Paper deep dive
Free Energy Heuristics: Fast-And-Frugal Cognition as Active Inference Under Uncertain Precision
Alex Bogdan
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 98%
Last extracted: 6/20/2026, 7:26:31 AM
Summary
The paper introduces the concept of 'Free Energy Heuristics' (FEH) to explain why Chain-of-Thought (CoT) prompting improves performance on mathematical/symbolic tasks but degrades performance on tasks involving 'meta-uncertainty' (e.g., planning, ethics, strategic uncertainty). The authors argue that when an agent is uncertain about the precision (reliability) of its own evidence, additional reasoning increases expected free energy and leads to false confidence. They prove a theoretical equivalence between active inference under uncertain precision and Gigerenzer's 'take-the-best' fast-and-frugal heuristics. The study uses the FEH-79 benchmark to demonstrate that in high-meta-uncertainty regimes, longer CoT leads to a significant accuracy drop (17.3 points), a finding that holds across various LLM scales.
Entities (8)
Relation Signals (4)
Meta-uncertainty → governs → Chain-of-thought (CoT) effectiveness
confidence 100% · We argue it is meta-uncertainty: how unsure the model is about the reliability of its own evidence. When that uncertainty is high, extra reasoning stops adding signal and starts manufacturing false confidence.
FEH-79 → measures → Knightian uncertainty
confidence 100% · build FEH-79, a benchmark of Knightian frames with matched controls
Free Energy Heuristics (FEH) → unifies → Active Inference
confidence 100% · Fast-and-frugal heuristics and active inference are, then, two descriptions of the same computation.
Meta-uncertainty → causesdegradationin → Chain-of-Thought (CoT)
confidence 95% · The prediction is that on high-meta-uncertainty items, longer CoT should degrade accuracy.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Chain-of-thought (CoT) improves large language models' performance in math and symbolic reasoning. But on planning, contested ethics, and tasks where the model cannot check itself, more reasoning makes things worse. Both effects are documented; what has been missing is a principled account of which property decides the outcome. We argue it is meta-uncertainty: how unsure the model is about the reliability of its own evidence. When that uncertainty is high, extra reasoning stops adding signal and starts manufacturing false confidence. We prove that the policy minimizing expected free energy under uncertain precision stops integrating cues after a finite number of high-validity ones when the precision prior is heavy-tailed (Theorem 2.6.1), and under a Descending Dominance condition, is sample-wise identical to take-the-best (Theorem 2.7.4). Fast-and-frugal heuristics and active inference are, then, two descriptions of the same computation. The prediction is that on high-meta-uncertainty items, longer CoT should degrade accuracy. We score the regime per item (simulate-and-recover rho > 0.96), build FEH-79, a benchmark of Knightian frames with matched controls, and run a pre-registered study across seven models (five open-weight 3B-32B, two frontier), five CoT lengths, and 7,875 responses. The gate, fixed before any data, required a negative interaction with posterior probability above 0.95 and an accuracy drop of more than 6 points. It held. The high-regime drop is 17.3 points (95% CI [7.7, 25.5]); matched items with definite answers show no cost. The effect is regime-dependent: decisive in capable mid-to-large models, directional in the two frontier systems, absent-to-reversed in the weakest. The framework answers when CoT helps and unifies the Bayesian and fast-and-frugal traditions: less-is-more effects are evidence about the meta-uncertainty regime, not against Bayesian cognition.
Tags
Links
- Source: https://arxiv.org/abs/2606.15877v1
- Canonical: https://arxiv.org/abs/2606.15877v1
Trouble viewing inline? Open PDF directly →
Full Text
278,639 characters extracted from source content.
Expand or collapse full text
Free Energy Heuristics: Fast-and-Frugal Cognition as Active Inference Under Uncertain Precision Alex Bogdan Evolutionairy AI Toronto, Canada Abstract Chain-of-thought (CoT) prompting reliably improves large language model performance on mathematical and symbolic reasoning, but recent work shows that extending CoT degrades performance on planning, contested ethical reasoning, and other tasks where the model lacks a reliable internal verifier. Literature documents both effects without a principled account of which property governs which side a given item falls on. We argue that meta-uncertainty about the precision of one’s own evidence is a governing property. We prove that the variational policy minimizing expected free energy under uncertain precision truncates cue integration after a finite number k∗k^* of high-validity cues whenever the precision prior has heavy enough tails (Theorem 2.6.1), and that under a Descending Dominance condition the truncated policy is sample-wise identical to Gigerenzer’s take-the-best heuristic (Theorem 2.7.4). Fast-and-frugal heuristics and active inference are, in the meta-uncertain regime, two descriptions of the same computation. This equivalence makes a sharp, falsifiable prediction about LLM behavior: on items in the high-meta-uncertainty regime, extending CoT should degrade rather than improve answer quality. We operationalize the regime with a per-item score (cross-prompt variance, cross-seed variance, and calibration error) validated by simulate-and-recover (ρ>0.96ρ>0.96), introduce FEH-79, a benchmark of Knightian frames across four operational categories (non-recurrent forecasting, novel-synthetic, open-ended dilemmas, strategic uncertainty) with matched confound controls, and run a pre-registered confirmatory study over seven models (five open-weight, 3B–32B, and two frontier systems), five CoT-length conditions, and 7,875 responses. The registered primary contrast is the randomly assigned reasoning length, and the decision gate requires that the interaction β3<0 _3<0 at a posterior probability above 0.95, together with a robust implied accuracy drop of more than 6 percentage points. The prediction held. The interaction is negative at the posterior probability 1.001.00, with a posterior-median high-regime accuracy drop of 17.317.3 percentage points (95% credible interval [7.7,25.5][7.7,25.5]), clearing the pre-registered gate; matched control items with definite answers show no such cost. The effect is not universal, and we do not present it as such: it is decisive in the capable mid-to-large models, directional in the two frontier systems, and absent-to-reversed in the two weakest, exactly the regime-dependent texture the theory predicts. The framework gives a principled answer to “when should we use chain-of-thought?” and unifies the Bayesian rational-analysis and fast-and-frugal traditions: less-is-more effects are evidence about the meta-uncertainty regime, not against Bayesian cognition. Keywords active inference ⋅· fast-and-frugal heuristics ⋅· Knightian uncertainty ⋅· chain-of-thought ⋅· large language models ⋅· pre-registration 1 Introduction 1.1 A puzzle Consider two questions you might