 0.518 (p < 0.001) in the wrong direction, driven by reporting quality and demographic confounds. The proper tests require daily financial return data.
Proved but with gaps. Unimodality of the decay force (algebraic error in proof; numerics confirm for all beta). Intermediate proportionalities in Proposition 4.1 (monotone but not tight).
Attempted and failed. The three-layer impossibility theorem (Part iii: model lacks coupling term). Structural parameter identification (Fisher information singular in theta).
Future work. Decomposed capacity allocation model. Stochastic extension for cliff-edge collapse. Dilution optimization via convex programming. Out-of-sample prediction on 2024-2030 panel data.
The economic covariance matrices that underlie portfolio theory, risk management, and macroeconomic forecasting are predominantly composed of Beier-space quantities. Their eigendecompositions are temporary. The stability of these decompositions depends on the information architecture that maintains the coordination of human decisions.
This paper shows that the information architecture and the stability it produces are reflexively coupled. The phase transition at S = 1, the saddle-node bifurcation at mu_crit, and the informational impossibility follow from this coupling. Cognitive suppleness unconditionally amplifies stability and conditionally expands the dimensionality of the tracked world. Three falsifiable predictions are stated. The framework's value will be determined by whether it predicts what existing frameworks do not.
The more flexibly you hold your model of the world, the more of the world your model can hold.