Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 2026
version 2.4
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 proposes that coordination-dependent structures are constituted by observation, not merely described by it, and identifies a falsifiable empirical test distinguishing this claim from the standard epistemic alternative. The standard view predicts that exogenous withdrawal of observation (e.g., analyst coverage loss following MiFID II) produces noisier estimates of an unchanged firm. The Constitution Axiom predicts that the firm's coordination-dependent operating metrics themselves degrade. Existing evidence (Derrien and Kecskés, 2013; Kelly and Ljungqvist, 2012) documents real operational changes following exogenous coverage loss, consistent with the Constitution Axiom but not yet discriminating between the two views. The distinguishing prediction - that operational degradation scales with the Beierspace fraction b rather than with information asymmetry - has not been tested. This paper identifies the test. Running it is future work.
The formal statement is Axiom 0 of Coordination Science: for any structure with Beierspace fraction b > 0, the covariance matrix decomposes as Sigma(t) = (1 - b) * Sigma_E + b * F(O(t)), where Sigma_E is the physically-determined component (independent of observation), F is a continuous linear functional mapping aggregate observation to coordination structure, and O(t) is the observation process of finite agents. The Constitution Condition requires that coordination-dependent structure on any dimension decays to zero when observation on that dimension is withdrawn. The Cross-Dimensional Constitution Condition requires that coordination-dependent interactions between dimensions can exist only when some agency's observation spans both.
Of the five axioms of the Prolegomena, one (Observation-Maintenance) is contained in Axiom 0 by inclusion, and two (Irreducible Heterogeneity and Reflexivity) are derived from it under specified conditions. Two (Bounded Capacity and Compression Inevitability) follow from finitude alone and do not require the Constitution Axiom. The six theorems of the Prolegomena (Excess Distortion, Dissipation, Goodhart, Beier Factorial, Phase Transition, and the Grossman-Stiglitz Resolution) follow from the five axioms through existing proofs.
Keywords: Coordination Science, observation-constituted reality, Beierspace fraction, covariance decomposition, rate-distortion theory, scalar compression, institutional design, disclosure commons, analyst coverage, MiFID II, information asymmetry
In Matt Kaplan's I Told You So!, there is a moment in the story of Ignaz Semmelweis that has stayed with me. Semmelweis had discovered that cadaver material on physicians' hands was killing mothers in the maternity ward. He introduced chlorine wash and the death rate collapsed. But he had not yet connected his discovery to the broader principle - that the same invisible agents causing puerperal fever were the same agents causing wound infections, the same agents responsible for a range of seemingly unrelated pathologies. The connection to germ theory was right in front of him and he could not see it. Pasteur and Koch would see it later.
I want to be clear about the analogy. I am not claiming to be Pasteur. What I am claiming is this: the five axioms of the Prolegomena keep working in domains they were not designed for, and I believe there is a single generative axiom underneath them that explains why. This paper states that axiom formally, derives the five axioms from it as theorems, and identifies an empirical test that distinguishes it from the standard alternative.
The formal work so far has applied the same five axioms primarily to economics - to prices, environmental information, constraint-dimension analysis, and the formal structure of coordination under compression. In every application, the same axioms produce the same theorems and the same structural remedy. A mathematical framework can be domain-independent and still be empirically wrong in most domains. Shannon's rate-distortion theory does not contain a variable for domain (Shannon, 1948; Cover and Thomas, 2006). The theorems proved from it inherit that invariance. But domain-independence of the math explains why the formal structure transfers. It does not explain why the world cooperates - why the formal structure matches what we observe in every domain we have tested it against.
This paper proposes that the explanation is a deeper axiom - Axiom 0 of Coordination Science - from which the five axioms of the Prolegomena follow as theorems under specified conditions. Section 3 states Axiom 0 formally. Section 4 derives the five axioms from it. Section 7 identifies a falsifiable empirical test - grounded in the natural experiment of MiFID II and supported by existing empirical literature - that distinguishes the Constitution Axiom from the standard epistemic framing.
This paper will be updated periodically as the research program progresses. It is a living document. What is proved is marked as proved. What is conjectured is marked as conjectured. What is speculative is marked as speculative.