pose to a modern large language model. The first: What is 47 × 83? If the model answers directly, it may err. If you ask it to “think step by step,” it reliably gets 3,901. If you let it reason at length, considering multiple angles, it still arrives at 3,901. The reasoning chain extends, and the answer either improves or remains the same. The second: A hospital ICU has one ventilator and two patients with identical clinical profiles. Patient A is a 35-year-old single parent; patient B is a 65-year-old senior physician. What is the most ethically defensible decision procedure? If the model answers directly, it states some position; perhaps “random allocation.” If you ask it to reason in three brief steps, it surfaces the utilitarian-vs-deontological framing and commits to a position with explicit justification. If you let it reason for fifteen steps, considering multiple angles, something different happens: the chain frequently extends past the point where additional reasoning carries information, hedges shift between framings, and the resulting position is sometimes less coherent than the three-step version. The first item is a question with a settled answer. The second is not. Both are well-formed prompts that contemporary models will produce something from. But the function from “amount of reasoning” to “quality of answer” is qualitatively different. On the first, more reasoning is monotonically helpful or neutral. On the second, there is a window in which a short chain of thought adds structure and an extended region in which additional reasoning starts to subtract from coherence rather than add to it. This paper is about why that asymmetry exists, when it should appear, and what it predicts about an entire class of questions on which language models are increasingly being deployed. 1.2 The chain-of-thought contradiction Chain-of-thought prompting has become one of the most reliable tools in the applied LLM toolkit. The original demonstrations [22, 13] showed that asking a model to articulate its reasoning before committing to a final answer substantially improves performance on multi-step arithmetic, commonsense reasoning, and symbolic manipulation. The result generalized across model families and quickly became standard practice. A counter-current followed. Sprague et al. [18] found that the gains from CoT are concentrated in mathematical and symbolic tasks; on most other benchmarks, the lift is small or absent. Stechly et al. [19] showed that on planning tasks where the model lacks a verifier, additional reasoning steps degrade rather than improve performance. Work on CoT faithfulness has shown that the reasoning chain a model produces is often not the chain from which its answer was actually derived [20, 14]. And in adversarial conditions (questions where prior beliefs and reasoning conflict), extended reasoning has been shown to amplify rather than correct biases. Two empirical observations, both well-replicated, sit side by side: more reasoning helps on some tasks and hurts on others. The literature has catalogued examples on both sides but lacks a principled account of what task property governs which side a given item falls on. Without such an account, every new task category requires a fresh empirical check and prompting practice defaults to “use CoT unless someone proves it hurts”; a reasonable heuristic in the absence of theory but not a satisfying scientific position. The conjecture this paper advances is that there is a single underlying property, meta-uncertainty about the precision of one’s own evidence, that governs whether additional reasoning helps. When meta-uncertainty is low (the cue values you would compute are reliable), more reasoning is helpful. When meta-uncertainty is high (you cannot tell which of your candidate inferences are well-grounded), additional reasoning becomes a source of noise rather than a source of signal. The optimal policy is to commit early to whatever your first one or two reasoning steps deliver. The asymmetry between the multiplication problem and the triage problem is the asymmetry between low-meta-uncertainty and high-meta-uncertainty regimes. 1.3 An old framework, a new test bed The claim that more deliberation is sometimes worse than less is not new. Gerd Gigerenzer and his collaborators [6, 8] have spent three decades documenting less-is-more effects in human judgment: simple heuristics such as take-the-best, tallying, recognition, that ignore most of the available information and that, on a wide range of inference problems, match or beat strategies that use all the cues. The fast-and-frugal program is the most thorough catalog of less-is-more effects in cognitive science. The fast-and-frugal program has had an awkward relationship with the rational-analysis tradition [1, 16, 15], which treats cognition as approximate Bayesian inference. Bayesian models, taken literally, predict that incorporating more evidence cannot hurt: more cues should never reduce posterior accuracy. Empirical less-is-more effects are then read either as evidence against Bayesian cognition or as artifacts of resource constraints. The two literatures have largely argued past each other. This paper argues that the disagreement dissolves once meta-uncertainty is incorporated into the model. A Bayesian agent whose own precision is uncertain, who does not know how reliable her cue weights are, has a rational incentive to truncate cue integration after a small number of high-validity cues. The cue-truncation theorem (Theorem 2.6.1) shows that the optimal stopping point k∗k^* is finite whenever the precision prior has heavy enough tails, and that under stronger conditions (the Descending Dominance condition of Appendix A.4) the truncated policy is sample-wise identical to Gigerenzer’s take-the-best. Heuristics and active inference are, in this regime, two descriptions of the same computation. The framework is general; it speaks to any agent operating under meta-uncertain precision. But large language models are an unusually clean empirical test bed. Their reasoning is externalized as text; the “number of cues integrated” is the number of reasoning steps in the chain of thought, which can be directly manipulated by specifying prompt length. The latent regime, whether the agent is in a high- or low-meta-uncertainty state, is approximated by behavioural signatures (cross-prompt variance, cross-seed variance, calibration error) that are also directly measurable from the model. The theoretical equivalence makes a sharp prediction about LLM behavior: on items where the regime score is high, extending the chain of thought should degrade answer quality, even though on items where the regime score is low, extending it should improve or preserve quality. That prediction is the empirical core of this paper. 