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 Scaling field in the Disclosure Incentive requires agents to assess interaction effects at aggregate scale - what happens when an activity is replicated by N agents over T periods. The companion paper on scaling risk (Beier, 2026ah) establishes that this information type is structurally absent from the price system. The present paper develops a further property: scaling interactions are not arbitrary. They follow mathematical regularities - threshold effects, power laws, network topologies, and conservation constraints - that are determined by the physical, biological, and engineering properties of the constraint environment being assessed. An aquifer's depletion dynamics follow hydrological laws. A grid's capacity constraints follow electrical engineering principles. Atmospheric loading follows thermodynamic and chemical kinetics. Each constraint environment has a geometry - a mathematical structure governing how individual actions interact at scale - and the geometry constrains what well-calibrated Scaling assessments can look like. Assessments that violate the geometry (because they are hollow, fabricated, or analytically careless) are detectable through the inconsistency, providing a detection channel for the adversarial threats the companion paper (Beier, 2026an) identifies. West (2017) documented the prevalence of scaling laws across biological, urban, and economic systems. While the existence of power-law scaling is robust across complex systems, specific exponents are domain-dependent and empirically contested across taxonomic groups. The structural point - that constraint environments impose geometric regularities detectable through weight-space analysis - does not depend on any particular exponent being universal. Bak's (1996) self-organized criticality framework holds that self-organized criticality produces power-law distributions in systems driven to threshold states. The present paper connects these mathematical regularities to the economics of the Disclosure Incentive: the geometry of constraint environments is both a source of structure for well-calibrated Scaling assessments and a detection tool for poorly calibrated ones.
Keywords: scaling geometry, power laws, threshold effects, constraint environments, detection, adversarial disclosure, self-organized criticality, network topology, interaction effects
JEL Codes: C65, Q54, D82, D85
When the companion paper on scaling risk (Beier, 2026ah) introduced the Scaling field as the fifth component of the disclosure format, the emphasis was on why scaling information constitutes a distinct information type that the price system cannot carry. The present paper asks a different question: what mathematical structure does scaling information have, and what does that structure imply for the economics of the Disclosure Incentive?
The question matters for two reasons. The first is that mathematical regularity in constraint environments provides structure that well-calibrated Scaling assessments must respect. An analyst assessing aquifer depletion at aggregate scale is not free to predict any trajectory - the hydrological properties of the aquifer (hydraulic conductivity, storage coefficient, recharge rate, boundary conditions) constrain the space of physically possible depletion paths. A Scaling assessment that predicts a trajectory outside this physically constrained space is either wrong or addressing a different aquifer than the one specified. The geometry of the constraint environment disciplines the Scaling field by narrowing the space of credible assessments.
The second reason connects to adversarial robustness. The companion paper on adversarial disclosure (Beier, 2026an) identifies AI-generated hollow disclosures as a major threat - structurally correct assessments that mimic analytical reasoning without containing it. The Scaling field was identified as the hardest field to generate hollowly because it demands domain-specific knowledge about interaction effects. The present paper strengthens this defense: the mathematical geometry of constraint environments provides a consistency check that hollow assessments are likely to fail. A generative model producing aquifer Scaling assessments without understanding hydrological dynamics will produce trajectories that violate the aquifer's geometric constraints - predicting depletion patterns inconsistent with the hydraulic properties, or interaction effects that violate conservation laws. The geometry is a detection surface.
Section 2 characterizes the mathematical structures that constraint environments exhibit. Section 3 develops the disciplining function. Section 4 develops the detection function. Section 5 examines three domains. Section 6 states limitations. Section 7 concludes.
Constraint environments exhibit four categories of mathematical regularity that the scaling geometry framework organizes.