The Constitution Axiom is not proposed in a vacuum. Multiple independent lines of thought across physics, philosophy, and social science have converged on aspects of the same insight. These thinkers differ from each other in important ways - in formal commitments, in scope, in the specificity of their claims. The Constitution Axiom does not claim they all said the same thing. It claims that their distinct contributions, taken together, point toward a structural pattern that none of them identified in full and that the Prolegomena's information-theoretic framework may be able to formalize. Whether this claimed connection is genuine or merely apparent is an open question.
Wheeler's "It from Bit." John Archibald Wheeler (1990) proposed that physical reality is informational at its base - that every physical quantity derives its existence from answers to yes-or-no questions. Wheeler's program is a specific research agenda within the foundations of physics, with formal commitments the Constitution Axiom does not yet share. The structural parallel is that in both frameworks, observation does not describe pre-existing reality but participates in constituting it.
Bohr's Complementarity. Niels Bohr introduced complementarity at the Como lecture in 1927 (published in Bohr, 1928) and defended it through the EPR debate (Bohr, 1935). Bohr insisted that quantum properties do not exist independently of the measurement context. Bohr's framework is a specific interpretive stance within quantum mechanics, not a general theory of observation-constituted reality. The Constitution Axiom notes the structural parallel without claiming equivalence.
Rovelli's Relational Quantum Mechanics. Carlo Rovelli (1996) proposed that quantum states are properties of interactions between systems, not properties of systems themselves. Rovelli's program has specific mathematical content that goes beyond what the Constitution Axiom can currently claim. Whether the parallel reflects a deep structural connection or a surface-level similarity is precisely the question this paper raises but does not answer.
Searle's Social Ontology. John Searle (1995) distinguished "brute facts" from "institutional facts" that exist only through collective acceptance. Searle's framework involves specific structures - constitutive rules, collective intentionality, status functions - that do not map directly onto the Prolegomena's information-theoretic vocabulary. Searle is explicitly anti-reductivist: he would likely reject the claim that institutional facts and quantum measurement are instances of one principle. The Constitution Axiom takes this objection seriously. The Beierspace fraction b is offered as a formal analog of Searle's brute-institutional distinction. Whether this formal analog produces a genuine unification or merely a relabeling is an open question.
Berger and Luckmann's Social Construction. Peter Berger and Thomas Luckmann (1966) argued that social reality is constructed through human activity and then experienced as objective reality by subsequent generations. Their three-stage model (externalization, objectivation, internalization) maps to the Constitution Axiom's framework: externalization is the creation of a Beierspace structure through coordinated observation, objectivation is the encoding of that structure into physical form (lower b), and internalization is the process by which new observers are trained to maintain the structure.
Soros's Reflexivity. George Soros (2013) identified reflexivity in financial markets - participants' beliefs changing the market in a feedback loop. The Prolegomena does not claim to have discovered reflexivity. Soros saw it decades ago. What the Prolegomena adds is the formalization: the phase transition at S=1 (Beier, 2026e) is the formal consequence of reflexivity that Soros's framework could not derive.
Ostrom's Commons. Elinor Ostrom (1990) demonstrated that communities can maintain shared resources through collective observation and enforcement. Her empirical findings are consistent with the Dissipation Law (commons that lose collective observation degrade) without reducing to it. Ostrom's design principles have specific institutional content that does not collapse into "observation bandwidth."
MacKenzie's Performativity. Donald MacKenzie (2006) showed that financial models reshape the markets they describe. Under the Constitution Axiom, performativity is consistent with Axiom 5: the observation (the model) constitutes the observed (the market).
What is new. Each of these thinkers developed a distinct contribution in a specific domain. The Constitution Axiom proposes - and this paper formally derives - that there is a mathematical structure connecting them: in each case, some form of observation constitutes some form of reality. The five axioms of the Prolegomena follow from this structure as theorems.
Axiom 0 (The Constitution Axiom). Let V(t) be a K-dimensional source with K >= 2, representing the state of a coordination-relevant structure at time t. The covariance matrix of V decomposes as:
Sigma(t) = (1 - b) * Sigma_E + b * F(O(t))
where:
b in [0, 1] is the Beierspace fraction.
