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.
When a single agent discloses structured judgment about a constraint environment, the information produced is one assessment - valuable, but bounded by the agent's expertise and vantage point. When N agents disclose structured judgments about the same constraint environment, the information produced is not N times a single disclosure. Each disclosure interacts with every other: one analyst's aquifer depletion assessment changes the meaning of another's irrigation efficiency assessment, which changes the meaning of a third's crop yield projection, which in turn alters the implications of the first analyst's aquifer assessment under the updated demand trajectory. The interaction is combinatorial rather than additive - the number of pairwise interactions grows as N(N-1)/2, the number of triples as N-choose-3, and the full set of possible interaction patterns grows factorially with N. This paper develops the economics of this combinatorial growth and argues that it is the formal basis for the network effects that the companion paper (Beier, 2026b) identifies in the Disclosure Incentive's externality application. Weitzman's (1998) recombinant growth model is the closest predecessor. In that framework, the combination of existing ideas produces new ideas at a rate that can exceed the rate of diminishing returns to capital accumulation, generating sustained growth. The present paper extends Weitzman's logic from idea recombination to disclosure interaction, with a key difference: ideas recombine as objects (a new technology combines principles from two existing technologies), while disclosures interact as assessments of a shared constraint environment (one analyst's assessment changes what another analyst's assessment implies). The interaction structure is richer than recombination because it is mediated by the constraint environment, which links disclosures that would otherwise be independent.
Keywords: network effects, combinatorial growth, recombinant growth, disclosure interaction, constraint environments, information architecture, coordination, Weitzman
JEL Codes: O40, D85, D82, D83, C72
The non-rivalry of ideas - a core result of Romer (1990) - means that one person using an idea does not diminish its availability to others - and that this non-rivalry is the engine of endogenous growth. The non-rival character of ideas means that the growth rate does not diminish as the stock of ideas increases, breaking the diminishing-returns logic that limits growth driven by capital accumulation. Weitzman (1998) pushed this further with a specific combinatorial mechanism: new ideas are produced by recombining existing ideas, and the number of potential recombinations grows faster than the stock of ideas from which they are drawn. In the limit, recombinant growth can sustain rates that exceed what the standard Romer model produces.
The present paper applies a related but distinct logic to the Disclosure Incentive. When N agents disclose structured judgments about constraint environments, the information value of the disclosure network is not the sum of N individual disclosures. It is the sum of the individual disclosures plus the interaction information that emerges when disclosures are read together - and the interaction information grows combinatorially with N.
The distinction from Weitzman is important. Weitzman modeled the recombination of ideas as objects: idea A combines with idea B to produce idea C, and the combination is a new idea that did not exist before. The present paper models the interaction of assessments as interpretations: assessment A changes the meaning of assessment B when both address the same constraint environment, and the changed meaning is not a new disclosure but a richer understanding of what the existing disclosures imply jointly. The aquifer analyst's depletion estimate does not "combine" with the irrigation engineer's efficiency assessment to produce a third idea. Rather, the depletion estimate contextualizes the efficiency assessment - it reveals that the irrigation efficiency matters only if the aquifer sustains the projected extraction, which the depletion estimate says it may not - and the contextualization is information that neither disclosure alone contains.
Section 2 develops the formal structure of disclosure interaction. Section 3 distinguishes interaction from recombination. Section 4 examines the role of the constraint environment as interaction medium. Section 5 derives implications for network value. Section 6 states limitations. Section 7 concludes.
Consider a constraint environment - a shared aquifer in a California agricultural region - assessed by N agents, each with different expertise, different data, and a different analytical vantage point.
Agent 1 (hydrologist) discloses: "At current aggregate extraction rates, the aquifer's recharge-depletion balance crosses a critical threshold within seven years. Confidence: 65%."
Agent 2 (irrigation engineer) discloses: "Current irrigation efficiency in the watershed is 62%, with demonstrated technology available to increase it to 78%. Adoption at scale requires capital investment of $X per acre. Confidence in the efficiency estimates: 80%."
Agent 3 (agricultural economist) discloses: "Under current commodity prices and subsidy structures, the dominant crop rotation in the region is water-intensive and will remain so unless water prices increase by at least 40%. Confidence: 70%."
Each disclosure has individual information value. The hydrologist's assessment informs anyone making decisions about the aquifer. The engineer's assessment informs anyone considering irrigation investment. The economist's assessment informs anyone modeling agricultural land values.
But the joint information exceeds the sum of the parts. The hydrologist's seven-year depletion estimate, combined with the engineer's finding that efficiency can increase from 62% to 78%, implies that efficiency adoption could extend the aquifer's viable life - but only if the economist's assessment of crop rotation stickiness is wrong, or if water prices rise enough to trigger adoption. The three disclosures together reveal a conditional structure that none of them contains alone: the aquifer's fate depends on whether the capital investment in irrigation efficiency occurs before the depletion threshold, which depends on whether commodity prices and subsidies shift enough to change crop rotation, which depends on policy trajectories that none of the three analysts assessed.
