Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 2026
version 2.3
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 an axiom unifying two prior results: that price formation destroys information through lossy compression (Beier, 2026c) and that economic covariance structures are unstable when the information architecture maintaining them falls below a critical threshold (Beier, 2026e). The axiom: observation of a coordination-dependent structure is inseparable from the maintenance of that structure. From this axiom and a bounded-capacity condition, the paper derives a dissipation law, a capacity budget, a minimum-dissipation bound (the rate-distortion function reinterpreted), a mixing theorem (the eigenspace-Beierspace distinction is a spectrum, not a partition), a regress theorem for institutional maintenance hierarchies, and a formal derivation of Goodhart's Law. The Beier Factorial - the combinatorial process by which autonomous agents form agencies through transparent interaction - is introduced as the creation-side counterpart to the dissipation law, with proved superlinear growth (at minimum quadratic under dense pairwise coupling) and increasing marginal returns to each additional disclosing agent. Two falsifiable conjectures are stated: a disclosure trap and a liquidity-instability pathway. Speculative extensions to quantum measurement, AI alignment, and civilizational encoding are collected in an appendix.
Keywords: observation-maintenance identity, coordination thermodynamics, Beierspace, eigenspace, lossy compression, dissipation, rate-distortion theory, Goodhart's Law, Beier Factorial, institutional regress
| Symbol | Definition | Introduced |
|---|---|---|
| K | Coordination complexity (number of independent dimensions) | Def. 2.6 |
| V = (V_1, ..., V_K) | Source vector of coordination-relevant quantities | Def. 2.6 |
| Sigma(t) | Time-varying covariance matrix of V | Sec. 3.1 |
| lambda_k | Eigenvalue (variance) of dimension k | Sec. 3.1 |
| R | Channel capacity (bits per unit time) | Def. 2.4 |
| m_k | Maintenance bandwidth on dimension k | Def. 2.5 |
| M = sum m_k | Total maintenance bandwidth | Thm. 4.1 |
| D*(M) | Minimum dissipation at bandwidth M (rate-distortion function) | Thm. 7.1 |
| theta* | Water level (reverse water-filling parameter) | Thm. 7.1 |
| D_price | Price system's dissipation | Cor. 7.2 |
| Delta | Excess dissipation: D_price - D*(R_price) | Cor. 7.2 |
| R_price | Price system's information rate | Cor. 7.2 |
| SNR_1 | Signal-to-noise ratio on consensus dimension | Cor. 7.2 |
| C_total | Total interpretively coherent modeling capacity | Thm. 4.2 (Sec. 5) |
| S | Stability ratio: LD/(sigma_e + sigma_b(LD)) | Sec. 3.1 |
| sigma_e | Rate of physical structural change | Sec. 3.1 |
| sigma_b | Rate of human-generated structural change (Beier drift) | Sec. 3.1 |
| V_{k,ss} | Steady-state disagreement on dimension k | Prop. 3.1 |
| N | Number of agents (sensor nodes) | Sec. 9 |
| K*(N) | Discoverable constraint dimensions from N disclosing agents | Sec. 9 |
| rho_m | Constraint interaction density at order m | Sec. 9 |
| B_n, P_n | Beierspace content, physical-substrate content at layer n | Thm. 10.2 |
| e_n, d_n, c_n | Encoding rate, dissipation rate, creation rate at layer n | Thm. 10.2 |
Two independently derived results share a common structure they do not acknowledge.
The Lossy Compression paper (Beier, 2026c) establishes that the price system introduces excess distortion Delta > 0 by concentrating all information rate on the consensus dimension while leaving constraint dimensions uncompressed. The result uses rate-distortion theory (Shannon, 1959; Berger, 1971) within a multivariate Gaussian framework.
The Beierspace paper (Beier, 2026e) establishes that economic covariance structures are predominantly composed of human-decision-determined quantities whose stability depends on a ratio S = LD/(sigma_e + sigma_b(LD)), with a phase transition at S = 1 and an informational impossibility in the collapse regime.
The coupling between them - excess distortion degrades information capacity, degraded capacity destabilizes the eigenstructure against which distortion is measured - was established in proof notes (Beier, 2026-PN1; Beier, 2026-PN2). The present paper identifies the axiom from which both results, their coupling, and several further consequences follow.
The observation that economic structures are partially constituted by the beliefs of participants has a substantial intellectual history, and the present formalization should be situated within it.
Hayek (1945) made the canonical case that prices aggregate dispersed information. The present framework accepts this and adds: the aggregation is lossy (Beier, 2026c formalizes the loss), and the loss has structural consequences because the information destroyed is the information the structure needs to persist (the Observation-Maintenance Identity).
Keynes (1936, Chapter 12) described asset pricing as a beauty contest - investors pricing based on beliefs about others' beliefs. This is the Beierspace observation stated without the formal apparatus of information theory, which did not exist in 1936.
Soros (1987) articulated reflexivity: market participants' biased perceptions affect fundamentals. The Beierspace paper (Beier, 2026e) formalizes this as sigma_b(LD) decreasing in LD and derives the phase transition that Soros's verbal framework could not reach. The present paper grounds both in the Observation-Maintenance Identity.
MacKenzie (2006) demonstrated that financial models reshape the markets they describe (performativity). Callon (1998) developed the broader performativity framework in economic sociology. Performativity describes how observation reshapes coordination structures - how models and measurements actively constitute the markets they claim to describe. The present framework describes the complement: how the absence of observation degrades coordination structures. MacKenzie's Black-Scholes story is about a model that made the world more like itself. The present paper's story is about dimensions that decay because nobody is looking at them. The two are complements, not equivalents. The present framework adds quantitative structure that the performativity literature does not provide: the rate-distortion bound on minimum dissipation, the water-filling threshold that determines which dimensions receive maintenance, and the excess distortion that measures the gap between actual and optimal observation.
