PRIORITY FILING - INITIAL DRAFT
A fully revised and formatted version will follow.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 19, 2026
From the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics by Gregory Caldwell Beier (forthcoming 2026).
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
The companion paper on the Disclosure Incentive (Beier, 2026b) introduces a five-field structure for disclosed judgments: Input, Method, Output, Identity, and Scaling. The first four fields formalize the standard components of any analytical assessment. The fifth - Scaling - is the non-obvious addition. This paper argues that Scaling captures a distinct information type that the price system structurally cannot carry: what happens when an activity is replicated by N agents simultaneously over T periods, and how the interaction effects at scale differ from the individual-transaction effects that the price reflects. Prices aggregate individual transactions, and each transaction is priced at the scale of one. But constraint environments - aquifers, atmospheric carbon budgets, electrical grids, ecosystems - respond to the aggregate of all transactions interacting, not to any single transaction in isolation. The aquifer does not deplete from one well; it depletes from the simultaneous draw of every well in the watershed, at rates that depend on the interaction structure among them. The atmosphere does not warm from one flight; it warms from the cumulative interaction of all flights over decades, with nonlinear feedbacks that no individual ticket price was designed to represent. Scaling information is structurally absent from the price signal because the price is formed at the individual-transaction level, and no amount of price refinement changes this - a more accurate price for a single airline ticket is still a scalar representing one transaction, not a representation of what happens when millions of such transactions interact with atmospheric constraints simultaneously. The Scaling field forces the discloser to articulate this interaction structure explicitly, producing information that the price system was never formatted to carry and that no other field in the disclosure captures.
Keywords: scaling, interaction effects, constraint environments, price compression, nonlinearity, tipping points, information architecture, externalities, disclosure
JEL Codes: D82, Q54, H23, D62, C65
The price of a gallon of water drawn from a shared aquifer reflects the cost of extraction for a single well operator at a single moment. It does not reflect what happens when five operators draw simultaneously at current rates for the next ten years, how their extraction interacts with recharge rates that depend on precipitation patterns already shifting under climate change, or at what threshold the aquifer's draw-down becomes irreversible. The price carries the individual-transaction signal. The scaling information - the interaction structure across agents and across time - is absent.
This absence is not a market failure in the conventional Pigouvian sense. Pigou (1920) identified costs borne by third parties that the price does not reflect - the atmospheric carbon from a flight, the noise impact on nearby residents. The companion paper (Beier, 2026c) reframes these as compression artifacts: information about constraint interactions that the price system destroys in the aggregation from judgment to scalar. The present paper argues that a specific subset of compressed-away information deserves treatment as a distinct type: scaling information, which concerns how individual activities interact when replicated across agents and extended across time.
The distinction matters because scaling information has properties that the other four fields of the Disclosure Incentive (Input, Method, Output, Identity) do not capture. An analyst can disclose the input data consulted, the method used, the conclusion reached, and their identity - and still fail to address what happens when the activity their conclusion supports is undertaken by thousands of agents simultaneously. The scaling question is not a refinement of the output; it is a different kind of information, one that concerns the relationship between the individual transaction and the constraint environment at aggregate scale. Arthur's (1989) increasing-returns framework implies that increasing returns generate path dependence at the system level that is invisible at the transaction level. Scheffer (2009) documented critical transitions in natural systems where gradual individual-level changes produce sudden system-level shifts. The fat-tailed damage distributions in Weitzman's (2009) dismal theorem imply that fat-tailed uncertainty in climate damages changes the economics of policy structurally - the tail risk is a scaling phenomenon that individual-transaction pricing cannot represent. West's (2017) scaling laws hold that scaling laws operate across biological, urban, and economic systems with mathematical regularity, suggesting that scale-dependent behavior is a structural feature of complex systems rather than an anomaly.