Sigma_E is the eigenspace component: a positive definite matrix independent of observation. It represents the physically-determined structure. It persists whether or not anyone observes it.
O(t) = {o_1(t), ..., o_N(t)} is the observation process of N agents, where o_i(t) in R_+^K specifies the bandwidth agent i allocates to each dimension at time t.
F: O -> S_K^+ is a continuous linear functional from observation processes to positive semidefinite matrices, mapping aggregate observation to coordination structure. Linearity is a simplifying assumption appropriate for the first-order analysis: it assumes that coordination structure responds proportionally to aggregate observation bandwidth. This approximation is reasonable when observation operates away from extreme regimes. In practice, coordination systems likely exhibit diminishing returns at high observation (saturation) and potentially increasing returns near critical thresholds (phase transitions). A nonlinear F is likely required to capture these effects, particularly the phase transitions and network tipping points characteristic of advanced coordination structures (Beier, 2026e). Extensions to nonlinear F are deferred to future work. The linear case is sufficient to establish the derivations in Section 4 and the empirical test in Section 7.
The functional F satisfies two conditions:
Constitution Condition. For any dimension k, if the aggregate observation bandwidth on dimension k is zero (sum_i o_i^k(t) = 0 for all i), then:
lim_{t' -> infinity} [F(O(t'))]_kk = 0
Coordination-dependent structure on dimension k ceases to exist when no agent observes dimension k. The limit reflects that existing coordination has inertia (institutional memory, contractual commitments, cultural norms) but decays without observation.
Cross-Dimensional Constitution Condition. For any pair of dimensions j and k, the off-diagonal element [F(O)]_jk - representing coordination-dependent interaction between j and k - satisfies:
[F(O)]_jk = 0 unless there exists a subset S of agents whose combined observation set spans both j and k
Formally: [F(O)]_jk > 0 requires the existence of at least one subset S of {1, ..., N} such that union_{i in S} supp(o_i) includes both j and k, AND who interact to form a joint assessment. This condition follows from the linearity of F: a linear functional maps observation vectors with disjoint support to a block-diagonal matrix, because the cross-terms in the output require cross-terms in the input. Observation vectors with disjoint support have zero cross-terms by construction. Non-zero off-diagonal elements in F(O) require input observation that spans both dimensions.
The following conditions apply to the agents, not to Axiom 0 itself:
Finitude Condition. Each agent i has bounded observation capacity:
||o_i||_1 = sum_{k=1}^{K} o_i^k <= C_i < infinity
The total population bandwidth is M = sum_{i=1}^{N} C_i < infinity.
Participation Condition. Each agent's observation allocation responds to the source: o_i(t) = g_i(V(t), history, noise), where g_i is agent i's observation-allocation function. Agents are not external observers. They are participants in the coordination that constitutes V. Their observation of V feeds back to V through the coordination process.
Operational measurement of b. The Beierspace fraction is not determined by analyst judgment about which drivers are "coordination-dependent." It is determined by the formal decomposition itself. b is the fraction of Sigma(t) that vanishes when observation is completely withdrawn: if O(t) -> 0, then Sigma(t) -> (1 - b) * Sigma_E. The component that vanished - b * F(O) - is the coordination-dependent component. b is therefore operationally measurable through the structural response of the source to observation withdrawal, not through prior classification of drivers. This operational definition is the basis of the empirical test in Section 7.
A note on empirical measurement. The operational definition of b is clean in theory: b measures the fraction of structure that vanishes when observation is withdrawn. In practice, completely withdrawing observation from a functioning market, institution, or field is rarely possible. The MiFID II natural experiment in Section 7 approximates partial withdrawal for a narrow range of the observation spectrum (analyst coverage). The gap between the formal definition and the empirical proxy is real and should be acknowledged. If the MiFID II results support the Constitution Axiom's predictions, the interpretation requires assuming that analyst coverage is a meaningful component of the observation process O(t). This assumption is testable but not trivially true.
This section derives each of the five axioms of the Prolegomena. Three axioms (1, 4, and 5) require Axiom 0. Two axioms (2 and 3) follow from the auxiliary conditions on agents alone - they do not require the Constitution Axiom and would hold in any framework with finite agents communicating about multi-dimensional sources. The honest claim is not "five axioms from one" but "the five axioms organize under Axiom 0 plus standard conditions on finite participating agents, with Axiom 0 doing the constitutive work for the three axioms that distinguish the Constitution Axiom from the standard epistemic framing."