North (1990) analyzed institutional change through the distinction between formal institutions (written rules enforced by third parties) and informal institutions (conventions, norms, self-enforced). The eigenspace-Beierspace spectrum (Theorem 8.1) overlaps with North's distinction but is not isomorphic to it: North's distinction concerns the enforcement mechanism (third-party vs. self-enforced), while the eigenspace-Beierspace distinction concerns the statistical properties (physically determined vs. belief-constituted). A written statute is physically persistent (eigenspace in the substrate sense) while its enforcement requires coordinated action by courts and officials (Beierspace in the maintenance sense). North's framework identifies the institutional maintenance hierarchy that the Regress Theorem (Section 10) formalizes, though the information-theoretic structure (convergence condition, encoding rates) is new.
Ostrom (1990) identified design principles for long-enduring common-pool resource institutions. Several of her principles - monitoring, graduated sanctions, conflict-resolution mechanisms - are observation channels in the present framework's language: mechanisms for maintaining coordination on specific dimensions of the resource governance structure. The convergence condition (Theorem 10.2) provides an information-theoretic criterion that Ostrom's empirical principles instantiate.
The rational inattention literature (Sims, 2003; Mackowiak and Wiederholt, 2009; Matejka and McKay, 2015) models agents as bandwidth-constrained information processors. The present framework extends this to the population level, connecting agent-level bandwidth to the maintenance of coordination structures, and adds the reflexive feedback (the maintained structures are constituted by the agents' coordinated actions) that rational inattention does not model.
The information design literature (Kamenica and Gentzkow, 2011; Bergemann and Morris, 2019) studies how a sender optimally chooses what to reveal. The disclosure channel analyzed in the companion papers (Beier, 2026c, Section 6; Beier, 2026d) is an information design problem; the present paper characterizes the structural consequences of what any disclosure design must leave unrevealed.
What is new in the present paper is the axiom itself and what it produces: the identity of observation and maintenance (not merely their correlation), the derivation of inevitable dissipation from this identity, the mixing theorem dissolving the eigenspace-Beierspace partition into a bandwidth-parameterized spectrum, the regress theorem and convergence condition for institutional hierarchies, the formal derivation of Goodhart's Law, and the Beier Factorial as the creation-side counterpart to dissipation.
Definition 2.1 (Coordination-dependent structure). A persistent pattern whose continuation requires ongoing coordinated action by multiple agents. The coordination may be explicit (contractual, regulatory) or implicit (convention, expectation, market participation). The distinguishing feature is that withdrawal of coordination degrades the structure's Beierspace components even in the absence of any physical change to the underlying substrate.
Remark. Many structures combine coordination-dependent and physically determined components. A road network requires physical maintenance (eigenspace: asphalt degrades regardless of beliefs) and coordination maintenance (Beierspace: traffic conventions, funding priorities, land-use coordination). Definition 2.1 applies to the coordination-dependent component. The physically determined component enters Theorem 8.1 as the inertial substrate.
Definition 2.2 (Eigenspace quantity). A quantity whose statistical properties - variance, covariance, and temporal dynamics - are determined by physical processes independent of human observation and action.
Definition 2.3 (Beierspace quantity). A quantity whose statistical properties are constituted by the beliefs and actions of agents participating in the structure that produces the quantity.
Remark. Definitions 2.2 and 2.3 are presented as a partition for expository clarity. Theorem 8.1 establishes that the distinction is a continuous spectrum.
Definition 2.4 (Observation channel). Any mechanism that produces signals about a structure's state and transmits those signals to agents whose actions contribute to the structure's persistence. The channel has capacity R, measured in bits per unit time (Shannon, 1948), and operates on a subset of dimensions determined by its institutional design.
Definition 2.5 (Maintenance bandwidth). The effective information rate m_k directed at sustaining coordination on dimension k. Effective means: bits received by agents, interpreted through shared models, and integrated into coordinated action.
Definition 2.6 (Coordination complexity). The number of independent dimensions K on which coordinated agent action is required for the structure to persist.
Definition 2.7 (Interpretively coherent modeling capacity, ICMC). The aggregate rate at which a population of N agents can receive information, integrate it into models that are both internally consistent and mutually compatible, and translate updated models into coordinated action. ICMC is bounded above by cognitive resources and below by shared interpretive infrastructure.
Remark on the status of ICMC. ICMC is a conceptual primitive motivating the bounded-capacity axiom, not a precisely measurable scalar. The formal results operate on channel capacity R and maintenance bandwidth m_k, both of which are well-defined. ICMC's role is to explain why total capacity is finite. The formal results depend on the finiteness (Axiom 2), not on measuring ICMC in absolute units. The relationship of ICMC to the formal apparatus is analogous to the relationship of utility to demand curves: the concept motivates the structure; the operational quantities carry the mathematics.
Definition 2.8 (Dimensional coherence). The degree of agreement among agents about the state of coordination on dimension k. Related to but distinct from structural persistence: see Proposition 3.2 below.
The proofs of the dissipation law and the mixing theorem rely on a result from the Beierspace paper. For self-containment, it is stated here with sufficient context.
Proposition 3.1 (Reflexive Beier Drift; Beier, 2026e, Proposition 4.1). In a market of N CARA agents with heterogeneous beliefs M_i(t) about the K-dimensional return covariance structure (in the framework of Grossman and Stiglitz, 1980; Hellwig, 1980), updating through a disclosure channel at rates determined by learning capacities l_i and disclosure rates d_j, the following holds. The aggregate disagreement V(t) = (1/N) sum_i ||M_i - M_bar||^2_F contracts exponentially toward a steady state V_ss proportional to sigma_v^2 / (LD), where sigma_v^2 is the variance of the innovation process driving belief change and LD is aggregate information capacity. Model disagreement produces action dispersion W through the CARA-normal allocation rule a_i = (1/gamma) M_i^{-1} mu. Action dispersion produces Beier-space covariance change sigma_b proportional to V_ss through heterogeneous order flow. Therefore sigma_b is a strictly decreasing function of LD: d(sigma_b)/d(LD) < 0.
The stability ratio S = LD/(sigma_e + sigma_b(LD)) has a unique critical threshold x* where S(x*) = 1. The threshold is an unstable fixed point: trajectories above x* converge to x_max (self-stabilization); trajectories below x* converge to 0 (self-destabilization). Under corrected dynamics with bounded re-entry, the system exhibits bistability below a critical re-entry intensity mu_crit: a collapse trap and a well-functioning state separated by an unstable fixed point.