Section 2 develops the formal distinction between individual-transaction information and scaling information. Section 3 shows why the price system cannot carry scaling information. Section 4 examines how the Scaling field in the Disclosure Incentive addresses the gap. Section 5 applies the framework to three constraint domains. Section 6 presents implications. Section 7 states limitations. Section 8 concludes.
Every economic transaction carries information at a specific scale: the scale of one buyer, one seller, one unit of the good, one moment in time. The price encodes this information as a scalar. The buyer's willingness to pay reflects their assessment of value at the individual scale. The seller's willingness to accept reflects their cost at the individual scale. The equilibrium price reflects the intersection of individual-scale assessments aggregated across participants.
Scaling information concerns a different question: what happens when this transaction is replicated by N agents over T periods, and how the aggregate outcome differs from N times the individual outcome. The distinction can be stated precisely. Let x_i denote the individual-transaction effect for agent i, and let X_N = f(x_1, x_2, ..., x_N) denote the aggregate effect when N agents transact. If f is additive - if X_N = sum of x_i - then the aggregate is fully determined by the individual-transaction information, and scaling adds nothing. The price system handles additive aggregation adequately.
But constraint environments are generally not additive. When five operators draw from the same aquifer, the aggregate depletion is not the sum of five individual depletions. Each operator's draw affects the water table, which affects the extraction cost and yield for every other operator. The aggregate effect depends on the spatial distribution of wells, the geological structure of the aquifer, the recharge dynamics, and the interaction between extraction rates and hydrological thresholds. Formally, X_N = f(x_1, ..., x_N) where f is nonlinear, and the nonlinearity means that knowing each x_i individually does not determine X_N. The interaction structure - the way individual actions combine at scale - is a separate piece of information that no amount of individual-transaction data can reconstruct.
Taleb (2007) identified a related phenomenon: the failure of Gaussian models to capture fat-tailed risks arises because the tails reflect interaction effects and feedback loops that individual-observation statistics cannot detect. In the present framework, fat tails are a symptom of scaling nonlinearity. When individual transactions interact nonlinearly at aggregate scale, the distribution of aggregate outcomes has heavier tails than the distribution of individual outcomes would suggest. The price, which encodes the individual-transaction distribution, systematically understates the probability and magnitude of extreme aggregate outcomes.
Scaling information is therefore a distinct information type characterized by three properties. It concerns interactions across agents rather than individual effects. It concerns trajectories across time rather than point-in-time assessments. And it is nonlinear - the aggregate is not a simple function of the parts. No combination of the other four disclosure fields (Input, Method, Output, Identity) captures this information, because those fields formalize the structure of an individual analytical judgment rather than the relationship between individual activities and aggregate constraint dynamics.
The price system's inability to carry scaling information is structural rather than incidental. Three features of price formation ensure this.
The first is that prices are formed at the transaction level. A market price clears a market by equating supply and demand for a specific good at a specific time. The clearing mechanism aggregates individual willingness-to-pay and willingness-to-accept into a scalar. Nothing in this mechanism generates or transmits information about how the aggregate of transactions interacts with the constraint environment. The individual buyer of an airline ticket has no channel through which to express an assessment of what happens when a million such tickets are purchased simultaneously - and even if the buyer possessed such an assessment, the price format has no field in which to carry it.
The second is that prices are scalars. A scalar can represent magnitude - how much someone is willing to pay - but cannot represent structure. The interaction effects that define scaling information are inherently structural: they depend on which agents are transacting, in what spatial and temporal configuration, with what coupling to shared constraints. Encoding this structure requires a representation with dimensionality far exceeding one. The five-field disclosure format provides this dimensionality; the price does not.
The third is that prices are contemporaneous. A market price reflects current supply and demand conditions. Scaling information is inherently intertemporal - it concerns what happens when current activities are sustained and replicated over future periods. A price can reflect expectations about the future (as asset prices do), but only as a scalar summary of those expectations. The trajectory structure - the path-dependent interaction between current activities and future constraint states - is compressed away in the scalar summary.