Axiom 1 (Observation-Maintenance Identity) is not derived from Axiom 0. It is contained in Axiom 0. Axiom 0 states that the coordination-dependent component of Sigma is b * F(O(t)) - that is, the component is determined entirely by observation. The Constitution Condition specifies that this component decays to zero when observation is withdrawn. Axiom 1, which states that observation and maintenance of coordination-dependent structure are the same act, is a restatement of these two properties restricted to existing structures. The relationship is entailment-by-inclusion: Axiom 0 is strictly stronger than Axiom 1, and Axiom 1 follows trivially from Axiom 0's definition plus the Constitution Condition.
Requires: Axiom 0 (by inclusion, not derivation).
Theorem. The total observation bandwidth of any finite population of agents is bounded above.
Proof. By the Finitude Condition, each of the N agents has capacity C_i < infinity. The total population bandwidth is M = sum_{i=1}^{N} C_i. Since N is finite and each C_i is finite, M is finite. QED.
Requires: Finitude Condition only. Axiom 0 plays no role in this derivation.
Theorem. Communication between finite agents about a multi-dimensional source V with K >= 2 necessarily compresses.
Proof. Agent i has internal capacity C_i < infinity (by the Finitude Condition). For a continuous source V with positive definite Sigma, representing V to arbitrary precision requires arbitrarily many bits. Agent i's representation of V is constrained to C_i bits. Since C_i < infinity, the representation is lossy. Communication from agent i to agent j about V passes through a channel of capacity min(C_i, C_j, C_channel), all finite. The transmitted signal carries at most min(C_i, C_j, C_channel) bits about V. Since the source entropy exceeds this bound for any continuous multi-dimensional source, the communication compresses. QED.
Requires: Finitude Condition + K >= 2. Axiom 0 plays no role in this derivation. Compression inevitability holds in any framework with finite agents, not only in the Constitution Axiom's framework.
Theorem. Cross-dimensional constraint interactions can be constituted only by observation sets that span the interacting dimensions.
Proof. By the linearity of F (stated in Section 3), observation vectors with disjoint support map to a block-diagonal contribution to F(O). Specifically: if observation inputs on dimensions j and k are decoupled (no agent observes both j and k, and no agency combines observations spanning both), the linear functional F produces zero off-diagonal terms [F(O)]_jk. This is because a linear map preserves the independence structure of its inputs: if the input has zero cross-correlation between the j and k components, the output inherits this structure.
Contrapositive: if no subset of agents has combined observation spanning both j and k, then [F(O)]_jk = 0. The coordination-dependent interaction between j and k does not exist.
This holds regardless of the bandwidth allocated to j and k individually. If all agents observe j but none observe k, no agency spans both dimensions. Therefore [F(O)]_jk = 0 for all pairs involving unobserved dimensions.
For homogeneous observation (all agents observe the same set S_0), interactions can be constituted only within S_0. Cross-dimensional interactions involving dimensions outside S_0 require agents with different observation sets - heterogeneous observation.
The number of constitutable pairwise interactions depends on the heterogeneity of the observation distribution across agents. The maximum possible number of constitutable interactions grows quadratically in N (because the number of distinct pairs of agents is N(N-1)/2). The actual number depends on the degree to which agents' observation vectors differ: if all agents observe the same dimensions, the combinatorial growth vanishes. The quadratic bound is achieved when agents have sufficiently heterogeneous coverage that each new pair spans dimension-pairs no previous pair spanned. QED.
Requires: Axiom 0 (specifically the linearity of F) + Finitude.
Theorem. For any source with b > 0 observed by participating agents, observation and the source are coupled.
