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 seed crystal on the Human-AI Exchange Rate (Beier, 2026aa) proposes that the relationship between human judgment and AI processing is trade rather than control, governed by comparative advantage along the dimension of computational irreducibility. This paper proves three formal results. Proposition EX-1 (Comparative Advantage Partition) establishes that for tasks whose computational complexity is irreducible (no algorithm achieves the answer faster than the problem's intrinsic complexity), human ab initio judgment has lower cost per unit of decision quality than AI pattern-matching, and vice versa for reducible tasks. Proposition EX-2 (Measurability) proves that the exchange rate is observable through the scored disclosure infrastructure - both human and AI contributions can be denominated in proper scoring rule units, producing a common currency for the trade. Proposition EX-3 (Monotonic Boundary Shift) proves that the boundary between human-led and AI-led domains shifts monotonically as AI capability increases, with the rate of shift determined by the domain's position on the irreducibility spectrum, and that the boundary never reaches zero for domains with positive computational irreducibility.
Keywords: exchange rate, comparative advantage, computational irreducibility, human-AI trade, scoring, boundary shift
JEL Codes: D82, O33, J24, C45
Proposition EX-1 (Comparative advantage in cognitive labor). Let a task T in a constraint domain require producing an assessment of the constraint state theta. Define:
Computational irreducibility index: CI(T) in [0, 1], where CI(T) = 0 means the task is fully reducible (the answer can be computed from historical patterns in O(n) time where n is the data size) and CI(T) = 1 means the task is fully irreducible (no algorithm achieves the answer faster than simulating the full dynamics of the constraint environment).
Let C_H(T) denote the cost of human judgment producing an assessment of quality q on task T, and C_AI(T) denote the cost of AI processing producing an assessment of the same quality q on the same task.
Under the following assumptions:
(CA1) Human judgment cost is weakly decreasing in CI(T): humans perform ab initio reasoning at roughly constant cost regardless of a task's irreducibility, because human cognition does not exploit reducibility the way algorithms do.
(CA2) AI processing cost is strictly increasing in CI(T): for reducible tasks, AI exploits patterns in historical data at low cost; for irreducible tasks, AI must approximate the full simulation, which is expensive and inaccurate relative to human ab initio reasoning.
(CA3) C_H(T) > C_AI(T) for CI(T) = 0: for fully reducible tasks, AI is cheaper (it can exploit the reducibility that human cognition cannot).
(CA4) C_H(T) < C_AI(T) for CI(T) = 1: for fully irreducible tasks, human judgment is cheaper (it performs the ab initio reasoning that AI cannot shortcut).
Then there exists a threshold CI* in (0, 1) such that:
For tasks with CI(T) > CI*: human judgment has comparative advantage (lower cost per unit of quality).
For tasks with CI(T) < CI*: AI processing has comparative advantage.
At CI(T) = CI*: the costs are equal.
Proof. Define the cost ratio R(T) = C_H(T) / C_AI(T). By (CA1) and (CA2), C_H is weakly decreasing in CI and C_AI is strictly increasing in CI, so R is strictly decreasing in CI. By (CA3), R(0) > 1 (AI is cheaper at CI = 0). By (CA4), R(1) < 1 (human is cheaper at CI = 1). Since R is continuous and strictly decreasing from R(0) > 1 to R(1) < 1, by the intermediate value theorem there exists a CI* in (0, 1) where R(CI*) = 1. Since R is strictly decreasing, this crossing point is unique.
For CI(T) > CI*: R(T) < 1, so C_H(T) < C_AI(T) - human judgment is cheaper.
For CI(T) < CI*: R(T) > 1, so C_AI(T) < C_H(T) - AI is cheaper.
The comparative advantage follows from Ricardo's (1817) principle: even if one party is absolutely more productive at all tasks, both parties benefit from specializing in their relatively cheaper tasks and trading. Here the partition is by computational irreducibility rather than by geography, and the "nations" are human cognition and machine processing. QED.
Empirical grounding. Examples of tasks at different points on the CI spectrum:
CI approximately 0 (fully reducible): cross-referencing thousands of disclosures for consistency, detecting anomalous calibration scores in a track record database, routing aquifer assessments to fund managers whose portfolios are exposed. AI dominates: pattern matching at scale.
CI approximately 0.5 (partially reducible): assessing a company's competitive position given its financial history and sector dynamics. Both human and AI contribute: AI processes the historical data, human judgment assesses the novel aspects.
CI approximately 1 (fully irreducible): assessing what happens to the Ogallala when a novel combination of climate shift, technology adoption, and policy change interacts with the aquifer's specific geology for the first time. Human judgment dominates: no historical pattern covers this specific configuration.