Per-dimension decomposition. The steady-state disagreement decomposes as V_ss = V_{1,ss} + V_{c,ss}, where V_{1,ss} is disagreement about the consensus dimension and V_{c,ss} = sum_{k>=2} V_{k,ss} is constraint-dimension disagreement. The price signal reduces V_{1,ss} toward lambda_1/(1 + SNR_1) but provides no information about constraint dimensions, so V_{k,ss} for k >= 2 remains at sigma_v^2 / m_k, where m_k is the per-dimension maintenance bandwidth from non-price channels. Beier drift sigma_b loads primarily on V_{c,ss} (the proof is in Beier, 2026-PN2, Lemma 0): constraint-dimension disagreement drives heterogeneous actions that the price cannot correct.
Axiom 1 (Observation-Maintenance Identity). For any coordination-dependent structure, observation through an information channel and maintenance of the structure's dimensional coherence are the same act:
(a) Every bit transmitted about dimension k increases dimensional coherence on dimension k, by updating receiving agents' models toward mutual consistency on that dimension.
(b) Every bit of maintenance bandwidth directed at dimension k constitutes an observation of dimension k, because the maintenance act requires generating and transmitting information about the current state.
(c) Dimensions receiving zero observation bandwidth receive zero coherence maintenance. Coordination on those dimensions is sustained only by inertia, which decays at a rate determined by the Beierspace content of the dimension.
(d) Increased dimensional coherence maintains the accuracy of shared models, not necessarily structural persistence. When the accurate shared model supports continued coordination, observation stabilizes the structure. When it reveals coordination failure, observation can trigger rational withdrawal. In both cases, observation maintains coherence (agents converge on the truth); the structural consequences depend on what the truth is.
Proposition 3.2 (The Coherence-Coordination Chain). The formal apparatus, the axioms, and the economic interpretation operate on three related but distinct quantities. In the Gaussian framework that underlies the companion papers, they are formally linked:
(i) Variance reduction (the information-theoretic quantity in D*(M)): m_k bits reduce the posterior variance on dimension k from lambda_k to lambda_k * 2^{-2m_k}.
(ii) Dimensional coherence (Definition 2.8): the reduction in steady-state disagreement V_{k,ss} = sigma_v^2 / m_k from higher m_k produces convergence of agents' models on dimension k.
(iii) Coordination maintenance (the economic outcome): reduced disagreement produces reduced action dispersion W_k (Proposition 3.1, step ii), which reduces Beier drift sigma_{b,k}, which stabilizes the covariance structure on dimension k.
The chain is: m_k -> V_{k,ss} -> sigma_{b,k} -> lambda_k stability. Each link is monotone under the Gaussian framework.
Outside the Gaussian framework, the three quantities can diverge. Agents can disagree about a dimension while still coordinating on it through institutional inertia (the enforced contract case). Agents can agree about a dimension while failing to coordinate (the bank-run case: everyone agrees the bank is failing, but agreement accelerates the failure).
The distinction matters for the paper's results. Results that depend only on coherence - Goodhart's Law (Corollary 6.1), the Mixing Theorem (Theorem 8.1) - are robust to departures from the Gaussian framework. They require only that higher m_k produces higher dimensional coherence, which holds under any framework where observation reduces model divergence. Results that depend on the full chain - the coupling with the Beierspace phase transition (Section 11.2), the quantitative minimum-dissipation bound (Theorem 7.1) - require the Gaussian link or a suitable non-Gaussian substitute. The qualitative claim (undermaintained dimensions dissipate) holds generally; the quantitative claim (the specific dissipation rate V_{k,ss} = sigma_v^2 / m_k) is CARA-normal-specific.
Axiom 2 (Bounded Capacity). The total ICMC of any finite population is bounded above at any moment. The bound depends on population size, individual cognitive resources, and shared interpretive infrastructure. It can change over time but is finite at every instant.
Scope Condition (Dimensional Excess). The results produce non-trivial consequences when K exceeds the effective bandwidth of the dominant observation channel. For the price system (a scalar channel), K > 1 suffices: any transaction involving more than one value-relevant dimension satisfies this.
Remark on the magnitude of K. The empirical question is not whether K > 1 but how much K exceeds M. Factor models in finance estimate the number of priced risk factors in equity returns at 3-15, depending on methodology (Fama and French, 1993; Ross, 1976; the PCA literature estimates 5-10 significant components in large equity covariance matrices). These are factors detectable through prices - dimensions on which the price system transmits information. The Scope Condition's K counts all coordination-relevant dimensions, including those the price system does not reach: environmental constraint interactions, social externalities, supply chain dependencies, regulatory regime stability. These are the dimensions the Lossy paper's V_2 through V_K represent. The claim that K is in the hundreds for complex transactions is based on the enumeration in Beier (2026c, Section 2): each agent's judgment encodes assessment of current value, future trajectory, risk factors, constraint interactions, scaling effects, and confidence levels across each dimension. The empirical estimation of K for specific asset classes is noted as an open problem (Section 13).
Theorem 4.1 (Inevitable Dissipation). Let a coordination-dependent structure have coordination complexity K, and let the total maintenance bandwidth be M = sum_{k=1}^{K} m_k. Define the full-maintenance bandwidth for dimension k as m_k^* = sigma_v^2 / V_{k,target}, where V_{k,target} is the maximum steady-state disagreement compatible with negligible Beier drift on dimension k. Under the Scope Condition, there exist dimensions k with m_k < m_k^*. Dimensional coherence on these undermaintained dimensions decays toward the zero-maintenance steady state at a rate bounded below by a positive function of the maintenance deficit.
Proof. The Scope Condition asserts that K exceeds the effective bandwidth of the observation channels. By rate-distortion theory, at total bandwidth M, the reverse water-filling solution leaves dimensions with lambda_k <= theta* below the water level, allocating them zero bandwidth. When K exceeds the number of dimensions above the water level (the formal content of the Scope Condition), at least one dimension k receives m_k < m_k^*. By Axiom 1, observation bandwidth and maintenance bandwidth are identical, so the undermaintained dimension receives insufficient coherence maintenance.