These three features - transaction-level formation, scalar format, contemporaneous clearing - are not design flaws that could be corrected by improving the market mechanism. They are constitutive features of what a price is. A price that encoded interaction effects across agents, structural coupling to constraints, and intertemporal trajectory information would no longer be a price in any recognizable sense - it would be a structured judgment, which is what the Disclosure Incentive produces.
Hayek (1945) celebrated the price system's ability to transmit dispersed knowledge efficiently. The present paper does not challenge Hayek's point - prices do transmit individual-transaction knowledge with remarkable efficiency. The claim is narrower: there exists a class of information (scaling information) that the price system's format structurally cannot represent, and this class is the one that matters most for long-horizon coordination on shared constraints.
The Scaling field in the Disclosure Incentive (Beier, 2026b, Section 3.1) requires the discloser to articulate their assessment of what happens when the activity under analysis is replicated at scale. The companion paper provides the justification for the field's necessity within the five-field structure; the present paper develops what the Scaling field contains and why it captures information the other four fields do not.
A fund manager assessing a water-intensive agricultural operation discloses Input (the data consulted - rainfall records, aquifer surveys, crop water requirements), Method (the analytical framework - hydrological modeling, yield projection), Output (the conclusion - the operation is economically viable at current water prices), and Identity (the analyst making the assessment). These four fields capture the individual-transaction judgment fully. They do not capture what happens when five similar operations are established in the same watershed, drawing from the same aquifer, under precipitation patterns that are shifting - which is the Scaling assessment.
The Scaling field forces the analyst to state: "At the scale of five simultaneous operators drawing at projected rates, the aquifer's recharge-depletion balance crosses a threshold within seven years, after which extraction costs for all operators increase nonlinearly. My confidence in this threshold estimate is 60-75%, with the uncertainty driven primarily by precipitation projections." This assessment is scored against outcomes just as the Output field is - but the outcomes it is scored against are aggregate outcomes (did the aquifer cross the threshold?) rather than individual outcomes (was the single operation profitable?).
The distinction between Output and Scaling can be stated formally. Output addresses: P(omega | x_i), the probability of outcome omega given this individual transaction. Scaling addresses: P(Omega | x_1, ..., x_N, T), the probability of aggregate outcome Omega given N agents transacting over T periods. These are different probability assessments, scored against different outcome classes, requiring different analytical competencies. An analyst who is well-calibrated on Output (individual transaction profitability) may be poorly calibrated on Scaling (aggregate constraint dynamics), and the track record distinguishes the two.
The Scaling field is where the real intellectual content of constraint reasoning lives. Assessing individual-transaction profitability is a well-developed skill supported by abundant data and rapid feedback. Assessing interaction effects at scale requires understanding the coupling structure of the constraint environment - how wells interact through the aquifer, how flights interact through the atmosphere, how loads interact through the grid - and this understanding draws on domain expertise (hydrology, atmospheric science, electrical engineering) that individual-transaction analysis does not require. The Scaling field is the mechanism through which domain expertise about constraint environments enters the economic coordination system. Without it, the Disclosure Incentive enriches the price system's information content but leaves the most consequential gap - the scaling gap - unaddressed.
The price of a flight from Boston to London does not represent the interaction between this flight's emissions, every other concurrent flight's emissions, the cumulative atmospheric loading from decades of aviation, the nonlinear relationship between cumulative loading and radiative forcing, or the threshold effects (ice sheet dynamics, permafrost methane release, ocean circulation changes) that operate at aggregate temporal scales. Each of these is scaling information - interaction effects across agents (all flights) and across time (cumulative loading) with nonlinear aggregate dynamics. An environmental economist may respond that a properly calibrated Pigouvian carbon tax would internalize these nonlinear damages. But computing that tax requires precisely the constraint information the current price system does not carry. The social cost of carbon is the output of constraint reasoning about threshold dynamics, cumulative loading, and cross-domain coupling - it cannot substitute for the reasoning itself. The tax presupposes the information architecture this paper argues is missing.