Many constraint environments exhibit threshold behavior: gradual individual-level changes accumulate until a critical point is reached, at which the system transitions abruptly to a qualitatively different state. Scheffer (2009) documented critical transitions across ecological, climatic, and social systems, showing that threshold dynamics are a pervasive feature of complex systems rather than an anomaly. Aquifers exhibit threshold behavior when draw-down reaches the level at which the saturated zone disconnects from surface water, after which recharge rates drop discontinuously. Atmospheric systems exhibit threshold behavior at tipping points where positive feedbacks (ice-albedo, permafrost methane, forest dieback) become self-reinforcing. Electrical grids exhibit threshold behavior when aggregate load exceeds transformer capacity, producing voltage collapse rather than proportional degradation.
The threshold structure constrains Scaling assessments. An analyst assessing aggregate aquifer dynamics must specify where the threshold lies (what draw-down level triggers the regime shift), how the system behaves on either side of the threshold (gradual depletion below, accelerated decline above), and what the interaction effects among users do to the trajectory relative to the threshold. A Scaling assessment that predicts smooth, continuous depletion in a system known to have threshold behavior is geometrically inconsistent - it violates the mathematical structure of the constraint environment.
In West's (2017) scaling framework, scaling relationships across biological, urban, and economic systems follow power laws with remarkable regularity. Metabolic rate scales as the 3/4 power of body mass across species spanning twenty orders of magnitude. Urban infrastructure scales sublinearly with city size (economies of scale), while innovation output scales superlinearly (increasing returns to density). Bak's (1996) analysis of sandpile models confirms that systems driven to critical states produce power-law distributions of event sizes - the sandpile model whose avalanche size distribution follows a power law regardless of the specific details of the sand grains.
Power-law scaling constrains interaction effects. If aquifer depletion scales as a power function of the number of users (because interaction effects among wells are governed by the physics of pressure diffusion through porous media), then a Scaling assessment of aggregate depletion must be consistent with this power-law relationship. An assessment that predicts linear scaling (doubling users doubles depletion) in a system governed by superlinear scaling (doubling users more than doubles depletion because of interaction effects) is geometrically inconsistent. The power law provides a quantitative constraint on the shape of the interaction function that the Scaling field must respect.
Many constraint environments have interaction structures that are determined by physical or institutional networks. Electrical grids have network topologies - the graph of connections among generators, transformers, and load points - that determine how disturbances at one node propagate to others. Water systems have network topologies determined by the hydrological connectivity of surface water and groundwater. Supply chains have network topologies that determine how disruptions cascade from one firm to others.
The network topology constrains Scaling assessments because the interaction effects the Scaling field captures propagate through the network. An analyst assessing how aggregate electrical load affects grid stability must account for the network's topology - which nodes are connected, what the capacity of each link is, where the bottlenecks are. A Scaling assessment that treats the grid as a single aggregate (total load versus total capacity) ignores the topological structure and may dramatically understate or overstate the interaction effects that operate through the network's specific configuration.
Physical constraint environments obey conservation laws that impose hard boundaries on what scaling trajectories are possible. The atmosphere's carbon budget is conserved - every ton emitted either remains in the atmosphere, is absorbed by oceans, or is taken up by terrestrial sinks, and the conservation constraint means that cumulative emissions determine atmospheric concentration regardless of the temporal path. Aquifer water budgets are conserved - extraction plus natural discharge equals recharge plus storage change, and the conservation constraint means that sustained extraction above recharge depletes storage regardless of how the extraction is distributed among users.
Conservation constraints provide the tightest geometric discipline on Scaling assessments. An analyst whose Scaling assessment implies a trajectory that violates the mass balance or energy balance of the constraint environment has made an error that the geometry identifies unambiguously. Unlike threshold identification (which involves uncertainty about parameter values) or power-law estimation (which involves statistical uncertainty about the exponent), conservation violations are categorical - they indicate that the assessment is internally inconsistent with the physics of the constraint environment.
The geometry of constraint environments disciplines the Scaling field by narrowing the space of credible assessments. An analyst producing a well-calibrated Scaling assessment must demonstrate consistency with the mathematical structure of the constraint environment being assessed - the thresholds must be acknowledged and their locations estimated, the scaling relationships must follow the relevant power laws, the network topology must be accounted for, and the conservation constraints must be satisfied.