For any such dimension k, Axiom 1(c) establishes that coherence maintenance is below the full-maintenance level. By Proposition 3.1 (per-dimension decomposition), the steady-state disagreement on dimension k is V_{k,ss} = sigma_v^2 / m_k. When m_k < m_k^*, V_{k,ss} > V_{k,target}, and the Beier drift on dimension k is strictly positive: sigma_{b,k} > 0.
The minimum aggregate dissipation across all dimensions at total bandwidth M is the rate-distortion function D*(M) (Section 7). D*(M) > 0 strictly under the Scope Condition. Any allocation other than reverse water-filling produces dissipation exceeding D*(M). QED.
Remark 4.1.1 (Dissipation is not collapse). The Second Law claims some dimensions are always dissipating, not that all structures eventually collapse. Structures can persist if dissipating dimensions are non-load-bearing.
Remark 4.1.2 (Entropy export). The coordination analog of entropy export in physical dissipative structures (Prigogine, 1967) is suggestive: the price system maintains coordination on V_1 by allowing coordination on V_2 through V_K to decay. The system chooses what to remember by choosing what to forget. Whether this analogy has the full mathematical structure of thermodynamic entropy production (minimum entropy production principle, Onsager reciprocal relations) is an open question. The analogy is noted as structurally suggestive, not claimed as isomorphism.
Theorem 4.2 (Capacity Budget). Under Axiom 2, total ICMC at any moment is allocated exhaustively:
C_total = C_maintain + C_create + C_process
where C_maintain sustains existing coordination, C_create establishes new coordination dimensions, and C_process is consumed by receiving and interpreting signals.
Proof. Exhaustive partition of activities by coordination effect. QED.
Remark. The Capacity Budget is a constraint on allocation, not a conservation law. ICMC can increase (education, institutional development) and decrease (institutional decay, interpretive fragmentation). The Budget says: at any instant, the total is finite and the allocation is exhaustive. The Second Law needs this finiteness to produce undermaintenance.
Remark (C_process and the Disclosure Incentive). The price system's lossy compression forces agents to devote capacity to reconstructing information the price destroyed. This reconstruction is C_process: it consumes ICMC without directly maintaining any coordination dimension. The Disclosure Incentive shifts allocation from C_process toward C_maintain by providing structured constraint-dimension signals directly.
Goodhart's Law is foregrounded because it is the framework's cleanest applied result and the one most directly relevant to an HDSR audience. Campbell's Law (Campbell, 1979) - that quantitative social indicators used for decision-making corrupt the processes they are intended to monitor - is the same phenomenon stated for social programs rather than monetary policy. Every ML evaluation metric, every KPI dashboard, every governance scorecard is a scalar observation channel on a high-dimensional target.
Corollary 6.1 (Goodhart's Law from the Dissipation Law). Let a coordination-dependent structure of complexity K be observed through a scalar measure (a one-dimensional channel). When the measure becomes a target - when agents redirect capacity toward optimizing the measured dimension (the standard Goodhart behavioral assumption):
(i) Capacity directed at the measured dimension is withdrawn from unmeasured dimensions (Capacity Budget, given the behavioral redirection).
(ii) The unmeasured dimensions dissipate at an increased rate (Dissipation Law).
(iii) The measure remains accurate on its own dimension - the scalar channel is an optimal codec on V_1 (Beier, 2026c, Section 4.3).
(iv) The structure degrades on unmeasured dimensions while the measure reports stability or improvement. The observed regularity collapses once pressure is placed upon it for control purposes (Goodhart, 1975) - not because the measure loses accuracy on its dimension but because the structure has changed on the dimensions it cannot see.
Proof. Step (i) combines the behavioral assumption (targeting redirects capacity) with the Capacity Budget (total is finite, so redirection toward the target withdraws capacity from elsewhere). Steps (ii)-(iv) follow from the Dissipation Law and Axiom 1 applied to the capacity allocation produced by (i). QED.
Remark 6.1.1 (Relationship to existing formalizations). Manheim and Garrabrant (2018) categorize four variants of Goodhart's Law: regressional, causal, extremal, and adversarial. The present derivation is closest to their regressional variant (the proxy-target relationship breaks down when the proxy is optimized) but provides a different mechanism: not statistical regression to the mean but information-theoretic dissipation on unobserved dimensions. The derivation also unifies the variants: all four operate through the same channel - a scalar measure cannot maintain a high-dimensional structure, and optimizing the measure concentrates maintenance on the measured dimension at the expense of the unmeasured ones. The rate of degradation is bounded below by D*(M), which Manheim and Garrabrant's taxonomy does not provide.
Remark 6.1.2 (Implications for ML evaluation). A machine learning model evaluated on a scalar metric (accuracy, F1 score, perplexity) is a coordination-dependent structure (its "alignment" with the intended task is maintained by the evaluation process) observed through a scalar channel. The Dissipation Law predicts degradation on task dimensions the metric does not capture. The standard response - adding metrics - is an attempt to increase M. The Dissipation Law says this succeeds only if the additional bandwidth covers all dimensions above the water level theta*, which for complex tasks may require more dimensions than any practical evaluation suite can provide.
Theorem 7.1 (Minimum-Dissipation Bound). Let a coordination-dependent structure have coordination complexity K with eigenvalue spectrum lambda_1 >= ... >= lambda_K > 0. Let total maintenance bandwidth be M bits per unit time. The minimum achievable dissipation rate, under the Gaussian source model with MSE distortion, is:
D*(M) = sum_{k=1}^{K} min(lambda_k, theta*)
where theta* solves:
M = sum_{k=1}^{K} max(0, (1/2) log(lambda_k / theta*))
Under the optimal allocation (reverse water-filling), dimensions with lambda_k > theta* receive positive bandwidth. Dimensions with lambda_k <= theta* receive zero bandwidth and dissipate at their full coordination intensity lambda_k.
