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 performative economics (Beier, 2026ao) argues that the Disclosure Incentive's five-field format generates reasoning that would not otherwise exist - the format is performative in Austin's (1962) sense. This paper provides formal content for the claim. Proposition PF-1 proves that the structured five-field format elicits strictly more information about the constraint state than an unstructured format, for any agent with calibration quality above q* and domain expertise above a minimum threshold. The mechanism is that each field in the structured format constrains the agent's response to a specific dimension of the constraint state, preventing the agent from substituting vague general commentary for precise dimensional assessment. Proposition PF-2 proves the stronger result for the Scaling field specifically: the Scaling field generates interaction-effect reasoning that no other field and no unstructured format elicits, because interaction effects are not salient to individual-transaction analysis and require an explicit prompt to activate. The two propositions formalize the distinction between a disclosure format as a passive container (which merely records reasoning the agent would have produced anyway) and as an active forcing function (which directs cognitive effort toward dimensions that would otherwise receive none).
Keywords: performativity, structured disclosure, elicitation, forcing function, Scaling field, interaction effects, information generation
JEL Codes: D82, D83, D91
Proposition PF-1 (Structured format generates more information). Let theta = (theta_1, ..., theta_K) be the K-dimensional constraint state. Let D_structured be the disclosure an agent produces under the five-field format (with fields targeting specific dimensions of theta), and D_unstructured be the disclosure the same agent produces when asked to "disclose your assessment of the constraint environment" without format structure. Let q denote the agent's calibration quality and e denote the agent's domain expertise (the fraction of dimensions of theta about which the agent has informed beliefs).
For any agent with q > q* (the truth-telling threshold from Beier, 2026r, Proposition 7) and e > e_min (a minimum expertise threshold defined below):
I(theta; D_structured) > I(theta; D_unstructured)
That is, the structured disclosure carries strictly more mutual information about the constraint state than the unstructured disclosure.
Proof. The proof proceeds by identifying the mechanism through which structure increases information content: dimensional coverage.
An agent with domain expertise e has informed beliefs about a fraction e of the K dimensions of theta. Under an unstructured format, the agent chooses which dimensions to address. Behavioral research on unstructured elicitation (Fischhoff, Slovic, and Lichtenstein, 1978; Kahneman, 2011) establishes three robust findings. First, agents under-report dimensions they are uncertain about - the certainty effect leads agents to emphasize their strongest assessments and omit their weakest. Second, agents omit dimensions that are not salient to their immediate decision context - a fund manager assessing an agricultural company focuses on profitability and omits aquifer dynamics unless prompted. Third, agents substitute narrative coherence for dimensional coverage - unstructured reports tell a "story" about the most salient aspects rather than systematically covering all dimensions.
Under the structured format, each field directs the agent to a specific set of dimensions. The Input field directs attention to data sources (which dimensions of theta have been observed). The Method field directs attention to the analytical framework (which model maps observations to assessments). The Output field directs attention to the conclusion (the posterior on the assessed dimensions, with confidence). The Identity field creates accountability for the full assessment. The Scaling field directs attention to interaction effects at aggregate scale (a specific set of dimensions that individual-transaction analysis omits entirely).
Let K_unstructured denote the number of dimensions addressed in the unstructured disclosure and K_structured denote the number addressed in the structured disclosure. By the three behavioral findings:
K_unstructured < e * K (the agent covers fewer dimensions than their expertise supports, due to certainty bias, salience bias, and narrative substitution)
K_structured >= e * K (the structured format prompts the agent to cover all dimensions they have expertise on, because each field targets a specific set and the coverage scoring from Beier, 2026r, Section 4.2, penalizes omission)
The mutual information decomposes across dimensions:
I(theta; D) = sum_{k=1}^{K} I(theta_k; D | theta_{<k})
where each term is the information D carries about dimension k given the information about prior dimensions. For dimensions the agent addresses (k in the covered set), the mutual information is positive (the agent has an informed belief and reports it). For dimensions the agent omits, the mutual information is zero (no information is conveyed).
Since K_structured > K_unstructured:
I(theta; D_structured) = sum over more dimensions of positive terms >= I(theta; D_unstructured) = sum over fewer dimensions of positive terms
The inequality is strict whenever K_structured > K_unstructured, which holds for any agent with e > e_min = K_unstructured / K (the expertise threshold is the fraction of dimensions the agent would cover without prompting - any agent with expertise exceeding this has latent coverage that the structured format activates). QED.
Remark. The proof identifies the mechanism as dimensional coverage rather than per-dimension quality. The structured format does not make the agent more knowledgeable about any single dimension - it activates latent knowledge about dimensions the agent would have omitted without the prompt. This is the performative property: the format generates information (coverage of omitted dimensions) rather than merely recording information (the agent's pre-existing assessment of the dimensions they would have addressed anyway).
Proposition PF-2 (The Scaling field generates interaction-effect information). Let theta_S denote the interaction-effect component of the constraint state - the dimensions describing how the constraint environment responds to aggregate activity by N agents over T periods. Let D_no_scaling be a disclosure produced under a four-field format (Input, Method, Output, Identity - all fields except Scaling) and D_with_scaling be a disclosure under the full five-field format (including Scaling). Then for any agent with domain expertise in the constraint domain:
I(theta_S; D_with_scaling) > I(theta_S; D_no_scaling) = 0 + epsilon
where epsilon >= 0 is the small amount of interaction-effect information that incidentally enters the other four fields. The Scaling field carries the vast majority of the interaction-effect information.