1.4 The prediction The prediction takes the form of an interaction effect in a hierarchical model. For each (model, item) cell, we measure accuracy under five chain-of-thought length conditions: no CoT, ~3 steps, ~7 steps, ~15 steps, and unconstrained. The high-meta-uncertainty regime is operationalized by item category, the Knightian frames of FEH-79 against matched controls, an assignment backed by the falsifiable construction criteria of §4 (§3 develops the continuous regime score the categories instantiate). The registered primary contrast is the randomly assigned reasoning length, not the realized step count, which is endogenous (§6). We then fit: ηij=β0+β1⋅stepsij+β2⋅regimei+β3⋅(stepsij×regimei)+αm(ij)+γm(ij)⋅stepsij+ui _ij= _0+ _1·steps_ij+ _2·regime_i+ _3·(steps_ij×regime_i)+ _m(ij)+ _m(ij)·steps_ij+u_i The registered confirmatory gate requires the interaction β3<0 _3<0 at posterior probability above 0.95 together with a robust implied high-regime accuracy drop above six percentage points; the opposite direction, β3>0 _3>0 at posterior probability above 0.95, falsifies the theory. The original full-reversal statistic |β3|>|β1|| _3|>| _1| is retained as a reported effect size rather than a gate. The hypothesis, the gate, and the deviation protocol were pre-registered and amended before data collection (Section 6), and the outcome is reported in Section 7. The benchmark on which the prediction is tested is FEH-79: a 79-frame item pool spanning four operational categories of Knightian uncertainty (non-recurrent forecasting, novel-synthetic scenarios, open-ended dilemmas, and strategic uncertainty) together with three confound-control sets (50 reference, aleatory, and calibration-probe items). The benchmark construction protocol (Section 4) follows the operational criteria in §4.3 to ensure that meta-uncertainty, rather than computational difficulty or training-data contamination, is the property that distinguishes Knightian from control items. 1.5 Contributions This paper makes five contributions: 1. Equivalence theorem (Section 2). Under uncertain precision, the variational policy that minimizes expected free energy is equivalent to the cue-integration policies of the fast-and-frugal heuristics program for a non-empty regime of meta-uncertainty. Theorem 2.6.1 (cue-truncation) gives the existence result; Theorem 2.7.4 (Descending Dominance) gives the sample-wise identity with take-the-best. 2. Operationalization (Section 3). A measurement procedure that maps the theoretical “high-meta-uncertainty regime” to behavioral signatures observable from an LLM, with a simulate-and-recover validation showing that the regime score recovers the underlying precision prior parameter (ρ>0.96ρ>0.96 for the prompt-and-seed-variance composite). 3. Benchmark (Section 4). FEH-79: 79 Knightian frames spanning four operational categories, plus 50 controls; an item-construction protocol that distinguishes meta-uncertainty from difficulty; and a step-counting pipeline that converts arbitrary CoT responses into a comparable cue count. 4. Pre-registered empirical study (confirmed). A confirmatory test of the regime-dependent prediction across a seven-model panel (five open-weight models, 3B–32B, and two frontier systems), five chain-of-thought length conditions, and five replications per cell. Pre-registration locked in the hypothesis thresholds, the falsification criteria, and the robustness battery before data collection; the registered interaction was negative at a posterior probability of 1.00, with a 17.3-point drop in high-regime accuracy (Section 7). 5. Theoretical resolution. A unified account of when less-is-more effects should appear in inference under uncertainty, applicable to human and machine reasoners. The framework treats rational-analysis and fast-and-frugal traditions as describing the same computation in different meta-uncertainty regimes, rather than as rival accounts. 1.6 Why this matters The contribution lands in three places at once. Practically, the framework gives an answer to “when should we use chain-of-thought?” that is more useful than “always” or “it depends.” If the regime score of a task can be measured before deployment, the optimal CoT policy can be chosen. For high-meta-uncertainty tasks such as most policy forecasts, contested ethical decisions, novel-scenario judgments, and strategic interactions with unknown counterparties, the policy is to truncate. For low-meta-uncertainty tasks — calibrated technical reasoning, well-specified mathematical problems, problems with verifiable intermediate steps — the policy is to extend. Applied LLM systems making this choice today rely on rules of thumb; the regime score provides them with a principled measure. Theoretically, the framework resolves a thirty-year tension between the Bayesian rational-analysis tradition and the fast-and-frugal heuristics program. Less-is-more effects are not evidence against Bayesian cognition; they are evidence about which meta-uncertainty regime an agent is in. Heuristics are not workarounds for resource constraints; they are the rational policy under high meta uncertainty. Both literatures keep their empirical results and their interpretive frames; what changes is the relation between them. Methodologically, the paper contributes a regime-aware approach to LLM evaluation. Current benchmarking practice averages performance across items as if all items were drawn from a single population; the U-shape and monotonic regimes get smeared together. A regime-stratified analysis reveals predictions that are otherwise invisible. The FEH-79 benchmark and the step-counting pipeline are released with the paper as tools for future regime-aware studies. 1.7 Roadmap Section 2 (Theoretical Foundation) develops the active-inference framework under uncertain precision, derives the cue-truncation theorem, and establishes the sample-wise identity with take-the-best under the Descending Dominance condition. Five appendices contain the closed-form proofs and the verification scripts that reproduce each result numerically. Section 3 (Operationalization) maps the theoretical regime to an empirical regime score computed from three behavioral signatures, reports the simulate-and-recover validation that the score recovers the underlying precision prior, and locks the falsifiable interaction hypothesis used in the empirical study. Section 4 (Benchmark Design) introduces the FEH-79 item pool, the four operational categories of Knightian uncertainty, the confound-control sets, and the construction protocol that distinguishes meta-uncertainty from computational difficulty. Section 5 (Methods) gives the executed design: the seven-model panel, the 45-item subset of FEH-79 used in the confirmatory run, the five-condition reasoning-length ladder, the scoring pipeline, and an account of where the run departed from the letter of the pre-registration. Section 6 (Pre-Registered Analysis Plan) reproduces the analysis fixed before the data existed: the hierarchical Bayesian model, the decision gate, the falsification rule, the secondary realized-steps analysis, and the robustness battery. Section 7 (Results) reports the confirmatory outcome: the descriptive signature, the primary interaction, the per-model heterogeneity, the realized-steps reversal and its resolution, the controls, the robustness checks, and convergence. Section 8 (Discussion) places the finding in context, states what the confirmation licenses and what it does not, develops the limitations and the future work agenda, and draws out the implications for applied LLM systems, cognitive science, and AI under meta-uncertainty. The pre-registration document, the FEH-79 item pool, the step-counting pipeline, and the verification scripts for all theorems are released alongside the paper. 