Proof. The mathematical object is the standard Gaussian rate-distortion function (Berger, 1971; Cover and Thomas, 2006, Chapter 10). The reverse water-filling derivation is standard. The interpretive step is licensed by Axiom 1: distortion (information destroyed by the codec) and dissipation (coherence lost on undermaintained dimensions) are the same quantity under the Observation-Maintenance Identity. QED.
Remark 7.1.1 (Robustness). For non-Gaussian sources with the same covariance structure, D*_Gaussian(M) >= D*_other(M) (Cover and Thomas, 2006, Chapter 10). The Gaussian is the hardest source to compress. The Gaussian bound is an upper bound on minimum dissipation, not a lower bound. The qualitative result (positive dissipation under the Scope Condition) is distribution-free.
Remark 7.1.2 (Structural parallel to thermodynamics). D*(M) is convex and decreasing in M. The inverse function R*(D) is convex and decreasing in D. The water level theta* equilibrates marginal maintenance return across active dimensions, playing a role analogous to temperature in statistical mechanics. Whether this structural parallel constitutes a formal isomorphism (state functions, Legendre transforms, entropy as concave function of extensive variables) is an open question. The rate-distortion literature develops the Legendre structure (Berger, 1971, Chapter 4). The present paper claims the structural parallel, not the isomorphism.
Corollary 7.2 (The Price System's Dissipation). The price system at rate R_price = (1/2) log(1 + SNR_1), concentrated on V_1, achieves dissipation D_price = lambda_1/(1 + SNR_1) + sum_{k=2}^{K} lambda_k. The excess dissipation Delta = D_price - D*(R_price) > 0 under conditions of Proposition C of Beier (2026c).
Corollary 7.3 (The Phase Transition). The Beierspace phase transition at S = 1 (Beier, 2026e, Theorem 5.2) occurs when dissipation on constraint dimensions accumulates to the level where the eigenstructure loses coherence. The self-destabilizing dynamics below S = 1 involve a positive feedback: dissipation degrades maintenance infrastructure (reduced LD), which increases dissipation (higher sigma_b). This has the qualitative character of a critical threshold in a high-dimensional system maintained by an ordering force, with a transition to disorder when the ordering force is insufficient.
Theorem 8.1 (Spectral Continuity of the Ontological Partition). The eigenspace-Beierspace distinction is a continuous spectrum parameterized by maintenance bandwidth. For any dimension k:
(i) Persistence rate is continuous and strictly increasing in m_k.
(ii) As m_k -> infinity, the dimension behaves as eigenspace: its covariance contribution is invariant to belief perturbations.
(iii) As m_k -> 0, persistence is determined by inertia (physical substrate plus cognitive convention), decaying at a rate set by the ratio of physical to cognitive inertia.
(iv) At intermediate m_k, the dimension is partially stable and partially dissipating.
Proof. Part (i): From Axiom 1 and Proposition 3.1, V_{k,ss} = sigma_v^2 / m_k is continuous and strictly decreasing in m_k. Lower disagreement means lower Beier drift (Proposition 3.1), meaning higher persistence. Part (ii): As m_k -> infinity, V_{k,ss} -> 0, beliefs converge, action dispersion vanishes, Beier drift vanishes. Part (iii): At m_k = 0, V_{k,ss} is at its maximum. The claim that inertial coordination decomposes into a physical component (eigenspace: does not decay through belief divergence) and a cognitive component (Beierspace: decays as models diverge) is an additional modeling assumption beyond the axioms and Proposition 3.1. It is motivated by the observation that physical substrate (a written contract, a built road, a codified law) constrains behavior independently of current beliefs, while cognitive coordination (conventions, expectations, taken-for-granted assumptions) does not. The decay rate at m_k = 0 is then determined by the relative weight of these two components, with the physical fraction providing a floor on persistence. Part (iv): Intermediate value of V_{k,ss} at finite positive m_k. QED.
Corollary 8.2 (Institutional Encoding). Physical infrastructure, legal codification, and institutional formalization shift dimensions toward eigenspace by converting coordination-dependent maintenance into physical-substrate persistence. The shift is never complete: enforcement and interpretation of codified structures are Beierspace.
The Dissipation Law says coordination decays on undermaintained dimensions. The Capacity Budget says total maintenance bandwidth is finite. Conjecture 12.3 below argues that dimensional creation generically outpaces channel extension. Together these suggest an increasingly losing race: the coordination structure grows in complexity faster than the observation channels can cover it.
The Beier Factorial identifies a class of observation architecture whose capacity grows faster than scalar channels. The term "factorial" refers to the mechanism - the combinatorial process by which autonomous agents form agencies through transparent interaction - not to a specific proven growth rate (Beier, 2026j, Section 4). The formal results establish superlinear growth (at minimum quadratic under dense pairwise coupling) and increasing marginal returns to each additional disclosing agent. Whether the growth rate achieves factorial scaling (N!) depends on the depth of the recursive agency structure and the constraint interaction density at each level, which is an empirical question the formal results do not resolve. The mechanism is named for the combinatorial richness it exploits, not for a proven asymptotic rate. The proved rates are polynomial (at minimum quadratic) and potentially much faster under deep recursive structure, with the binding constraint being the attention budget LD rather than the combinatorial space itself.
Minsky (1986) described intelligence as emerging from the interaction of simple agents organized into agencies. In Minsky's framework, the agents are computational subprocesses within a single mind. The Beier Factorial extends the agent/agency structure to a population of autonomous humans interacting through disclosure.
An agent in the Beier Factorial is a human sensor node - a person who observes, models, and reports on some subset of the system's state vector V. The agents are heterogeneous: different training, different physical observation access, different cognitive styles, different models. This heterogeneity is the source of the combinatorial richness.
An agency is a composite inference structure that emerges when multiple agents combine observations through disclosure. The agency produces joint inferences - conclusions about constraint interactions that no constituent agent could reach alone, because the interaction lives in the space between their respective observation sets.
Definition 9.1 (Agency). An agency is a set of agents whose joint inference produces an observation not derivable from any proper subset of the agents' individual observations. A productive agency's output functions as a new data source for further inference.