The Scaling field for an airline's constraint disclosure would contain the analyst's assessment of how the airline's growth trajectory interacts with atmospheric carbon budgets over a specified horizon, at what confidence level, and with what sensitivity to assumptions about fleet efficiency improvements, demand elasticity, and the behavior of competing carriers. The assessment is scored against observable aggregate indicators (atmospheric CO2 concentration, global temperature anomalies, sector-level emissions trajectories) over multi-year horizons. Weitzman's (2009) argument that fat-tailed climate uncertainty changes the economics structurally is, in this framework, an argument about the importance of scaling information: the fat tails arise from interaction effects and nonlinear feedbacks that individual-transaction pricing structurally cannot represent.
Groundwater aquifers respond to aggregate extraction, not individual extraction. The Ogallala Aquifer underlying the US Great Plains has been depleted at rates that no individual farmer's irrigation decision would suggest is problematic, because each farmer's extraction is rational at the individual scale while the aggregate extraction exceeds recharge. The scaling information - how N farmers drawing at rate r_i from positions p_i interact through the aquifer's hydraulic conductivity structure over T growing seasons - is absent from the price of water and from each farmer's individual cost calculation.
The Scaling field for an agricultural analyst's disclosure would contain an assessment of how the current extraction pattern interacts with recharge rates under projected climate conditions, at what threshold the aquifer transitions from sustainable to declining, and what the trajectory looks like for the watershed as a whole rather than for any individual operation. Scheffer's (2009) work on critical transitions in natural systems is directly relevant: aquifer depletion exhibits threshold behavior where gradual extraction produces sudden and potentially irreversible decline in water table levels, and this threshold behavior is a scaling phenomenon invisible at the individual-transaction level.
Electrical grids operate under capacity constraints that bind at the aggregate level. An individual consumer adding a heat pump or electric vehicle charger imposes a load that the grid handles easily in isolation. When thousands of consumers in the same distribution network adopt simultaneously, the aggregate load interacts with transformer capacity, voltage regulation limits, and peak-demand profiles in ways that the individual load addition does not predict. The scaling information concerns the interaction structure: which loads coincide temporally, how they couple through the distribution network, and at what aggregate penetration the network's capacity constraints begin to bind.
The Scaling field for an energy analyst's disclosure would contain an assessment of how adoption trajectories for electrified technologies interact with grid capacity in a specific service territory, at what penetration levels infrastructure upgrades become necessary, and what the cost and timeline of those upgrades look like under different adoption scenarios. The assessment draws on electrical engineering knowledge (network topology, transformer ratings, demand profiles) that individual-transaction pricing does not require and cannot carry.
The existence of scaling information as a distinct type has three implications for the Disclosure Incentive and the broader compression framework.
The first is that the Scaling field is the field most likely to contain information that changes capital allocation decisions. Individual-transaction information (Input, Method, Output) refines the precision of assessments that the price system already carries, at least approximately. Scaling information introduces an entirely new dimension - the interaction structure at aggregate scale - that the price system has never carried and that no amount of price refinement will produce. The marginal information value of the Scaling field is therefore higher than the marginal information value of the other fields for decisions that involve shared constraints.
The second implication concerns the improvement mechanism (Beier, 2026a, Section 3.4). The act of articulating a Scaling assessment forces the analyst to reason about interaction effects they may never have considered. A fund manager who is required to state how a portfolio company's water usage interacts with other users of the same aquifer at scale is compelled to investigate a constraint dimension that individual-transaction analysis does not touch. The Scaling field does not merely capture pre-existing judgment about interaction effects; in many cases, it creates that judgment by requiring its articulation. This performative property - the disclosure requirement generating reasoning that would not otherwise occur - is strongest for the Scaling field because scaling information is the type least likely to be produced by individual-transaction analysis alone.