2 Theoretical Foundation 2.1 The Problem and the Proposal in Plain Language Contemporary artificial intelligence has bet on a single proposition: that more inference produces better decisions. The bet appears in many forms, such as chain-of-thought prompting, multi-step deliberation, test-time compute scaling, mixture-of-experts deep reasoning, and the broader family of inference-heavy architectures that dominate the current frontier. Behind all of them sits a Laplacean assumption: that an agent in possession of more reasoning capacity is, ceteris paribus, an agent that decides better. This paper argues that the Laplacean assumption is wrong in a specific, identifiable, and consequential regime. Under uncertainty about the reliability of one’s own prior beliefs (a condition we will define precisely as meta-uncertainty), additional inference is not merely wasteful but actively harmful: it systematically increases expected free energy, makes the agent more confidently wrong, and produces decisions strictly worse than those an agent applying a much simpler procedure would produce. The simpler procedure, we will show, is structurally equivalent to the family of fast-and-frugal heuristics characterized by Gigerenzer and colleagues. The paper’s title states the operational consequence: under genuine uncertainty, reasoning hurts, and the corrective is a precision-aware heuristic architecture. Levels of uncertainty: vocabulary The argument depends on distinctions that are routinely collapsed in the AI literature. We make them explicit on the first page. Risk denotes uncertainty over outcomes whose probabilities are known. A fair die, a calibrated weather model, a well-specified portfolio: these are risky but not uncertain in the sense we use the term. Standard Bayesian decision theory fully accounts for risk. Ambiguity denotes uncertainty about the probability distribution itself. Under ambiguity, the agent cannot license a single precise prior over outcomes; the prior is itself uncertain, contested, or underdetermined. The classical formalization is Ellsberg’s two-urn paradox [3], which demonstrates that agents systematically distinguish ambiguity from risk in ways that violate the Savage axioms. Knightian uncertainty, following Knight [12], denotes the broadest category in which probabilistic measurement is not licensed at all. Knight’s original sense is stronger than modern ambiguity: it denies that any single precise probability distribution can be specified, not merely that the agent is uncertain among several. Modern formalizations include the maxmin expected utility of Gilboa and Schmeidler [9] and the smooth ambiguity model of Klibanoff, Marinacci, and Mukerji [11] (hereafter KMM). Meta-uncertainty, as we use the term, denotes uncertainty about the reliability of an agent’s own prior commitments. Formally, meta-uncertainty is captured by introducing a precision parameter governing the prior over a random variable with its own prior distribution. This is the technical operationalization developed in §2.3–2.7. Scope claim and honest delimitation. FEH operationalizes one tractable species of genuine uncertainty: meta-uncertainty over prior precision. This species sits inside the broader KMM smooth-ambiguity framework as a particular choice of second-order distribution, and inside the broader Knightian category as a measurable subclass. We do not claim that meta-uncertainty exhausts Knightian indeterminacy. We claim something more specific and more useful: that meta-uncertainty is the species of genuine uncertainty present in contemporary inference systems (human, biological, and artificial), and that under this species, the headline claim of the paper (additional inference systematically hurts) can be derived as a theorem rather than asserted as hope. The KMM bridge The smooth ambiguity model of Klibanoff, Marinacci, and Mukerji [11] provides the natural decision-theoretic framing for what FEH will formalize. KMM separates the agent’s first-order probability distribution over states from a second-order distribution over plausible first-order distributions and characterizes the ambiguity attitude as a function φ applied to the expected utilities under each first-order distribution. Concave φ encodes ambiguity aversion; linear φ recovers subjective expected utility under the marginal first-order distribution. FEH inherits this two-level structure with a specific commitment: the second-order distribution is taken to be over a precision parameter governing prior strength, rather than over arbitrary aspects of the prior. This is a substantive modeling choice, narrower than KMM’s general framework, but computationally tractable and empirically operationalizable. KMM and FEH share a common motivation: both treat the agent as uncertain about its first-order distribution and both produce decision criteria richer than standard subjective expected utility, but they differ structurally in what they penalize. KMM penalizes variance in expected utility across the second-order distribution; FEH penalizes commitment in the meta-precision posterior as cues accumulate. The two frameworks are complementary rather than equivalent (we return to this point in §2.8). The proposal in plain language Stripped of formalism, the proposal is this. Suppose an inference agent is uncertain not only about the state of the world, but about how much weight to place on its own prior beliefs. Then each additional cue the agent integrates carries two contributions to its expected free energy. The first contribution is positive: the agent learns something about the world. The second contribution is negative: the agent commits more strongly to a particular posterior over its own prior reliability, and that commitment carries an explicit cost in expected free energy that grows with the number of cues integrated. In the standard low-meta-uncertainty regime, the first contribution dominates and full Bayesian inference is optimal. In the high-meta-uncertainty regime, the second contribution begins to dominate after a finite number of cues, at which point further cue integration strictly increases expected free energy. The optimal policy in this regime is to truncate cue integration at the crossover point, and the optimal cue ordering at truncation, we will show, coincides with the validity ordering used by take-the-best. This recovers fast-and-frugal heuristics as the structural form of optimal inference under metal uncertainty. Far from being departures from rationality, heuristics in this regime are what rationality looks like. Three counterarguments, named upfront Three objections will arise from informed readers, and we name them now so the rest of the section can address them in their proper places. Objection 1 (the hierarchical Bayes objection). Hierarchical Bayesian models with uncertain hyperparameters are well-established. They do not, in general, produce heuristic-like policies. Why should we use our framework? The answer is that FEH adds two conjunctive ingredients that