The departure from Minsky is fourfold. (i) Agents are autonomous humans, not subprocesses, so agency formation is voluntary and driven by disclosure. (ii) Agencies persist, grow, or dissolve based on track record validation against outcomes - the pruning mechanism that prevents combinatorial explosion from being dominated by noise. (iii) Agencies can form higher-order agencies: a recursive structure that Minsky's single-mind framework did not require. (iv) The track record mechanism (Beier, 2026d) provides the quality filter: agencies that produce validated predictions accumulate track record capital; agencies that do not dissolve.
The mechanism is best seen through an example (from Beier, 2026j, Section 4.2). Three colonists on Mars work in different subsystems: water recycling, power generation, and food production. Over breakfast, they share observations from the previous cycle. Colonist A reports a 3% drop in water recycler efficiency. Colonist B reports a 20-minute shift in solar array output timing. Colonist C reports a growth anomaly in crop section 4.
No individual colonist can identify the cause. No pair of observations is sufficient. The three-person agency - the joint inference from combining all three observations - identifies a dust storm pattern simultaneously coating solar panels (shifting power timing), changing thermal cycling in the water system (reducing efficiency), and reducing photosynthetic light in the greenhouse (producing the crop anomaly). The constraint interaction lives in the joint space and requires all three heterogeneous observation sets.
The number of such interactions discoverable through disclosure grows with both the number of agents and the density of constraint coupling in the environment.
Proposition 9.2 (Superlinear growth; Beier, 2026j, Proposition 4.1). Let N sensor nodes observe a K-dimensional state vector with constraint interaction density rho_m at order m (the fraction of m-tuples of dimensions with non-trivial interaction). Under full disclosure, the number of productive agencies A*(N) grows as Omega(N^alpha) for alpha > 1, where alpha depends on the interaction density profile {rho_m} and the distribution of observation subsets across nodes.
Proof sketch. The (N+1)th sensor node forms potential agencies with every existing subset. When rho_m is bounded below by a constant for some m >= 2 (dense coupling), the expected number of productive new agencies containing the (N+1)th node grows at least linearly in N, because each existing node's observation set has positive probability of interacting with the new node's observations on at least one constraint dimension. The increment A*(N+1) - A*(N) >= c * N for constant c > 0 depending on rho, yielding A*(N) >= (c/2) * N^2. Under dense pairwise coupling (rho_2 bounded below), the growth is at least quadratic. Under dense higher-order coupling (rho_m bounded below for m >= 3), the growth is at least O(N^m).
Remark (Agency count vs. dimension count). The number of discoverable constraint dimensions K*(N) is bounded above by A*(N) (each productive agency discovers at most one new dimension) and below by A*(N) * p_distinct, where p_distinct is the probability that a productive agency discovers a dimension not yet discovered by another agency. When the total constraint dimensionality K is much larger than A*(N) - the Earth regime, where there are always more dimensions to find - overlap among agencies is negligible, p_distinct is close to 1, and K*(N) ~ A*(N). When A*(N) approaches K - the Mars saturation regime - overlap dominates and K*(N) saturates at K regardless of further agency growth. The superlinear growth claim for K*(N) therefore holds in the regime where K >> A*(N), which is the regime of interest for the creation-dissipation analysis.
Remark. On Mars (N = 50-500), K*(N) is bounded above by the habitat's finite constraint dimensionality. Eventually the colony discovers all interactions its environment contains and K* saturates. On Earth, the constraint dimensionality so vastly exceeds current K* that the binding constraint is the number of disclosing agents, not the dimensionality of the problem.
Proposition 9.3 (Increasing marginal returns; Beier, 2026j, Proposition 4.2). Under full disclosure and dense constraint coupling, the marginal contribution to the productive agency count A* of the (N+1)th disclosing agent is strictly increasing in N whenever rho_m exceeds a computable threshold for some m >= 2 and all existing agents disclose fully. In the regime K >> A*(N), the marginal contribution to K* inherits this property (by the dimension-count remark on Proposition 9.2).
Proof. The (N+1)th agent forms potential new agencies with every existing subset. The theoretical number of subsets is 2^N - 1, strictly increasing in N. In practice, the (N+1)th agent interacts productively only with subsets whose observation sets overlap with theirs on active constraint dimensions - a much smaller number determined by the interaction graph's local structure. Under the dense-coupling assumption (rho_m bounded below), the expected number of productive new agencies containing the (N+1)th agent grows at least linearly in N (the same argument as Proposition 9.2: each existing agent has positive probability of productive interaction). The marginal contribution A*(N+1) - A*(N) is therefore at least c * N for constant c > 0, which is strictly increasing in N. Under hoarding, the marginal contribution is bounded by |O_{N+1}| (the individual's observation capacity), which is independent of N. The ratio of marginal A* under disclosure to marginal A* under hoarding therefore grows at least linearly in N. Under the complete-graph upper bound (every subset accessible), the ratio grows exponentially in N (2^N / |O_{N+1}|), but the linear lower bound under sparse interaction is sufficient to establish the increasing-returns property.
This is the increasing-returns information externality: each disclosing agent creates new agency possibilities with existing disclosers. The more agents disclose, the more valuable each additional disclosure becomes. The externality is specific to the combinatorial agency structure and does not arise in standard information economics, where the value of information production is determined by scarcity rather than by the combinatorial richness of the agency space.
At depth 1 (pairwise agencies only), the proved growth rate is quadratic. Greater depth produces faster growth. At depth L, agencies of agencies of agencies compose hierarchically, and the pool of combinable entities at each level includes the outputs of the previous level. Under the dense-coupling assumption at each level, the agency count at level L is Theta(N^{2^L}).