This disciplining function serves two economic purposes. The first is quality assurance. A Scaling assessment that is consistent with the constraint environment's geometry is more likely to be well-calibrated than one that is not, because geometric consistency requires the analyst to have engaged with the physical, biological, or engineering properties of the system rather than producing an assessment from general principles or pattern-matching alone. The geometry acts as a filter that separates assessments grounded in domain knowledge from assessments that lack it.
The second purpose is comparability. When two analysts produce Scaling assessments of the same constraint environment, the geometry provides a shared framework for evaluating where they agree and where they disagree. If both analysts agree on the threshold location but disagree on the scaling exponent, the disagreement is localized to a specific parameter rather than diffused across the entire assessment. The geometry makes the disagreement structure legible, connecting to the companion paper on aggregation economics (Beier, 2026c), which shows that preserved disagreement carries information that the price system's scalar aggregation destroys.
The geometry of constraint environments provides a detection channel for adversarial threats to the judgment data stream. The companion paper (Beier, 2026an) identifies four attack categories: Sybil attacks, coordinated poisoning, AI-generated hollow disclosures, and adversarial selection. The geometry is most relevant for detecting the third: hollow disclosures that are structurally correct but analytically empty.
A generative AI model producing aquifer Scaling assessments without understanding hydrological dynamics will produce assessments whose form mimics genuine analysis (they will mention depletion rates, interaction effects, and confidence levels) but whose substance is not constrained by the aquifer's geometry. The assessment may predict smooth depletion in a system with threshold behavior. It may imply scaling relationships inconsistent with the physics of pressure diffusion. It may violate the water budget's conservation constraint by implying that aggregate extraction exceeds recharge plus storage change without the storage depletion that conservation requires. Each inconsistency is detectable by comparing the Scaling assessment's implied trajectory against the geometric constraints that domain expertise identifies.
The detection operates at the intersection of the computer science and physics entrances that the companion paper on adversarial disclosure describes. The computational methods (statistical fingerprinting, complexity analysis) detect AI-generated text. The geometric methods detect assessments whose physical implications are inconsistent with the constraint environment's mathematical structure. An attack sophisticated enough to pass linguistic fingerprinting - generating text that reads as genuine domain analysis - may still fail the geometric consistency check, because producing trajectories consistent with hydrological dynamics, electrical network physics, or atmospheric chemistry requires understanding the geometry rather than merely mimicking the vocabulary.
The detection is not infallible. A sophisticated attacker who incorporates the constraint environment's geometry into the generative model can produce assessments that pass both linguistic and geometric checks. The companion paper's adversarial analysis identifies this as the most demanding attack and argues that the cross-disciplinary detection architecture makes it prohibitively expensive. The present paper's contribution is to identify the geometric consistency check as one specific detection surface whose strength depends on the depth of the constraint environment's mathematical structure - deeper geometry (more conservation laws, more precisely characterized thresholds, better-estimated scaling exponents) provides a harder surface to mimic.
Aquifer geometry is governed by Darcy's law (flow is proportional to hydraulic gradient), the continuity equation (mass conservation), and the aquifer's specific yield and hydraulic conductivity parameters. Interaction effects among wells are mediated by pressure diffusion - each well creates a cone of depression whose extent depends on the pumping rate, the aquifer properties, and the distance to neighboring wells. The scaling geometry specifies that aggregate depletion from N wells is generally superlinear in N (because overlapping cones of depression reduce each well's yield), that threshold behavior occurs when the water table drops below confining layers or disconnects from surface water, and that the water budget is conserved. A well-calibrated Scaling assessment must be consistent with all of these geometric constraints.