This conditional structure is the interaction information. It exists in the joint reading of the disclosures, not in any individual disclosure. And it points to a fourth assessment that the disclosure network has not yet produced but that the interaction reveals is needed: what are the policy trajectories for water pricing and agricultural subsidies in this region?
Weitzman's (1998) recombinant growth model treats ideas as objects that combine pairwise (or in higher-order tuples) to produce new objects. The number of pairwise combinations of N ideas is N(N-1)/2, the number of k-tuples is N-choose-k, and the total number of possible combinations is 2^N - 1. Growth is sustained because the combination space grows exponentially with the stock of ideas.
Disclosure interaction has a similar combinatorial structure but operates through a different mechanism. Ideas combine as components: the transistor and the integrated circuit combine to produce the microprocessor - a new object that inherits properties from both components. Disclosures interact as perspectives: the hydrologist's assessment and the engineer's assessment do not combine to produce a third assessment. They interact to reveal what the constraint environment implies when both assessments are true simultaneously. The output of the interaction is not a new disclosure but a conditional structure - an understanding of how the constraint environment behaves under the joint assessment that no individual assessment conveys.
The distinction matters because the interaction mechanism is richer than the combination mechanism. Two ideas combine in one way (A+B = C). Two disclosures about a shared constraint environment interact in multiple ways: A contextualizes B, B contextualizes A, and the joint {A, B} reveals implications about the constraint environment that neither A alone nor B alone implies. When a third disclosure C enters, it interacts with A, with B, with {A, B}, and the joint {A, B, C} reveals further conditional structure. The number of distinct interaction patterns grows faster than the number of combinations because each subset of disclosures generates its own conditional implications about the shared constraint.
Formally, for N disclosures about a shared constraint environment, the number of non-empty subsets is 2^N - 1. Each subset generates conditional implications that are specific to that subset and not contained in any proper subset. The information content of the disclosure network is:
I_network = sum over all non-empty S of I_interaction(S)
where I_interaction(S) is the conditional information revealed by reading the disclosures in subset S jointly. For independent disclosures (no shared constraint environment), I_interaction(S) = 0 for |S| > 1, and the network information equals the sum of individual disclosures. For disclosures that address a shared constraint environment, I_interaction(S) > 0 for subsets S whose members address related aspects of the environment, and the network information exceeds the linear sum.
The growth rate of I_network with N depends on how densely the disclosures are connected through the constraint environment. In the densely connected case (all N disclosers address aspects of a single shared constraint), the interaction terms dominate and I_network grows combinatorially. In the sparsely connected case (disclosers address unrelated constraints), the interaction terms are small and I_network grows approximately linearly.
What makes disclosure interaction combinatorial rather than additive is the constraint environment. Two disclosures about unrelated topics (one about aquifer depletion in California, another about semiconductor supply chains in Taiwan) do not interact meaningfully - reading them together reveals nothing that reading them separately does not. Two disclosures about the same constraint environment interact because the constraint environment links them: the aquifer connects the hydrologist's depletion estimate to the engineer's irrigation assessment to the economist's crop rotation analysis, and the connections generate conditional structure that the individual disclosures do not contain.
The constraint environment functions as an interaction medium - a shared structure through which the information in individual disclosures combines to produce joint implications. Arthur's (1989) analysis of increasing returns and lock-in provides a related insight: when agents' decisions interact through a shared environment (a technology standard, a network, a resource base), the aggregate dynamics are path-dependent and nonlinear in ways that individual decisions do not predict. The disclosure interaction mechanism adds an information-theoretic dimension to Arthur's observation: when agents' assessments interact through a shared constraint environment, the aggregate information content is nonlinear in the number of assessments.
The practical implication is that the Disclosure Incentive's value grows disproportionately with the density of disclosure in a specific constraint domain. A disclosure ecosystem with ten analysts covering different aspects of a single aquifer is worth substantially more than ten analysts covering ten unrelated topics, because the constraint environment (the aquifer) mediates interactions that amplify each disclosure's information value. This is the formal basis for the network effects claim in the companion paper (Beier, 2026b, Section 8): each disclosure enriches the information environment for every other participant, and the enrichment is combinatorial when the disclosures address a shared constraint.
The implication for platform design is that the Disclosure Incentive's value concentrates in constraint-specific clusters. An early-stage disclosure platform produces the most value by achieving critical mass in a small number of constraint domains (one aquifer, one grid service territory, one atmospheric loading region) rather than by spreading thin across many unrelated domains. The combinatorial growth depends on density within a shared constraint environment, not on breadth across unrelated ones.
The combinatorial growth of disclosure interaction has three implications for the economics of the Disclosure Incentive.