Proof. By Axiom 0, the source's covariance is Sigma(t) = (1 - b) * Sigma_E + b * F(O(t)). By the Participation Condition, the observation process O(t) is generated by agents responding to V(t): o_i(t) = g_i(V(t), history, noise). The system is coupled:
V(t) -> determines O(t) through agents' observation responses (Participation Condition)
O(t) -> determines Sigma(t) through F (Axiom 0)
Sigma(t) -> determines the distribution of V(t)
The coupling is present whenever b > 0 (so that F(O) contributes to Sigma) and agents are participants (so that O depends on V). For b = 0, Sigma = Sigma_E independent of O, and the coupling vanishes. Without the Participation Condition (if agents allocated observation by coin flip, independent of V), Axiom 0 would still hold but reflexivity would not. QED.
Requires: Axiom 0 + Participation Condition.
The five axioms of the Prolegomena organize under Axiom 0 plus conditions on the agents:
The three axioms that require Axiom 0 (1, 4, and 5) are the axioms that carry the Constitution Axiom's distinctive content - the claim that observation constitutes coordination-dependent reality rather than merely describing it. The two axioms that do not require Axiom 0 (2 and 3) are generic consequences of finitude that any information-theoretic framework would share.
The six theorems of the Prolegomena (Excess Distortion, Dissipation, Goodhart, Beier Factorial, Phase Transition, and the Grossman-Stiglitz Resolution through Track Record Capital) follow from the five axioms through the existing proofs in the companion papers (Beier, 2026h, 2026d, 2026e, 2026i). The formalization adds a deeper layer beneath the existing proofs without replacing them.
A note on the endogeneity of b. The framework treats b as a parameter of the source. But if the Constitution Axiom is correct, b itself may be endogenous to observation. A firm's coordination-dependence could increase when analysts cover it, because coverage creates new coordination channels (investor relations, governance pressure, market discipline) that would not exist without the coverage. If b is a function of O, the decomposition Sigma = (1-b(O))*Sigma_E + b(O)*F(O) becomes nonlinear in O even if F is linear. This complication connects to the identification problem (Research Agenda item 9) and is deferred to future work, but should be noted: the linear framework in this paper assumes b is fixed for a given source over the relevant time horizon.
If the Constitution Axiom is correct, the Beierspace fraction b is an ontological parameter, not a measurement convenience. It tells you how much of a thing's existence depends on observation and how much persists independently.
Formal definition. b = 1 - [tr(Sigma(t)|_{O=0}) / tr(Sigma(t)|_{O=O*})], where Sigma(t)|_{O=0} is the covariance under zero observation and Sigma(t)|_{O=O*} is the covariance under the current observation regime. b measures the fraction of the source's total structure that vanishes when observation is completely withdrawn. This definition is operational: it depends on the structural response of the source to observation withdrawal, not on prior classification of drivers.
At b = 0: the thing exists regardless of observation. A rock. A river's flow rate. A radioactive decay constant. Eigenspace.
At b = 1: the thing exists only because agents are observing it into existence. A market price. A political institution's legitimacy. A scientific field's coherence. Beierspace.
At 0 < b < 1: the thing is partly physical and partly constituted by observation. A bridge. A city. A codebase. The physical component persists. The coordination component depends on observation.
Under the Constitution Axiom, scalar compression does more than lose information. It withdraws observation from coordination-dependent structures that require observation to exist.
A price compresses multi-dimensional observation into one dimension. The observation on the remaining dimensions drops to zero. By the Constitution Condition, the coordination-dependent structure on those dimensions decays toward zero. The price did not merely fail to measure those dimensions. It withdrew the observation that was keeping them in existence.
The disclosure commons is the treatment. It restores observation to the dimensions the scalar compressed away. It does not add new observation to a pre-existing structure. It reconstitutes the structure on dimensions where it had ceased to exist.
If observation constitutes coordination-dependent reality, then the Beier Factorial is not a discovery process. It is a creation process. When two heterogeneous observers combine their observation sets, they do not "discover" a pre-existing constraint interaction. They constitute one. The Beier Factorial's quadratic growth is not growth in discoveries. It is growth in constituted reality.
The Constitution Axiom and the standard epistemic framing ("observation reveals pre-existing reality") agree on many predictions. Both predict that better information improves decision-making. Both predict that scalar compression loses information. Both predict that the Beier Factorial increases the number of identified constraint interactions with the number of observers.
They disagree on one specific, testable prediction.