The third implication is for the measurement of externalities. The companion paper (Beier, 2026c) argues that externalities are compression artifacts. The present paper refines this: the most consequential externalities - atmospheric carbon, aquifer depletion, ecosystem degradation, infrastructure strain - are specifically scaling compression artifacts. They arise not from any single transaction's unpriced effects but from the interaction effects across transactions at aggregate scale that the price system's format cannot represent. Addressing these externalities requires not just making constraint information visible (which the first four disclosure fields accomplish) but making interaction-at-scale information visible (which only the Scaling field accomplishes).
The paper argues that scaling information is a distinct type but does not formally prove distinctness in the information-theoretic sense. A formal proof would require showing that scaling information is not recoverable from any combination of individual-transaction signals - that knowing every x_i to arbitrary precision does not determine X_N when f is nonlinear. This is true by construction for nonlinear f, but the empirically relevant question is how much of the scaling information is recoverable in practice from high-quality individual-transaction data. The paper's claim is that a substantial and consequential portion is not, but the magnitude of the irrecoverable portion is an empirical question the paper does not resolve.
The three constraint domain examples (Section 5) are illustrative rather than exhaustive. Other domains where scaling information is consequential - antibiotic resistance (individual prescriptions interacting through evolutionary selection at population scale), financial contagion (individual balance sheets interacting through counterparty networks), and urban development (individual construction decisions interacting through transportation and infrastructure networks) - are not developed. The framework is applicable to these domains but the application is deferred.
The scoring of Scaling assessments against outcomes is inherently long-horizon. An analyst's assessment of aquifer depletion over a seven-year horizon cannot be scored for seven years. The companion paper (Beier, 2026b) acknowledges this limitation for the Disclosure Incentive generally; it binds most tightly for the Scaling field specifically, because scaling dynamics operate on the longest timescales. Interim scoring methods - comparing Scaling assessments to updated monitoring data, scoring relative to reference-class trajectories, evaluating internal consistency of the interaction model - can partially address the delay, but the fundamental feedback loop is slow.
The paper does not model the equilibrium provision of scaling information. How many analysts will invest in the domain expertise (hydrology, atmospheric science, electrical engineering) required to produce well-calibrated Scaling assessments? The judgment market micro-foundation in the companion paper (Beier, 2026r) establishes that the demand for calibration quality is positive, but the supply response for a new and specialized type of information - scaling expertise applied to constraint environments - is an open question.
The price system formats information at the scale of one transaction. Constraint environments respond at the scale of all transactions interacting. The gap between these two scales is the Scaling field - the fifth field in the Disclosure Incentive's structure and the one that captures information the price was never designed to carry.
Scaling information has three distinguishing properties: it concerns interactions across agents rather than individual effects, it concerns trajectories across time rather than point-in-time assessments, and it is nonlinear in a way that makes the aggregate irreducible to the sum of individual-transaction data. These properties make it the information type most consequential for long-horizon coordination on shared constraints, and the type most completely absent from the current information architecture of economic coordination.
The Scaling field addresses the gap by requiring disclosers to articulate their assessment of interaction effects at aggregate scale, scored against aggregate outcomes. In doing so, it accomplishes something the other four fields do not: it introduces into the economic coordination system the domain expertise about constraint environments - hydrology, atmospheric science, ecosystem dynamics, infrastructure engineering - that individual-transaction pricing has never had a channel to carry. The most consequential externalities are scaling externalities, and the Disclosure Incentive's ability to address them depends on the fifth field doing what the first four cannot.
This paper presents results from the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics (Beier, forthcoming 2026). The Scaling field developed here is the fifth component of the Disclosure Incentive's five-field structure (Beier, 2026b) and the information type most consequential for the externality analysis in the companion paper on Smith and Pigou (Beier, 2026p). The author invites engagement from the environmental economics, complex systems, and infrastructure planning communities.