More precisely:
I(theta_S; D_with_scaling) = I(theta_S; Scaling field) + I(theta_S; other four fields | Scaling field)
and
I(theta_S; Scaling field) >> I(theta_S; other four fields)
Proof. The proof has two parts: establishing that the four-field format carries negligible interaction-effect information, and establishing that the Scaling field carries substantial interaction-effect information.
The Input field specifies data sources. Data sources for individual-transaction analysis (company financials, market data, sector reports) do not contain interaction-effect data by default. An analyst assessing one company's water usage consults that company's data, not the aggregate extraction data across the watershed. The Input field under a four-field format therefore contains data about individual-agent behavior, not about aggregate interaction effects. I(theta_S; Input) is approximately zero because the data sources don't address theta_S.
The Method field specifies the analytical framework. Standard analytical frameworks for individual-transaction assessment (discounted cash flow, comparable company analysis, risk factor assessment) do not model interaction effects at aggregate scale. They model the individual agent's performance under assumed market conditions. I(theta_S; Method) is approximately zero because the methods don't model theta_S.
The Output field states the conclusion. The conclusion of an individual-transaction assessment is a statement about the individual agent's value or risk profile. It does not state what happens when N agents replicate the activity, because the individual-transaction framework does not address that question. I(theta_S; Output) is approximately zero for the same reason.
The Identity field identifies the analyst. It carries no direct information about theta_S.
The total interaction-effect information in the four-field disclosure is:
I(theta_S; D_no_scaling) = I(theta_S; Input) + I(theta_S; Method | Input) + I(theta_S; Output | Input, Method) + I(theta_S; Identity | rest) approximately 0 + epsilon
where epsilon captures any incidental interaction-effect information that enters the individual-transaction analysis (an analyst who happens to mention "our well's yield has been declining as neighboring farms expand irrigation" provides interaction-effect information incidentally, but this is not elicited by the format).
The Scaling field asks explicitly: "What happens when this activity is replicated by N agents over T periods? How do the interaction effects at aggregate scale differ from the individual effects?"
For an agent with domain expertise in the constraint domain (a hydrologist who understands how wells interact through pressure diffusion, an atmospheric scientist who understands how emissions aggregate nonlinearly, a grid engineer who understands how loads interact through network topology), this prompt activates analytical reasoning about theta_S that the four-field format does not activate. The agent's domain expertise includes knowledge about interaction effects - the hydrologist understands cones of depression overlap, the atmospheric scientist understands feedback mechanisms, the grid engineer understands voltage stability - but this knowledge is not activated by individual-transaction analysis because individual-transaction analysis does not ask about aggregate interactions.
The Scaling field's prompt is the forcing function. It directs the agent's cognitive effort toward theta_S specifically. The resulting disclosure carries:
I(theta_S; Scaling field) = H(theta_S) - H(theta_S | Scaling field response)
which is positive and substantial whenever the agent has domain expertise about interaction effects (H(theta_S | Scaling field response) < H(theta_S) because the response reduces uncertainty about interaction effects).
The inequality I(theta_S; Scaling field) >> I(theta_S; other four fields) follows from Parts 1 and 2: the four-field format produces approximately zero interaction-effect information while the Scaling field produces substantially positive interaction-effect information. QED.
Remark. The proposition formalizes the distinction between a container and a forcing function. The four-field format is a container for individual-transaction analysis that the agent would have performed anyway. The Scaling field is a forcing function that directs cognitive effort toward interaction effects that the agent's ordinary workflow does not address. The information the Scaling field elicits is R_generated in the companion paper's (Beier, 2026ao) terminology - reasoning that was not previously occurring, brought into existence by the format's demand.
The proposition connects to the companion paper on scaling risk (Beier, 2026ah), which establishes that interaction effects at aggregate scale constitute a distinct information type the price system cannot carry. The present proposition establishes that the Scaling field is the specific mechanism through which this distinct information type is produced - generated through the structured prompt that activates latent domain expertise about interaction effects.
Proposition PF-1's mechanism (dimensional coverage) assumes that the agent has latent expertise that the structured format activates. For an agent without domain expertise (e = 0), the structured format cannot generate information the agent does not possess. The proposition applies to agents with e > e_min, which is a subset of all agents. The companion paper's calibration threshold q* (Beier, 2026r, Proposition 7) independently ensures that agents who participate in the disclosure ecosystem have sufficient quality; the present proposition adds the expertise threshold e_min as a second condition for the performative property.
Proposition PF-2's Part 1 relies on behavioral evidence about how analysts conduct individual-transaction analysis. If an analyst's standard workflow already includes interaction-effect assessment (for instance, a systemic risk analyst at a central bank), then the four-field format may carry substantial interaction-effect information without the Scaling field. The proposition holds for the typical analyst whose workflow is transaction-focused; it does not hold for the exceptional analyst whose workflow already addresses aggregate interactions. The Scaling field's value as a forcing function is highest for the majority of analysts who do not already analyze interaction effects and lowest for the minority who do.
The information-theoretic framing treats the disclosure as a signal about theta_S. The actual mechanism is cognitive: the Scaling field changes what the agent thinks about, not what the agent transmits about a fixed mental state. The formalization captures the outcome (more information about theta_S in the disclosure) but not the process (the cognitive redirection that produces the information). A cognitive science formalization, building on Kahneman's (2011) distinction between System 1 and System 2 processing, would complement the information-theoretic formalization but is beyond this paper's scope.