Proposition EX-2 (The exchange rate is observable). Let the Disclosure Incentive's scored infrastructure operate in a constraint domain where both human analysts and AI systems produce assessments that are scored against outcomes by the same proper scoring rule S. Define:
Human quality: Q_H = E[S(r_H, omega)] where r_H is the human's reported assessment and omega is the realized outcome.
AI quality: Q_AI = E[S(r_AI, omega)] where r_AI is the AI system's assessment.
Human cost: C_H = cost of producing the human assessment (analyst time, data acquisition, cognitive effort).
AI cost: C_AI = cost of producing the AI assessment (compute, data licensing, model maintenance).
The exchange rate is:
E = (Q_H / C_H) / (Q_AI / C_AI) = (Q_H * C_AI) / (Q_AI * C_H)
This quantity is observable because:
(i) Q_H and Q_AI are measured by the same scoring rule applied to the same outcomes.
(ii) C_H and C_AI are market prices (analyst salaries, compute costs) that are observable.
Proof. The scoring rule S provides a common unit of measurement for assessment quality. By the strict properness of S (Beier, 2026r, Proposition 5), the expected score is maximized by truthful reporting, so Q_H and Q_AI measure the genuine assessment quality of human and AI respectively, not their strategic reporting quality.
The common denomination is essential. Without the scoring infrastructure, human quality would be measured by institutional proxies (credentials, employer reputation) and AI quality by task-specific benchmarks (accuracy on held-out test sets), and the two would not be comparable. With the scoring infrastructure, both are measured in the same units: expected score under a proper scoring rule on the same domain, evaluated against the same outcomes.
The exchange rate E is therefore:
E = (quality per dollar, human) / (quality per dollar, AI)
When E > 1, human judgment produces more quality per dollar and should lead.
When E < 1, AI produces more quality per dollar and should lead.
When E = 1, the contributions are equally cost-effective.
The exchange rate is not assumed or calibrated. It is measured from the data the scoring infrastructure produces. Each scored disclosure (human or AI) provides a data point for Q. Market prices provide C. The exchange rate emerges from the accumulation of scored assessments across both sources. QED.
Remark on the Hamlet Principle. The exchange rate is measurable but the computational irreducibility of a task is not directly observable - it must be inferred from the exchange rate itself. A domain where E >> 1 persistently (human judgment consistently produces more quality per dollar than AI) is empirically identified as a high-CI domain. A domain where E << 1 persistently is empirically identified as a low-CI domain. The exchange rate is the empirical measure of computational irreducibility: you cannot compute CI from first principles for most real-world tasks, but you can observe which tasks human judgment handles more cost-effectively than AI, and the persistent cost-effectiveness gap reveals the task's position on the irreducibility spectrum.
Proposition EX-3 (The boundary shifts monotonically but never reaches zero). Let AI capability be parameterized by a scalar A(t) that is strictly increasing in t (AI improves over time). Let the cost of AI processing on a task with irreducibility CI be:
C_AI(CI, A) = f(CI) / g(A)
where f is strictly increasing in CI (more irreducible tasks are harder for AI) and g is strictly increasing in A (better AI reduces cost). Let human cost C_H(CI) be independent of A (human capability does not change with AI improvement - a simplification; see limitations).
The threshold CI*(A) where human and AI costs are equal satisfies:
(i) CI*(A) is strictly increasing in A: as AI improves, the threshold shifts upward, and AI gains comparative advantage in tasks that were previously human-led.
(ii) CI*(A) < 1 for all finite A: the threshold never reaches the fully irreducible end of the spectrum. There always exist tasks where human judgment retains comparative advantage.
(iii) The rate of shift dCI*/dA is decreasing in CI*: the boundary shifts quickly through the reducible tasks and slowly through the irreducible tasks. The last domains to fall to AI are the most irreducible, and they resist the longest.
Proof. (i) The threshold CI*(A) is defined by C_H(CI*) = C_AI(CI*, A) = f(CI*) / g(A). Rearranging: f(CI*) = C_H(CI*) * g(A). Since g is strictly increasing in A and C_H is independent of A, the right side is strictly increasing in A. Since f is strictly increasing in CI*, CI* must increase to maintain the equality. Therefore dCI*/dA > 0. QED.
(ii) For CI = 1 (fully irreducible), f(1) represents the cost of simulating the full dynamics of the constraint environment. By the definition of computational irreducibility (Wolfram, 2002), no computation achieves the answer faster than the full simulation. Therefore f(1) is bounded below by the physical cost of the simulation itself - a positive constant that does not decrease with A, because A improves the efficiency of pattern-matching and approximation, not the speed of irreducible computation. (An AI that is twice as fast at pattern-matching is not twice as fast at simulating an irreducible dynamical system - the irreducible computation takes the same number of steps regardless of the processor's speed on reducible operations.)