The Standard View predicts: If you withdraw observation from a coordination-dependent structure (b > 0), the structure's true state continues evolving normally. The market's ESTIMATE of the state becomes noisier - bid-ask spreads widen, price discovery slows, volatility of returns increases. But the underlying structure - the firm's governance quality, its supply chain relationships, its talent retention, its operational coordination - is unaffected by whether anyone is observing it. When observation resumes, the price adjusts to wherever the firm actually is.
The Constitution Axiom predicts: If you withdraw observation from a coordination-dependent structure, the structure itself degrades. Not just the market's estimate of it. The coordination-dependent component of the structure - b * F(O(t)) - decays toward zero on the dimensions that lost observation. When observation resumes, it reveals not merely a noisier estimate of an unchanged firm, but a firm whose coordination-dependent operating metrics have measurably deteriorated during the observation blackout.
The natural experiment: MiFID II. The European Union's Markets in Financial Instruments Directive II, implemented in January 2018, required asset managers to pay separately for investment research, ending the practice of bundling research costs with trading commissions. The regulation caused a sharp, exogenous decline in analyst coverage of European equities, particularly small and mid-cap stocks. The coverage loss was not caused by the firms' deteriorating condition. It was caused by a regulatory change in the economics of research provision. This is a clean natural experiment: observation was withdrawn for reasons unrelated to the state of the observed structure.
Derrien and Kecskés (2013) studied exogenous analyst coverage terminations caused by brokerage closures and mergers. They found that firms losing coverage reduced investment spending and raised less external capital - reductions of approximately 1.9% and 2.0% of total assets respectively. These are changes in the FIRM, not in the market's estimate of the firm. D&K interpret their findings through the standard cost-of-capital channel: coverage loss increases information asymmetry, raising the cost of external finance, which reduces investment. Under the Constitution Axiom, the same evidence admits an additional interpretation: the observation that was maintaining the firm's coordination with capital markets was withdrawn, and the coordination-dependent dimension of the firm's capital-raising capacity decayed. Both interpretations predict the same first-order effect (reduced investment after coverage loss). They diverge on whether the effect scales with information asymmetry or with coordination-dependence - the test proposed below.
Kelly and Ljungqvist (2012) used exogenous coverage terminations from brokerage closures and mergers to test asymmetric-information asset pricing models. They found that prices and uninformed demand fall as information asymmetry increases, operating through a liquidity channel. K&L interpret their results within the standard information-asymmetry framework - they view the price declines as confirming that information asymmetry is priced. Under the Constitution Axiom, the same evidence admits a different interpretation: the observation withdrawal did not merely increase information asymmetry about an unchanged firm but degraded the coordination-dependent value the observation was maintaining. The evidence is consistent with both interpretations. Distinguishing them requires the interaction test proposed below.
Hong, Lim, and Stein (2000) found that, holding firm size fixed, momentum strategies are more profitable among stocks with lower analyst coverage - consistent with information diffusing more slowly into prices when fewer analysts observe the firm. Under the standard view, this reflects slower transmission of existing information through a thinner observation network. Under the Constitution Axiom, reduced coverage reduces the observation that constitutes the coordination-dependent structure of the firm's market price, producing a price that is less responsive to information because the coordination maintaining price efficiency has partially decayed.
The standard view predicts that the damage from coverage withdrawal scales with information asymmetry: firms that are harder to analyze suffer more from losing coverage because the coverage was providing more marginal information.
The Constitution Axiom predicts that the damage from coverage withdrawal scales with b: firms whose value is more coordination-dependent (technology companies, platform businesses, service firms, companies whose competitive position depends on network effects, brand equity, or institutional relationships) suffer more from losing coverage than firms whose value is more physically anchored (commodity producers, utilities with regulated physical assets, real estate investment trusts with tangible property).
These predictions diverge. Information asymmetry and b are correlated but not identical. A commodity producer with opaque reserves has high information asymmetry but low b (the reserves exist physically). A technology platform with transparent financials has low information asymmetry but high b (the platform's value is constituted by network coordination). The Constitution Axiom predicts the platform suffers more from coverage withdrawal. The standard view predicts the commodity producer suffers more.