The first concerns the incentive to be the marginal discloser. Under linear information growth, the Nth discloser adds the same information value as the first. Under combinatorial growth, the Nth discloser adds more information value than the (N-1)th, because the Nth disclosure interacts with all N-1 existing disclosures. The marginal value of disclosure is increasing in the size of the disclosure network for the relevant constraint domain. This creates a positive feedback loop: as the network grows, the incentive to join increases, which accelerates growth - the mechanism through which Admati and Pfleiderer's (2000) disclosure externality is converted from a barrier (each firm's disclosure benefits competitors) into an accelerant (each firm's disclosure increases the value of joining the network).
The second concerns the comparison with price-based information aggregation. Prices aggregate information linearly: each informed trader's signal contributes additively to the price's information content (in the CARA-normal framework of the companion paper). Disclosure networks aggregate information combinatorially: each disclosure contributes its individual information plus its interaction with every other disclosure in the constraint domain. The disclosure channel produces more information per participant than the trading channel, and the gap widens as the number of participants grows. This is an additional reason - beyond the Grossman-Stiglitz resolution - why the cooperative channel produces information that the competitive channel cannot replicate at scale.
The third concerns the dynamics of adoption. Combinatorial network effects generate tipping-point dynamics: below a critical mass of disclosers in a constraint domain, the interaction information is sparse and the individual return to disclosure may not exceed cost. Above the critical mass, the interaction information explodes and the return to disclosure rises sharply. The prediction is that disclosure adoption in any given constraint domain will follow an S-curve: slow initial uptake as early disclosers bear the cost without the benefit of dense interaction, followed by rapid adoption once the critical mass triggers the combinatorial growth, followed by saturation as the constraint domain is densely covered and the marginal interaction information from additional disclosures diminishes.
The paper argues for combinatorial growth of information in disclosure networks but does not prove a specific growth rate. The factorial characterization in the title is an upper bound that obtains only when every disclosure interacts meaningfully with every other - a condition that holds when all disclosers address a single, densely interconnected constraint environment. In practice, constraint environments have varying degrees of internal connectivity (the aquifer example is densely connected; a large atmospheric region is sparsely connected), and the actual growth rate depends on the constraint environment's interaction structure. A formal model specifying the growth rate as a function of the constraint environment's connectivity would strengthen the argument but is beyond this paper's scope.
The formal expression I_network = sum over all subsets of I_interaction(S) is a framework rather than a computable quantity. Computing I_interaction(S) for specific subsets requires specifying how disclosures interact through the constraint environment, which requires domain knowledge (hydrology, atmospheric science, grid engineering) that the general framework does not contain. The framework identifies what to measure and why it matters, but the measurement itself is domain-specific.
The comparison with Weitzman (1998) is structural rather than formal. Weitzman's model has an explicit production function for new ideas from recombination, calibrated parameters, and growth-rate implications. The present paper has a conceptual distinction (interaction vs. recombination) and a qualitative growth argument (combinatorial vs. linear) but does not produce a calibrated growth model. The formal connection between disclosure network size and economic growth - mediated through the TFP channel of the companion paper (Beier, 2026ae) - is an important open question.
The tipping-point prediction (Section 5) is qualitative. The critical mass of disclosers required to trigger combinatorial growth depends on the constraint domain's structure, the quality of the disclosers, and the capacity of other participants to read and interpret the interaction information. These parameters are domain-specific and not estimable from the general framework.
The information output of a disclosure network grows faster than the number of disclosers because disclosures interact through the constraint environments they assess. The hydrologist's depletion estimate changes the meaning of the engineer's efficiency assessment, which changes the meaning of the economist's crop analysis, and the conditional structure that emerges from reading them jointly is information that none of them individually contains. The growth is combinatorial because it arises from subsets of disclosures - pairs, triples, and higher-order groupings - each revealing implications specific to that combination of perspectives on the shared constraint.
This combinatorial structure is what gives the Disclosure Incentive its scaling property. A disclosure ecosystem with critical mass in a constraint domain produces information that exceeds what the same number of agents could produce through trading on private information, because the cooperative channel generates interaction information that the competitive channel - which transmits individual signals into a scalar price without preserving their conditional structure - structurally cannot. The constraint environment is the interaction medium, the structured disclosure format is the mechanism that preserves the conditional structure, and the combinatorial growth is the payoff for building the cooperative channel that the Grossman-Stiglitz framework, by excluding it, never allowed the economics profession to see.
This paper presents results from the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics (Beier, forthcoming 2026). The combinatorial growth mechanism developed here provides the formal basis for the network effects identified in the companion paper on the Disclosure Incentive (Beier, 2026b) and connects to the growth implications developed in the companion paper on TFP (Beier, 2026ae). The author invites engagement from the growth theory, network economics, and information economics communities.