Whether this tower-exponential growth is achievable depends on three empirical quantities: the interaction density at each level (does the complete-graph assumption hold, or is interaction sparse?), the productivity rate at each level (what fraction of higher-order compositions produce novel inferences rather than noise?), and the recursion depth (how many levels of productive composition are possible?). At one extreme - dense interaction, high productivity, logarithmic recursion depth - the growth reaches the superexponential regime N^{N^c}, which is Stirling-equivalent to factorial growth up to polynomial factors (N^N = e^N * N! * (1 + o(1)) / sqrt(2 pi N)). At the other extreme - sparse interaction, rapid productivity decay, shallow recursion - the growth may be merely quadratic or cubic.
The formal results do not resolve which regime obtains in practice. What they establish is a hierarchy of observation architectures ranked by growth rate:
Scalar channels (prices, ratings, scores): capacity O(log N). One dimension per channel. Dissipation dominates on all other dimensions.
Flat multi-channel systems (prices plus disclosure plus ratings): capacity O(N). Linear in participation. Dissipation may be contained for low-dimensional structures.
Combinatorial networks at depth 1 (the Beier Factorial under proved conditions): capacity Omega(N^2). Quadratic growth. Dominates flat channels for any N > 1.
Combinatorial networks at depth L > 1 (the Beier Factorial under recursive composition): capacity potentially Omega(N^{2^L}). Grows with recursion depth and interaction density. The regime in which the creation-dissipation race can be won.
The practical question is the productivity rate at depth - whether the track record capital mechanism (Beier, 2026d) provides sufficient quality filtering to sustain productive composition at higher levels. This is an empirical question about real disclosure networks, not a mathematical one. The mathematics establishes the possibility space; the architecture determines where in that space a given network operates.
Theorem 10.1 (Maintenance Regress). Every observation channel for a coordination-dependent structure is itself coordination-dependent and subject to the Dissipation Law. Maintenance therefore requires a hierarchy of channels, each maintaining the one below it, with each layer subject to its own dissipation.
Proof. By Definition 2.4, an observation channel requires coordinated agent action (operation, funding, interpretive framework maintenance). This makes it coordination-dependent (Definition 2.1) with its own complexity K_1 and undermaintained dimensions (Dissipation Law). Maintenance of the channel requires a further channel. By induction. QED.
Theorem 10.2 (Convergence Condition). The regress converges - produces a finite total maintenance cost - if at each layer n, the rate of encoding coordination into physical substrate exceeds the dissipation rate net of creation:
e_n > d_n(B_n^*) + c_n for all n
where e_n is the encoding rate, d_n is the dissipation rate on the Beierspace residual B_n, and c_n is the creation rate of new complexity at layer n, and the inter-layer contraction ratio B_{n+1}^*/B_n^* is uniformly bounded above by some r < 1 across all layers.
Proof. At each layer n, Beierspace content B_n satisfies dB_n/dt = -e_n + d_n(B_n) + c_n at steady state. The maintenance cost imposed on layer n+1 is the ICMC required to observe and maintain layer n's Beierspace residual B_n^*, which is proportional to B_n^* through the maintenance bandwidth relationship established in the Dissipation Law (higher B_n^* requires more bandwidth to maintain, drawing on the capacity available at layer n+1). The condition e_n > d_n + c_n ensures each layer's residual is strictly smaller than the previous layer's: B_{n+1}^* < B_n^*. Monotone decrease alone does not guarantee convergence of sum_n B_n^* (the harmonic series is a counterexample). The uniform contraction assumption B_{n+1}^* <= r * B_n^* for r < 1 gives B_n^* <= r^n * B_0^*, so sum_n B_n^* <= B_0^*/(1 - r) < infinity.
When the condition fails at some layer n_0, B_{n_0}^* exceeds the capacity available from layer n_0 + 1, and the shortfall propagates upward if subsequent layers cannot absorb the increased load. QED.
Remark (Uniform contraction). The uniform contraction ratio r < 1 is a modeling assumption about the institutional architecture. It holds when each layer's physical encoding achieves a consistent proportional reduction in Beierspace residual - when the institutional technology for converting coordination into substrate does not degrade as the hierarchy deepens. Whether this holds for specific institutional stacks is an empirical question. The condition is sufficient for convergence but may not be necessary: sum_n B_n^* could converge under weaker conditions (e.g., B_{n+1}^*/B_n^* -> r < 1 rather than uniform boundedness).
Remark. The convergence condition provides an information-theoretic criterion for institutional stability. The condition is falsifiable in principle: the prediction is that institutional hierarchies where encoding lags dissipation-plus-creation at any critical layer exhibit cascading instability, while those satisfying the condition with margin exhibit resilience. Independent measurement of e_n, d_n, and c_n is challenging but conceivable through institutional data (rate of norm codification, rate of institutional coherence loss, rate of new regulatory complexity creation).
Proposition C of Beier (2026c) follows from the Minimum-Dissipation Bound (Theorem 7.1) applied to a scalar channel concentrated on V_1. The excess distortion is excess dissipation under the Observation-Maintenance Identity.
The phase transition at S = 1 follows from the Dissipation Law applied to eigenstructure dynamics. Below the threshold, dissipation degrades maintenance infrastructure, producing positive feedback toward collapse.
The coupling is a direct consequence of Axiom 1: lossy compression destroys constraint-dimension information (Lossy result); destroyed information is withdrawn maintenance (Axiom 1); withdrawn maintenance causes dissipation (Dissipation Law); dissipation destabilizes the eigenstructure (Beierspace result); eigenstructure instability invalidates the distortion calculation (the codec drifts out of alignment). A single causal chain, not a coupling between independent phenomena.
The Disclosure Incentive is a supplementary maintenance channel targeting undermaintained dimensions. Its marginal value exceeds marginal price-precision improvement (Beier, 2026c, Proposition D) because undermaintained dimensions have higher residual variance and therefore higher marginal return to bandwidth. In the coupled system, disclosure has three value components: static distortion reduction, stability dividend, and threshold dividend (Beier, 2026-PN1, Finding F.2).
This section contains two conjectures tightly connected to the formal apparatus and producing specific falsifiable predictions.
Conjecture 12.1. Disclosure mandates targeting the consensus dimension V_1 (earnings guidance, analyst price targets) improve the stability ratio S without reducing excess dissipation Delta. The system appears healthier by S while the coupled threshold x*_eff (Beier, 2026-PN1) is unchanged.