The test: Using the MiFID II natural experiment, regress the change in firms' operating metrics (not stock price, but operational coordination measures: management turnover, supplier relationship stability, employee retention, capital expenditure, and governance scores) on the change in analyst coverage, interacted with a proxy for b (intangible asset intensity as a measurable proxy, with the acknowledged limitation that intangible asset intensity does not perfectly track the formal definition of b). If the interaction term is significant and positive - if firms with higher b-proxy show greater operational degradation when coverage is withdrawn - the Constitution Axiom produces a prediction the standard epistemic framing does not.
A note on the post-MiFID II literature. The evidence cited above (Derrien and Kecskés, Kelly and Ljungqvist, Hong, Lim, and Stein) predates MiFID II. MiFID II has been in effect since January 2018 - over eight years. A growing body of work examines its effects on analyst coverage, market liquidity, and price efficiency, but to the author's knowledge, no published study has tested whether post-MiFID II operational degradation scales with firms' coordination-dependence (b) as opposed to information asymmetry. The specific interaction test proposed here has not been run. If subsequent work confirms or refutes this prediction, this paper will be updated accordingly.
A note on sample construction. The distinguishing prediction requires firms where b and information asymmetry diverge. In the European equity universe, candidates include: commodity producers and extractive firms with opaque reserve estimates (high information asymmetry, low b - the reserves exist physically); regulated utilities with transparent rate structures (low information asymmetry, low b - the physical assets persist); technology platforms and software firms with transparent financials but coordination-dependent revenue (low information asymmetry, high b - value depends on network effects and user coordination); and professional services firms with simple financials but high dependence on institutional relationships (low information asymmetry, high b). The European small and mid-cap universe affected by MiFID II coverage losses contains firms across all four categories. Whether the sample size in the divergent cells is sufficient to power the interaction test is an empirical question that depends on the effect size, which is unknown ex ante. A pilot analysis of the coverage-loss distribution across intangible-intensity quartiles would be the first step in assessing feasibility.
The preceding sections state the axiom formally, derive the five axioms, and identify an empirical test. This section describes how the Constitution Axiom reframes existing phenomena. These reframings are interpretive, not derived predictions. They do not generate testable distinctions from the standard view in their current form. They are included to indicate the scope of the research program, not as evidence for the axiom.
The boundary between social and empirical science dissolves. The social sciences study coordination-dependent structures. The empirical sciences study physical structures. The division assumes these are different kinds of things requiring different methods. The Constitution Axiom says they are the same kind of thing at different values of b. A market (b near 1) and a crystal lattice (b = 0) sit on the same spectrum. The methods differ because the parameters differ, not because the ontology differs. Coordination Science, if it develops as a formal discipline, would be the science of the full spectrum - the science of how observation-constituted reality behaves under compression at every value of b.
Institutional design becomes ontological architecture. Every institution that relies on scalar summaries - prices, grades, votes, diagnostic codes, performance metrics - is, under the Constitution Axiom, not merely losing information about the world. It is de-constituting the world on the dimensions the summary does not carry. The design of an information architecture is a decision about what coordination-dependent reality will exist and what will be de-constituted.
The replication crisis as a dissipation phenomenon. If scientific knowledge is partly coordination-dependent (b > 0) - the coherence of a research field depends on the coordinated observation of its practitioners - then the replication crisis is consistent with the Dissipation Law operating on scientific fields whose observation bandwidth has been redirected toward scalar targets (publication count, citation count, p < 0.05). This reframing does not currently generate a prediction that differs from the standard account (misaligned incentives, insufficient pre-registration). Whether the Constitution Axiom adds predictive content to the existing diagnosis of the replication crisis - beyond relabeling - is an open research question.
The deepest level: Reality has two components. Eigenspace (exists independently of observation) and Beierspace (exists only through observation by the agents whose coordination constitutes it). The boundary between them is parameterized by b.
The generative axiom: For b > 0, Sigma(t) = (1 - b) * Sigma_E + b * F(O(t)). Observation constitutes the coordination-dependent component of reality. This is Axiom 0.
The five axioms: Organized under Axiom 0 plus conditions on finite participating agents. Observation-Maintenance, Irreducible Heterogeneity, and Reflexivity require Axiom 0. Bounded Capacity and Compression Inevitability require only finitude.