Grid geometry is governed by Kirchhoff's laws (conservation of current at nodes, conservation of voltage around loops), the capacity ratings of transformers and transmission lines, and the network topology connecting generators to loads. Interaction effects operate through the network - a load increase at one node affects voltage and current flow at every connected node, and the effects depend on the network's specific configuration. Threshold behavior occurs when a critical line or transformer reaches its capacity rating, at which point protective systems either shed load or allow cascading failure. A well-calibrated Scaling assessment of aggregate electrification impacts must account for the network topology, not merely the aggregate load-to-capacity ratio.
Atmospheric geometry is governed by the carbon cycle - the set of reservoirs (atmosphere, oceans, terrestrial biosphere, geological) and fluxes (emissions, absorption, weathering) that determine atmospheric concentration. Conservation is exact: every ton of carbon emitted either accumulates in the atmosphere or is absorbed by sinks, and the partitioning between accumulation and absorption follows well-characterized (though uncertain at the margins) geochemical dynamics. Threshold behavior is present in the carbon cycle through climate feedback mechanisms (ice-albedo, permafrost methane, ocean circulation changes) that produce nonlinear responses to cumulative loading. Weitzman's (2009) fat-tail argument is, in the geometric framework, an observation about the shape of the constraint environment's response function near potential thresholds.
The paper argues that constraint environments have geometric properties but does not develop a unified mathematical framework that encompasses all types of geometry (thresholds, power laws, network topologies, conservation constraints) within a single formalism. Each type of geometric regularity has its own mathematical treatment in the relevant domain science (hydrology, electrical engineering, atmospheric chemistry), and unifying these treatments is a mathematical project that the paper identifies as a direction for future work rather than accomplishing.
The detection function depends on the availability of domain expertise to specify the geometric constraints for each constraint environment. A detection algorithm that checks Scaling assessments against geometric constraints must be calibrated for each domain - the threshold locations, scaling exponents, network topology, and conservation laws differ across aquifers, grids, and atmospheric systems. The detection is not a general-purpose algorithm but a domain-specific one that requires the same expertise it is designed to protect, which creates a chicken-and-egg problem: the experts needed to calibrate the detection are the same experts whose disclosures the detection is designed to validate.
The paper treats geometric constraints as sharp boundaries between consistent and inconsistent assessments. In practice, many constraints are known only approximately - the threshold location has a confidence interval, the scaling exponent has estimation uncertainty, the conservation budget has measurement error. An assessment that falls within the geometric constraints' confidence intervals is consistent, but the confidence intervals may be wide enough that even substantially wrong assessments appear consistent. The geometric detection is strongest where the constraints are precisely characterized and weakest where they are uncertain.
Constraint environments are not arbitrary. They have mathematical structure - thresholds, power laws, network topologies, conservation constraints - that governs how individual actions interact at scale. The geometry disciplines the Scaling field by narrowing the space of credible assessments to those consistent with the constraint environment's physical, biological, or engineering properties. The same geometry provides a detection surface for hollow or fabricated disclosures, because assessments that mimic the vocabulary of constraint analysis without engaging its mathematics produce trajectories that violate the geometric constraints the domain imposes.
The Scaling field is the hardest field to fill candidly - it requires genuine domain expertise about interaction effects - and the hardest to fake convincingly, because the geometry of the constraint environment tests the assessment's substance rather than its form. Every constraint environment has a geometry. Every well-calibrated Scaling assessment must be consistent with it. Every hollow assessment risks violating it. The geometry is the Disclosure Incentive's deepest line of defense against the corruption of the judgment data stream, because it is grounded in the physics of the constraint environment rather than in the statistics of the disclosure record - and physics does not negotiate.
This paper presents results from the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics (Beier, forthcoming 2026). Scaling geometry connects the Scaling field of the Disclosure Incentive (Beier, 2026b) to the mathematical structure of the constraint environments the field is designed to assess, and provides a detection surface for the adversarial analysis developed in the companion paper (Beier, 2026an). The author invites engagement from the complex systems, applied mathematics, and domain science communities.