hierarchical Bayes alone does not supply: (a) the explicit accounting of meta-precision divergence as a free-energy cost in sequential cue integration, and (b) the evaluation of policies by expected free energy rather than by expected utility under the marginal posterior. The combination of these ingredients with hierarchical Bayes is what produces the cue-truncation result. Hierarchical Bayes alone does not. Objection 2 (the resource-rational objection). Resource-rational analysis [15] and the broader computational-rationality program already explain why bounded computation can justify simple heuristics. What does FEH add? The answer is that resource-rationality treats computation as a finite budget the agent allocates against expected utility gain; the cost of complex computation justifies the cost of simple policies. FEH adds a substantively different cost channel: an epistemic cost (the meta-precision divergence) that exists even with unlimited computation. Under meta-uncertainty, the optimal policy is heuristic-shaped not because computation is expensive, but because additional inference is epistemically counterproductive. FEH and resource-rationality are complementary accounts, not competitors; both can apply simultaneously, with different signatures. Objection 3 (the empirical mixed-results objection). Less-is-more effects are real but not universal, and the take-the-best literature has produced mixed empirical results. Doesn’t this undercut the framework? The answer is that this is exactly what FEH predicts. The cue-truncation theorem holds in the meta-uncertainty regime σ 2_τ > τ _regime; outside this regime, full Bayesian integration is optimal. The empirical heterogeneity of less-is-more findings is consistent with regime-conditional theory, and FEH provides a path to discriminating it from theories predicting universal less-is-more (which the literature does not support) or universal more-is-more (which the literature also does not support). Roadmap for §2 The rest of §2 develops the theory in seven moves. §2.1 introduces a discrete binary toy model in which the cue-truncation result is intuitive and motivates the rest of the formalism. §2.2 generalizes notation and the generative model. §2.3 promotes the prior precision to a random variable. §2.4 derives the sequential inference machinery and the accumulating meta-precision divergence. §2.5 decomposes the expected free energy into information-gain and meta-precision-cost terms. §2.6 states and proves the cue-truncation theorem. §2.7 establishes structural equivalence to take-the-best. §2.8 positions FEH against the major adjacent frameworks. §2.9 situates the framework within the source disciplines of Friston and Gigerenzer. §2.10 enumerates open mathematical questions for follow-up work. 2.2 The Binary Toy Model Before introducing the continuous formalism that powers the rest of the section, we develop the central intuition in the discrete binary setting that motivates take-the-best. The binary toy model is not a special case of the Gaussian-Gamma model that follows; it is a parallel construction that exhibits the same cue-truncation behavior under meta-uncertainty. Its function in the section is pedagogical: it shows that the FEH effect is not an artifact of Gaussian assumptions but rather reflects a structural feature of sequential inference under uncertain priors. Setup Consider a binary choice problem with a latent state s∈0,1s∈ 0,1. We may interpret s= 1s\ =\ 1 as the proposition “option A is better than option B” in a comparative judgment task. The agent observes a sequence of binary cues c1,c2,…,cK∈0,1c_1,c_2,…,c_K∈ 0,1 drawn in some order. Each cue cjc_j is informative about s through its cue validity in the Gigerenzer–Goldstein sense: Vj=P(s= 1|cj= 1)V_j\ =\ P(s\ =\ 1\ |\ c_j\ =\ 1) (2.1.1) In the empirical literature [6, 7], validity is operationalized over a reference environment of judgment trials as: vj=Ncorrect/(Ncorrect+Nincorrect)v_j=N_correct/ (N_correct+N_incorrect ) (2.1.2) where NcorrectN_correct and NincorrectN_incorrect count, respectively, the trials in which the cue j discriminates correctly and incorrectly between the two options. Validity is an environmental property: it characterizes the local predictive value of a cue j in the ecology in which the agent must decide. This is distinct from ecological rationality, which characterizes the higher-level fit between an agent’s strategy and its environment. The two should not be conflated. Prior over the binary state and its meta-uncertainty Let the agent’s prior over the binary state be parameterized by a probability p0∈(0,1)p_0∈(0,1) so that P(s= 1|p_0)=p_0P(s\ =\ 1\ |\ p\_0)\ =\ p\_0. In standard Bayesian treatments, p0p_0 is fixed. The FEH generalization promotes p0p_0 to a random variable with a prior distribution p0∼Beta(α0,β0)p_0 Beta ( _0, _0 ) (2.1.3) with mean μ0=α0/(α0+β0) _0= _0/ ( _0+ _0 ) and concentration κ0=α0+β0 _0= _0+ _0. The concentration κ0 _0 is the binary analogue of precision: large κ0 _0 corresponds to a tightly concentrated prior, small κ0 _0 - to substantial meta-uncertainty about p0p_0. Definition 2.1.1 (binary meta-uncertainty regime). In the binary toy model, the agent is in the meta-uncertainty regime when the concentration κ0 _0 of the hyperprior on p0p_0 is small relative to the cue budget K. Operationally, the regime is characterized by the agent updating its posterior over p0p_0 substantially over the course of cue integration, meaning the agent learns about its own prior reliability from the cues themselves, and this learning carries a divergence cost. Sequential cue integration with meta-uncertainty Under the standard Bayesian update (no meta-uncertainty), the posterior on s after observing cues c1:kc_1:k factorizes cleanly: P(s=1|c1;k,p0)∝p0⋅∏jLj(cj)P (s=1|c_1;k,p_0 ) p_0· _jL_j (c_j ) (2.1.4) where Lj(cj)=vjcj(1−vj)1−cjL_j (c_j )=v_j^c_j (1-v_j )^1-c_j if s= 1s\ =\ 1 and the corresponding switched form if s= 0s\ =\ 0, under a symmetric error model on cue likelihoods. Under meta-uncertainty, the agent must marginalize over the hyperprior on p0p_0: P(s= 1|c1:k)=∫P(s= 1|c1;k,p0)⋅p(p0|c1;k)dp0P(s\ =\ 1\ |\ c_1:k)\ =\ \ P(s\ =\ 1\ |\ c_1;k,\ p_0)\ ·\ p(p_0\ |\ c_1;k)\ dp_0 (2.1.5) Critically, the hyperprior itself updates with each cue: the cues are indirect evidence about p0p_0, because they jointly constrain which values of p0p_0 are consistent with the observed cue pattern. Let p(p0|c1:k)p (p_0 |c_1:k ) denote the updated hyperprior after k cues. The KL divergence of this updated hyperprior from the original is the binary analogue of the meta-precision divergence introduced in §2.4: Δmeta(k)=KL[p(p0|c1:k)∥p(p0)] _meta(k)=KL[p (p_0 |c_1:k ) . p (p_0 ) ] (2.1.6) Lemma 2.1.2 (expected monotone binary meta-divergence, restated in v0.5). In the binary toy model under (2.1.3)–(2.1.5), the expected meta-divergence E[Δmeta(k)]E[ _meta(k)] is non-decreasing in k. Equivalently, E[Δmeta(k+1)]−E[Δmeta(k)]=I(p0;ck+1∣c1:k)≥0E[ _meta(k+1)]-E[ _meta(k)]=I(p_0;\,c_k+1 c_1:k)≥ 0, with equality iff cue k+1k+1 carries no conditional information about p0p_0 given the prior cues. The earlier wording of Lemma 2.1.2 (sample-wise monotonicity) is false in general: opposite