Mechanism. Consensus-targeted disclosure adds bits to an already-compressed dimension (low marginal return per Proposition D). Delta is unchanged because constraint dimensions remain uncompressed. The threshold dividend requires Delta to fall.
Falsifiable prediction. Post-implementation of consensus-targeted disclosure mandates (e.g., Regulation FD), the stability ratio S should improve while covariance instability during subsequent stress events (measuring Delta's effect on eigenstructure) should be unattenuated relative to the pre-mandate period, controlling for shock magnitude.
Conjecture 12.2. An exogenous increase in price informativeness (higher SNR_1) can trigger self-destabilizing dynamics by pushing the coupled threshold x*_eff above current capacity x, through the mechanism: higher SNR_1 -> higher Delta -> higher psi(Delta) -> lower x_eff -> higher x*_eff.
Falsifiable prediction. Controlling for other factors, rapid exogenous improvements in market microstructure in systems near the stability boundary should precede covariance instability. Joint test: high SNR_1 with low constraint-dimension disclosure should exhibit the largest instability; high SNR_1 with high constraint-dimension disclosure should be attenuated.
Conjecture 12.3. The rate of coordination-dimension creation generically exceeds the rate of observation channel extension: creating a new Beierspace dimension (through distributed agent action) is cheaper and faster than building the institutional observation channel to maintain it. The Scope Condition's dimensional excess widens over time in the absence of countervailing investment.
Remark. The Beier Factorial (Section 9) is the countervailing mechanism: a recursively compositional disclosure network whose proved capacity is at least quadratic and potentially much faster under deep recursive structure. Whether this mechanism is sufficient to offset the dimensional growth asymmetry depends on the productivity rate of recursive composition at depth (Section 9.6).
Problem 1 (Thermodynamic Isomorphism). Does the structural parallel between D*(M) and thermodynamic state functions extend to a full mathematical isomorphism with Legendre transforms, entropy as concave function of extensive variables, and coordination analogs of exact differentials?
Problem 2 (The Creation Principle). Under what conditions do agents allocate capacity to creating new coordination dimensions rather than maintaining existing ones? What is the social welfare function that trades off the value of new coordination against the dissipation cost of increased dimensionality?
Problem 3 (Effective Recursion Depth and Productivity Rates). The Beier Factorial's proved growth rate (quadratic under dense pairwise coupling) is a lower bound. Higher-order growth depends on the recursion depth and productivity rate at each level of agency composition - both empirical quantities. Estimating these for specific disclosure networks (financial markets with structured ESG disclosure, scientific collaboration networks, organizational knowledge-sharing systems) would determine where in the hierarchy of observation architectures (Section 9.6) real networks operate. The gap between the proved quadratic lower bound and the potential superexponential upper bound is the Beier Factorial's main open empirical question.
Problem 4 (Non-Stationary Minimum-Dissipation Bound). The bound D*(M) is derived for a stationary source. Coordination structures are non-stationary. The single-letter property of Gaussian rate-distortion theory gives instantaneous validity, but the dynamic feedback (dissipation changes the spectrum, changed spectrum changes the dissipation rate) requires a dynamic bound. This is an open problem in information theory itself (Ephraim and Merhav, 2002).
Problem 5 (Empirical Calibration). Priority targets: eigenvalue spectrum estimation for specific asset classes, effective maintenance bandwidth of existing disclosure regimes, dimensional creation rates in specific sectors, productivity rates of recursive composition at depth.
The observation-maintenance identity and bounded capacity produce a framework with a dissipation law, a capacity budget, a minimum-dissipation bound, a mixing theorem, a regress theorem, a derivation of Goodhart's Law, and a characterization of recursive observation capacity (at minimum quadratic and potentially superexponential). The framework recovers two independently derived results - excess distortion in price formation and the Beierspace phase transition - as projections of a single structure.
The Goodhart derivation provides the information-theoretic foundation for a folk theorem in social science: scalar measures of high-dimensional structures degrade the unmeasured dimensions. The Beier Factorial identifies the class of observation architecture whose capacity can match the complexity growth that scalar channels cannot: recursively compositional networks whose proved capacity is at least quadratic and whose potential capacity, under deep recursive structure, reaches the superexponential regime.
The framework's discipline is in what it distinguishes. What is proved (dissipation, capacity budget, mixing, Goodhart) is separated from what is conjectured (disclosure trap, liquidity-instability, dimensional growth asymmetry) and from what is speculative (thermodynamic isomorphism, quantum parallel, civilizational encoding). The axioms identify the terrain. What is mapped, what is surveyed, and what is glimpsed from a distance are marked accordingly.
The following extensions are included for their generative value. They are not derived from the axioms and are not claimed as results. They are directions.
Quantum mechanics and coordination thermodynamics share a structural property: the observation channel's limitations determine the system's accessible state space, and the system's state is not fully independent of the observation. Both involve projection operators on high-dimensional state spaces through low-dimensional channels. Both produce irreducible information loss. Whether this reflects shared deep structure or mathematical generality is an open question.
AI alignment through scalar reward signals is a lossy codec on a Beierspace source (human values are high-dimensional and coordination-dependent). The Dissipation Law predicts degradation on value dimensions the reward signal cannot reach. This connects the framework to the alignment problem but requires formalizing what "Beierspace" means for a value structure.
Physical infrastructure converts Beierspace maintenance burden into physical-substrate persistence. A bridge does not destabilize because agents stopped believing in it. The history of material civilization may be readable as information architecture: each construction phase encodes coordination into matter, reducing the aggregate dissipation rate. The convergence condition (Theorem 10.2) provides the criterion: civilizations endure when encoding outpaces dissipation at every institutional layer.
Major institutional innovations (double-entry bookkeeping, insurance, central banking, securities regulation) add observation bandwidth. Each creates new coordination possibilities that create new unobserved dimensions. Whether the net effect is stabilizing depends on whether new bandwidth exceeds new dimensionality. The ratchet character (institutions persist after crises but new complexity also persists) means the system accumulates both bandwidth and dimensionality over historical time.