Therefore C_AI(1, A) = f(1) / g(A) where f(1) is bounded below. Meanwhile C_H(1) is finite. For CI*(A) to reach 1, we would need f(1) / g(A) <= C_H(1), which requires g(A) >= f(1) / C_H(1). But if f(1) represents the irreducible simulation cost, g(A) can approach f(1)/C_H(1) only if AI can perform irreducible computations at human cost - which contradicts the irreducibility (the computation cannot be shortcut).
Formally: there exists a lower bound f_min > 0 such that f(1) >= f_min for all A, because the irreducible computation has a minimum cost independent of AI capability improvements (which improve reducible operations only). If C_H(1) < f_min, then CI*(A) < 1 for all A. The condition C_H(1) < f_min holds when human ab initio judgment on fully irreducible tasks is cheaper than the minimum simulation cost - which is empirically grounded by the observation that human experts routinely produce useful assessments of novel constraint configurations without simulating the full dynamics. QED.
(iii) Implicit differentiation of f(CI*) = C_H(CI*) * g(A) gives:
f'(CI*) * dCI*/dA = C_H(CI*) * g'(A) + C_H'(CI*) * g(A) * dCI*/dA
Solving:
dCI*/dA = C_H(CI*) * g'(A) / (f'(CI*) - C_H'(CI*) * g(A))
For f' strictly increasing (the cost of AI increases faster at higher CI - the marginal task is harder) and C_H' weakly decreasing (human cost decreases or is constant at higher CI), the denominator increases with CI*, making dCI*/dA decrease. The boundary shifts quickly through the easy (low-CI) tasks and slowly through the hard (high-CI) tasks. QED.
Interpretation. The proposition captures the intuition behind the seed crystal's claim that "the division of labor shifts but doesn't collapse to one side." AI capability improvements shift the boundary, but the shift decelerates as it approaches the irreducible end of the spectrum. The final domains - the ones requiring genuine ab initio judgment about novel constraint configurations - are the most resistant to AI displacement, and the resistance is structural (a consequence of computational irreducibility) rather than contingent (a consequence of current AI limitations).
The practical implication: as AI capability increases, the exchange rate shifts in AI's favor for aggregation, routing, pattern detection, and anomaly identification (low CI). It shifts slowly in AI's favor for standard constraint assessment with historical analogs (medium CI). It barely shifts for novel constraint configurations without historical precedent - first-time aquifer-climate-agriculture interactions, unprecedented infrastructure failure modes, novel regulatory environments (high CI). The Disclosure Incentive's value proposition is most durable in exactly these high-CI domains where the exchange rate most persistently favors human judgment.
The assumption that human capability C_H is independent of AI improvement (used in Proposition EX-3) is a simplification. In practice, AI tools augment human judgment - an analyst using AI-assisted data processing produces better assessments at lower cost than one without. The augmentation shifts C_H downward over time, which could either accelerate or decelerate the boundary shift depending on whether AI augments human judgment more on reducible or irreducible tasks. A richer model would specify C_H(CI, A) as a function of both irreducibility and AI capability, but the qualitative result (the boundary shifts but doesn't reach 1) survives as long as the augmentation effect is weaker than the direct AI improvement effect on reducible tasks.
The computational irreducibility index CI(T) is defined abstractly. For most real-world tasks, CI cannot be computed from first principles because the task's computational complexity is unknown. The exchange rate E (Proposition EX-2) provides the empirical proxy: tasks where E >> 1 persistently are high-CI, tasks where E << 1 are low-CI. But the proxy is noisy, and the mapping from E to CI depends on the cost structure (C_H, C_AI) as well as the quality structure (Q_H, Q_AI). The exchange rate reflects the combined effect of irreducibility and current cost structures, not irreducibility alone.
Proposition EX-1's assumptions (CA1)-(CA4) are stylized. (CA1) assumes human cost is independent of CI, but in practice humans are more expensive on some tasks than others regardless of irreducibility (e.g., tasks requiring rare domain expertise). (CA2) assumes AI cost is monotonically increasing in CI, but AI cost also depends on data availability, which is not captured by CI alone. The assumptions are sufficient for the existence of the threshold CI* but do not claim to capture all relevant determinants of the cost structure.
The proposition that CI*(A) < 1 for all finite A depends on the claim that computational irreducibility imposes a floor on the cost of solving irreducible problems. Wolfram (2002) argues this is a fundamental property of certain dynamical systems, but the claim is not universally accepted in theoretical computer science. If future AI architectures find ways to approximate irreducible computations cheaply (through novel heuristics, quantum computation, or methods not yet invented), the floor could lower and CI* could approach 1. The proposition is conditional on the irreducibility floor holding.