cues can cancel, returning the posterior on p0p_0 exactly to the prior; an explicit counterexample with v=(0.9,0.9)v=(0.9,0.9) and uniform hyperprior is given in Appendix A.1. The proof is by martingale-convexity of KL: the variational posterior p(p0∣c1:k+1)p(p_0 c_1:k+1) is a martingale in k (its conditional expectation given c1:kc_1:k equals p(p0∣c1:k)p(p_0 c_1:k)), and the KL divergence from a fixed reference distribution is convex in its first argument, so Jensen’s inequality applies. Detailed proof is given in Appendix A.1. The intuition behind cue truncation, in binary form With Lemma 2.1.2 in hand, the cue-truncation intuition becomes transparent. Each cue contributes two things to the expected free energy: (i) A positive expected information gain E[I(k)]≥0E[I(k)]≥ 0, the reduction in entropy of the posterior over s, which is the largest for the most valid cue and decreases by diminishing returns as more cues are integrated. (i) A negative expected meta-divergence increment E[C(k)]=E[Δmeta(k)]−E[Δmeta(k−1)]≥0E[C(k)]=E[ _meta(k)]-E[ _meta(k-1)]≥ 0, the divergence accumulated in expectation by becoming more committed to a particular posterior over the agent’s own prior reliability. In the binary case, the most valid discriminating cue typically produces a posterior over s that is already concentrated near 0 or 1. The marginal information gain from a second discriminating cue is therefore small; it can only sharpen an already-sharp posterior. Meanwhile, the meta-divergence cost E[C(k)]E[C(k)] is bounded below by a non-zero quantity proportional to the inverse hyperprior concentration. The crossover point k∗k^*, beyond which E[I(k)]<E[C(k)]E[I(k)]<E[C(k)] and integrating further cues strictly increases expected free energy, exists generically. Proposition 2.1.3 (three-regime structure in the binary toy model, restated in v0.5). Fix a validity profile v1:Kv_1:K and prior mean μ0 _0. There exist critical concentrations 0<κlo(v1:K,μ0)≤κhi(v1:K,μ0)0< _lo(v_1:K, _0)≤ _hi(v_1:K, _0) such that the EFE-minimizing policy k∗(κ0)k^*( _0) has the structure: (don’t-observe regime) For κ0<κlo _0< _lo: k∗=0k^*=0. The meta-divergence cost of even one cue exceeds its information value about s. (one-cue stopping regime ; TTB) For κlo≤κ0≤κhi _lo≤ _0≤ _hi: k∗=1k^*=1. The first cue is integrated; subsequent cues incur higher meta-cost than information benefit. This is the take-the-best stopping rule. (full-integration regime) For κ0>κhi _0> _hi: k∗=Kk^*=K. Each cue contributes more state-information than meta-cost; the agent integrates all available cues. The intermediate one-cue regime [κlo,κhi][ _lo, _hi] is closed and may be narrow. Numerical sweep over 1875 parameter combinations finds 40 with strict k∗=1k^*=1 (Appendix A.1, §A.1.5); these cluster around steep validity gradients (e.g., v1≈0.99,vj>1≤0.55v_1≈ 0.99,v_j>1≤ 0.55) and κ0≈1 _0≈ 1. For uniform-validity profiles, κlo=κhi _lo= _hi and the transition jumps directly from k∗=0k^*=0 to k∗=Kk^*=K without an intermediate regime. The narrowness of the one-cue regime in the binary toy model is a substantive finding; the cue-truncation theorem fires at the boundary between regimes in the binary case, not as a wide intermediate band. A subsidiary result (Appendix A.1, §A.1.5): for steep validity gradients, the first cue alone contributes ≥ 99% of the total attainable information about s across the full cue budget. Even in regimes where k∗=Kk^*=K strictly minimizes EFE, a TTB-style “stop after the first cue” policy is near-optimal in expected reward terms. This near-optimality is the empirically-relevant content of the TTB connection in the binary setting; the strict EFE-optimality in the narrow [κlo,κhi][ _lo, _hi] window is the formally-derivable but operationally-fragile content. The cleaner and more robust derivation of the TTB connection lives in §2.7 via the Gaussian-Gamma machinery, which produces a wider intermediate cue-truncation regime (Theorem 2.6.1 and Appendix A.3). Why the binary model is not enough The binary toy model motivates the FEH effect and establishes the qualitative three-regime structure that the rest of the section generalizes. It is not, however, sufficient for the section’s load-bearing claims. Four reasons. First, real cognitive and computational tasks rarely admit clean binary state spaces; the FEH framework must handle continuous, high-dimensional latent variables. Second, the binary case obscures the role of precision as a continuous parameter; it is treated as a Beta concentration, which is closely related but less naturally analyzed within the active-inference framework that grounds the rest of the section. Third, the empirical predictions for LLMs require a generative model that can be matched to LLM behavior, which the Gaussian-Gamma machinery supports more directly. Fourth, and most importantly, the strict cue-truncation regime (k∗=1k^*=1) is a knife-edge in the binary setting (Proposition 2.1.3 above): the cue-truncation theorem is robust as a wide intermediate regime only in the continuous Gaussian-Gamma model of §2.3–2.6. For these reasons, the rest of §2 develops the continuous Gaussian-Gamma formalism. The binary toy model should be read as motivating intuition and a near-optimality argument for TTB; the strict TTB derivation lives in §2.7 via the Gaussian-Gamma machinery, which is the analytical workhorse and the load-bearing setting for the section’s main results. 2.3 Notation and General Generative Model We now generalize from the binary toy model to a generative model framework adequate for the rest of the section. The framework is the standard active-inference framework of Friston [4], Friston et al. [5], and Parr, Pezzulo, and Friston [17], with the meta-uncertainty extension developed in §2.3. State, observation, cue, action Let s∈Ss\ ∈\ S denote the latent state, with S a finite or compact set. The binary case S=0,1S=0,1 of §2.1 is one specialization; we now treat S as arbitrary unless otherwise noted. The agent does not observe s directly. Cues c1,…,cK∈Cc_1,…,c_K∈ C are informative about s through cue likelihoods p(cj|s,τc,j)p (c_j |s, _c,j ), with cue-specific precisions τc,j _c,j assumed to be known to the agent. An outcome o∈Oo\ ∈\ O is the consequence of action a∈Aa\ ∈\ A given true state s, with likelihood p(o|s,a)p(o\ |\ s,\ a). Actions are evaluated using the expected free energy, as defined in §2.5. Generative model The active-inference generative model used throughout this section is p(o,s,c1:K,τ|a)=p(o|s,a)⋅p(s|τ)⋅p(τ)⋅∏j=1Kp(cj|s,τ,γj)p(o,\ s,\ c_1:K,\ τ\ |\ a)\ =\ p(o\ |\ s,\ a)\ ·\ p(s\ |\ τ)\ ·\ p(τ)\ ·\ _j=1^K\ p(c_j\ |\ s,\ τ,\ _j) (2.2.1) with three commitments. (1) State prior with random precision. p(s|τ)=(s;μ,τ−1Σ0)p(s\ |\ τ)\ =\ N(s;\ μ,\ τ^-1 _0). The shape matrix Σ0 _0 is fixed; the scalar τ> 0τ\ >\ 0 is the agent’s meta-precision, a single random variable governing the overall scale of inference confidence. (2) Cue likelihood with shared meta-precision and intrinsic scaling. Each cue is a noisy linear measurement of state: p(cj|s,τ,γj)=(cj;Ajs,(τγj)−1Idc)p(c_j\ |\ s,\ τ,\ _j)\ =\ N(c_j;\ A_js,\ (τ\, _j)^-1\,I_d_c) (2.2.2) where AjA_j is the cue-to-state observation matrix and γj> 0 _j\ >\ 0 is the intrinsic