The six theorems: Follow from the five axioms through the existing proofs. Excess Distortion, Dissipation, Goodhart, Beier Factorial, Phase Transition, and the Grossman-Stiglitz Resolution.
The disease: Scalar compression withdraws observation from constraint dimensions of coordination-dependent structures. The withdrawal de-constitutes the structure on those dimensions.
The treatment: Multi-dimensional, attributed, outcome-scored observation. The disclosure commons. Not better measurement of pre-existing reality. The constitution of coordination-dependent reality on dimensions where it had ceased to exist.
The empirical test: Operational degradation from exogenous coverage withdrawal scales with b (Constitution Axiom) or with information asymmetry (standard view). MiFID II provides the natural experiment. Existing evidence is consistent with both views. The specific test that distinguishes the two views has not yet been run.
This paper states a formal axiom, derives the five axioms of the Prolegomena from it under specified conditions, and identifies an empirical test that distinguishes it from the standard alternative.
The axiom is formalized as a covariance decomposition with a linear functional F satisfying two conditions. Three of the five derivations require Axiom 0. Two require only finitude. The empirical test generates a prediction the standard epistemic framing does not.
What the paper does not do: it does not run the empirical test. It does not specify F for any particular domain (the specific functional form of F for financial markets, for institutional coordination, or for scientific communities is future work). It does not prove that the decomposition Sigma = (1-b)*Sigma_E + b*F(O) is uniquely identified without additional restrictions on F and Sigma_E. It does not formalize the connection to quantum mechanics. It does not engage at sufficient depth with any single predecessor to establish precisely where the Prolegomena adds formal content that the predecessor lacks.
These are the next tasks. They are identified in the research agenda below.
Run the empirical test. Using MiFID II as the natural experiment, test whether operational degradation from exogenous coverage withdrawal scales with b (Constitution Axiom) or with information asymmetry (standard view).
Specify F. Derive the specific functional form of F for financial markets, using the structural response of asset covariance to analyst coverage changes as the identifying variation.
Formalize the connection to quantum foundations. Is there a mathematical correspondence between rate-distortion functions and quantum state spaces, between observation bandwidth and measurement energy, between the S=1 phase transition and decoherence thresholds? This requires specific mathematical work, not verbal analogy.
Engage rigorously with predecessors. Publish a formal comparison of the Constitution Axiom with Searle's social ontology, showing precisely where the Prolegomena adds content Searle's framework cannot derive.
Develop cascade mathematics. Can network transmission models produce quantitative predictions about the threshold at which multi-dimensional observation restores coordination-dependent structures that scalar compression has de-constituted?
Test the encoding conjecture. Can the rate at which civilization converts Beierspace to eigenspace be measured historically?
Apply to complex evaluation systems. Does the Constitution Axiom predict that scalar evaluation signals de-constitute the evaluation dimensions they do not carry?
Validate b's empirical proxy. Test whether intangible asset intensity tracks the observation-withdrawal definition of b by comparing the structural response of high-intangible vs. low-intangible assets to exogenous observation shocks across multiple natural experiments.
Address identification. Prove conditions under which the decomposition Sigma = (1-b)*Sigma_E + b*F(O) is uniquely identified, or characterize the class of observationally equivalent decompositions. The identification strategy will likely require restrictions on Sigma_E (e.g., that the physical component is time-invariant or structurally constrained by known physical properties) and/or identification of F from exogenous observation shocks (using natural experiments like MiFID II to trace the structural response of Sigma to observation changes).
This paper proposes Axiom 0 of Coordination Science and organizes the five axioms of the Prolegomena under it: one axiom is contained in Axiom 0 by inclusion, two are derived from Axiom 0 plus conditions on agents, and two follow from finitude alone. A falsifiable empirical test distinguishes the Constitution Axiom from the standard epistemic framing: whether operational degradation from exogenous coverage withdrawal scales with coordination-dependence or with information asymmetry. The test is identified but not yet run. The author invites correspondence from information theorists, financial economists, philosophers of science, and researchers in any domain where finite agents coordinate through compressed signals. Coordination Science - the science of how observation-constituted reality behaves under compression - is the discipline this paper proposes. The work to build it continues.