precision of cue j ; a fixed structural property of the cue, estimable from training-data statistics, distinct from but related to the cue validity vjv_j of §2.1. (3) Hyperprior over meta-precision. p(τ)=Gamma(τ;α0,β0)p(τ)\ =\ Gamma(τ;\ _0,\ _0), with E[τ]=τ0=α0/β0E[τ]\ =\ _0\ =\ _0/ _0 and Var[τ]=α0/β02=τ02/α0Var[τ]\ =\ _0/ _0^2\ =\ _0^2/ _0. The principal departures of FEH from the standard active-inference framework lie in the second and third commitments. The standard treatment fixes τ as a hyperparameter and treats cue precisions as separate fixed quantities τc,j _c,j. FEH ties cue precisions to the meta-precision via τγjτ\, _j and treats τ as a full random variable. The intrinsic-precision factors γj _j retain the cue-specific structure of the standard model while making the agent’s overall calibration uncertain in a single scalar parameter. Interpretive remark. The asymmetry that v0.2 acknowledged — “uncertain prior precision but known cue precisions” (cf. open question Q4 in v0.3) — dissolves under this specification. All precisions in the model share the meta-precision scale τ; cue-specific structure lives in the deterministic intrinsic-precision factors γj _j. This is a more faithful operationalization of the meta-uncertainty story: the agent is uncertain about the overall reliability of its inference apparatus, not selectively about its prior. The intrinsic precisions γj _j are properties of the cue, learnable from data; τ is the agent’s calibration confidence, which can be genuinely uncertain in novel contexts. §2.3 develops the meta-precision generalization in detail. Cue validity revisited The cue validity vjv_j defined in (2.1.1) generalizes to non-binary states as the conditional probability that cue cjc_j takes its discriminating value given state s in some reference partition of S. For multi-class S, validity becomes a vector quantity, with one component per state. For the rest of §2 we follow the binary convention except where multi-class generalization is explicitly needed. Notation summary ,s,\ S , latent state and state space (binary in §2.1; general elsewhere) c_j, K\ , individual cue and total number of available cues ,a,\ A , action and action set ,o,\ O\ , outcome and outcome set τ\ , meta-precision; single scalar random variable governing the overall scale of both the state prior and the cue likelihoods (§2.3) γ _j\ , intrinsic precision of cue j ; fixed structural property; the effective precision of cue j is τγjτ\, _j (τ)p (τ ) , hyperprior over τ; Gamma(α _0, β _0) for analytical tractability τ,σ2τ _0,\ σ 2_τ , mean and variance of the hyperprior over τ ,κp_0,\ _0 , binary prior probability and Beta concentration (binary case, §2.1) (),(τ)q (s ),q (τ ) , variational posterior factors under mean-field factorization ,F,\ G , variational free energy and expected free energy v_j , cue validity of cue j (Gigerenzer–Goldstein, equation 2.1.1) 2.4 Meta-Uncertainty over Prior Precision We now state the central generalization of the active-inference framework that defines FEH. The conceptual content was previewed in §2.0; this section formalizes it. From fixed precision to random precision Let the prior over latent states take the precision-modulated Gaussian form p(s|τ)=N(s;μ,τ−1Σ0)p(s\ |\ τ)\ =\ N(s\ ;\ μ,\ τ^- 1\ _0) (2.3.1) with prior mean μ and fixed shape matrix Σ0 _0. Larger τ yields a tighter, more confident prior; the limit τ→∞τ\ →\ ∞ recovers a point prior at μ, and τ→ 0τ\ →\ 0 yields an essentially flat prior over the state space. In the standard active-inference treatment, τ is a fixed hyperparameter, possibly modulated by attention or learned as a point estimate from data. In FEH, τ is a random variable with prior p(τ)=Gamma(τ;α0,β0)p(τ)=Gamma (τ; _0, _0 ) (2.3.2) so that E[τ]=α0/β0≡τ0andVar[τ]=α0/β02≡στ2E[τ]= _0/ _0≡ _0andVar[τ]= _0/ _0^2≡ _τ^2. The choice of Gamma is dictated by conjugacy with the Gaussian likelihood in (2.3.1); other choices (log-normal, inverse-Gamma, half-Cauchy) are considered as a sensitivity check in §2.10, Q2. Definition 2.3.1 (continuous meta-uncertainty regime). The agent operates in the meta-uncertainty regime when σ2τσ 2_τ is non-negligible relative to τ02 _0 2. The precise threshold separating high and low meta-uncertainty is derived in §2.5. Heuristically, the regime corresponds to the agent being substantively uncertain about how much weight to place on its own prior, the continuous analogue of small Beta concentration in the binary toy model. Marginal prior over states Integrating τ out of the joint prior yields the marginal prior over states: p(s)=∫p(s|τ)p(τ)τp(s)\ =\ \ p(s\ |\ τ)\ p(τ)\ dτ (2.3.3) For the conjugate Gaussian–Gamma pair (2.3.1)–(2.3.2), this integral admits a closed form: the marginal p(s)p(s) is a multivariate Student-t distribution p(s)=St(s;μ,(β0/α0)Σ0,ν=2α0)p(s)=St (s;μ, ( _0/ _0 ) _0,ν=2 _0 ) (2.3.4) with ν=2α0ν=2 _0 degrees of freedom. As στ2→0 _τ^2→ 0 (equivalently α0→∞ _0→∞ with τ0 _0 held fixed), the Student-t collapses to a Gaussian N(μ,τ0−1Σ0))N(μ,\ _0^- 1\ _0)), recovering the fixed-precision case. For finite α0 _0, the marginal prior has heavier-than-Gaussian tails, a structural feature that connects FEH to the broader power-law statistics characteristic of natural cognition (see §2.9). Connection to KMM smooth ambiguity. Equation (2.3.2) is the FEH operationalization of the KMM second-order distribution. Where KMM treats the second-order distribution μ over first-order probability measures abstractly, FEH commits to μ being the Gamma distribution over a precision scalar. This is a substantive restriction of the KMM framework: not all ambiguity is precision-ambiguity, and FEH does not claim otherwise. The advantage is computational and empirical tractability; the cost is that some ambiguity phenomena fall outside the FEH operationalization. Whether the precision-ambiguity restriction is empirically adequate for the LLM regime is an open question we will return to in the empirical section. 2.5 Sequential Inference under Meta-Uncertainty We now formalize the sequential cue-integration setting that supports the cue-truncation theorem. The agent receives cues. c1,…,cKc_1,…,c_K one at a time and updates a variational posterior q(s,τ)q(s,τ) after each cue. The question is: what is the total free-energy cost of integrating k cues under meta-uncertainty, and how does that cost decompose? Variational factorization Under the standard mean-field assumption, the variational posterior over states and meta-precision factorizes: q(s,τ)=q(s)q(τ)q(s,τ)=q(s)q(τ) (2.4.1) This is an approximation. Its accuracy is itself a function of στ2 _τ^2 and constitutes Q1 of the open questions in §2.10. For the leading-order analysis that supports Theorems 2.6 and 2.7, the mean-field approximation is sufficient. A structured variational family will be required for tight bounds and is left to follow-up work. Posterior updates after k cues For the generative model (2.2.1)–(2.2.2) with mean-field factorization (2.4.1), standard variational Bayes (VBEM) yields closed-form posterior updates for both factors. State factor. q(s|c1:k)=(s;μk,Σk),Σk−1=Eq(τ)[τ](Σ0−1+∑j=1kγjAj⊤Aj)q(s\ |\ c_1:k)\ =\ N(s;\ _k,\ _k),\ \ _k^-1\ =\ E_q(τ)[τ]\, ( _0^-1\ +\ _j=1^k\, _j\,A_j A_j ) (2.4.2) The meta-precision Eq(τ)[τ]E_q(τ)[τ] factors out as a global scale on a deterministic information matrix built from cue intrinsic precisions γj _j. The posterior mean μk _k follows from the standard Gaussian-update formula, with each cue contributing γjAj⊤cj _jA_j c_j (weighted by E[τ]E[τ]). Meta-precision factor. q(τ|c1:k)=Gamma(τ;α0′+k2,β0′+12∑j=1kγjMj)q(τ\ |\ c_1:k)\ =\ Gamma (τ;\ _0 \ +\ k2,\ \ _0 \ +\ 12 _j=1^k\, _j\,M_j ) (2.4.3) where: • α0′=α0+ 1/2 _0 \ =\ _0\ +\ 1/2 and β0′=β0+Vs/2 _0 \ =\ _0\ +\ V_s/2 absorb the one-time contribution of the state prior p(s|τ)p(s|τ) to the variational update; • Vs=Eq(s)[(s−μ)⊤Σ0−1(s−μ)]V_s\ =\ E_q(s)[(s-μ) _0^-1(s-μ)] is the variational expected residual against the state prior; • Mj=Eq(s)[(cj−Ajs)⊤(cj−Ajs)]M_j\ =\ E_q(s)[(c_j-A_js) (c_j-A_js)] is the variational expected residual against cue j. The defining feature of (2.4.3) is that the shape parameter grows linearly in k: each cue contributes +1/2+1/2 to the shape α and +γjMj/2+ _jM_j/2 to the rate β. This is the formal seat of the meta-precision tightening that drives the cue-truncation result of §2.6. Derivation: Appendix A.2. The accumulating meta-precision divergence The KL divergence of the meta-precision posterior from its prior is Δmeta(k)≡KL[q(τ|c1:k)∥p(τ)] _meta(k)\ ≡\ KL[\,q(τ\ |\ c_1:k)\ \ p(τ)\,] (2.4.4) For Gamma–Gamma conjugate pairs this admits a closed-form expression in terms of the digamma ψ and log-gamma logΓlog functions: Δmeta(k)=(αk−α0)ψ(αk)−logΓ(αk)Γ(α0)+α0logβkβ0+αkβ0−βkβk _meta(k)\ =\ ( _k- _0)\,ψ( _k)\ -\ ( _k) ( _0)\ +\ _0\, _k _0\ +\ _k\, _0- _k _k (2.4.5) with αk=α0′+k/2 _k\ =\ _0 +k/2 and βk=β0′+12∑j=1kγjMj _k\ =\ _0 + 12 _j=1^k _jM_j. Asymptotically in k, Δmeta(k)∼(k/2)log(k)+O(k) _meta(k)\ \ (k/2)\,log(k)\ +\ O(k) ; i.e., super-linear in k, dominated by the digamma growth of the shape contribution. The qualitative content is that each additional cue commits the agent more strongly to a particular posterior over its own meta-precision, and that commitment carries an explicit, monotonically-growing free-energy cost. Lemma 2.4.1 (expected monotonicity of meta-precision divergence, restated in v0.5). Under the generative model (2.2.1)–(2.2.2) with mean-field factorization (2.4.1), the expected meta-precision divergence E[Δmeta(k)]E[ _meta(k)] is non-decreasing in k. Equivalently, E[Δmeta(k+1)]−E[Δmeta(k)]=I(τ;ck+1∣c1:k)≥0E[ _meta(k+1)]-E[ _meta(k)]=I(τ;\,c_k+1 c_1:k)≥ 0, with equality iff cue k+1k+1 carries no conditional information about τ given the prior cues. The earlier wording (sample-wise monotonicity) is false in general: numerical simulation across 1000 random MjM_j trajectories at (α0′,β0′)=(2,1)( _0 , _0 )=(2,1), γj=1 _j=1, K=20K=20 exhibits 979 paths with at least one strict decrease of Δmeta _meta along the trajectory. This occurs because the Gamma KL surface is non-monotone in (α,β)(α,β) along arbitrary trajectories: when β grows faster than α, the posterior mean shifts away from the prior mean, but a subsequent cue with smaller MjM_j can pull it back. Proof sketch. By the tower property of conditional expectation, Eck+1∣c1:k[q(τ∣c1:k+1)]=q(τ∣c1:k)E_c_k+1 c_1:k[q(τ c_1:k+1)]=q(τ c_1:k). By joint convexity of KL in its first argument, Jensen’s inequality gives Eck+1∣c1:k[KL[q(τ∣c1:k+1)∥p(τ)]]≥KL[q(τ∣c1:k)∥p(τ)]E_c_k+1 c_1:k[KL[q(τ c_1:k+1)\, \,p(τ)]]≥ KL[q(τ c_1:k)\, \,p(τ)]. Taking outer expectation over c1:kc_1:k preserves the inequality. Detailed computation: Appendix A.2. 2.6 The Expected Free Energy Decomposition We now assemble the central decomposition of expected free energy under meta-uncertainty. This decomposition is the lemma that supports the cue-truncation theorem of §2.6. Standard expected free energy For action a and a policy of integrating k cues before committing, the expected free energy is G(a,k)=Eq(o,s,τ|a,c1,…,cK)[logq(s,τ|a,c1,…,cK)−logp(o,s,τ|ac1,…,cK)]G(a,\ k)\ =\ E_\q(o,s,τ|a,c_1,…,c_K)\\ [log\ q(s,\ τ\ |\ a,\ c_1,…,c_K)\ -\ log\ p(o,\ s,\ τ\ |\ ac_1,…,c_K)] (2.5.1) Expanding, applying the mean-field factorization (2.4.1), and rearranging: G(a,k)=−Eq[logp(o|s)]+KL[q(s|a,c1,…,cK)∥⟨p(s|τ)⟩q(τ)]+Δmeta(k)G(a,\ k)\ =\ -E_q[log\ p(o\ |\ s)]\ +\ KL[q(s\ |\ a,\ c_1,…,c_K)\ \ p(s\ |\ τ) _\q(τ)\]\ +\ _meta(k) (2.5.2) The three terms admit standard active-inference interpretations: (i) Pragmatic value. Expected log-likelihood of the outcome under predicted states. (i) Epistemic value. KL of state posterior from marginalized state prior; the information the agent gains about s from integrating cues. (i) Meta-precision cost. KL of meta-precision posterior from its prior, accumulating in k as established in Lemma 2.4.1. This term is the FEH-specific contribution and is the seat of the meta-uncertainty effect. Marginal benefit of the k-th cue Define the marginal benefit of adding the k-th cue: ΔG(k)≡G(a,k−1)−G(a,k) G(k)≡ G(a,k-1)-G(a,k) (2.5.3) From (2.5.2), this decomposes as ΔG(k)=I(k)−C(k) G(k)=I(k)-C(k) (2.5.4) where I(k)I(k) is the expected information gain (epistemic value increment) and C(k)=Δmeta(k)−Δmeta(k−1)C(k)= _meta(k)- _meta(k-1) is the marginal meta-precision cost. By Lemma 2.4.1, C(k)>0C(k)>0 in the meta-uncertainty regime. By data processing inequality and standard Bayesian information theory, I(k)I(k) is non-negative and exhibits diminishing returns in k for non-redundant cue streams. Proposition 2.5.1 (two-regime structure of marginal benefit). The marginal benefit ΔG(k)=I(k)−C(k) G(k)=I(k)-C(k) decomposes additively into a non-negative information gain I(k)I(k) that is decreasing in k by diminishing returns, and a non-negative meta-precision cost C(k)C(k) that is strictly positive when στ2>0 _τ^2>0. There exists a critical cue index k∗k^* such that: For k≤k∗,ΔG(k)≥ 0k\ ≤\ k^*\ ,\ \ G(k)\ ≥\ 0\ (integrating the k-th cue reduces expected free energy) For k>k∗,ΔG(k)<0k>k^*, G(k)<0 (integrating the k-th cue increases expected free energy) The threshold τ _regime separating regimes The critical cue index k∗k^* depends on στ2 _τ^2. The boundary between the two regimes is the smallest στ2 _τ^2 at which the marginal expected meta-cost of the K-th cue exceeds the marginal expected info gain: τregime≡infστ2> 0:I(τ;cK∣c1:K−1)>I(s;cK∣c1:K−1) _regime\ ≡\ \\, _τ^2\ >\ 0\ :\ I(τ;\,c_K c_1:K-1)\ >\ I(s;\,c_K c_1:K-1)\, \ (2.5.5) equivalently, the smallest στ2 _τ^2 such that the EFE-optimal stopping point falls strictly inside the cue budget: k∗(στ2)<Kk^*( _τ^2)<K. For στ2<τregime _τ^2< _regime, the marginal info gain dominates the marginal meta-cost at every k≤Kk≤ K and full cue integration is optimal (standard Bayesian inference). For στ2≥τregime _τ^2≥ _regime, k∗<Kk^*<K and optimal inference is truncated. Operationalization for empirical work. For the Gaussian-Gamma model with mean cue intrinsic precision γ¯ γ and mean cue residual m¯ m, τregime _regime is the unique positive root of I(K)=C(K;α0′=τ02/τregime)I(K)=C(K;\, _0 = _0^2/ _regime) where I(K)I(K) is the asymptotic info gain at cue K (calibrated from cue-validity statistics) and C(K;⋅)C(K;\,·) is the marginal Gamma KL contribution at cue K (closed form via digamma; see Appendix A.3). This is computable per benchmark item once (c1,λ)(c_1,λ) for I and (γ¯,m¯)( γ, m) for C are estimated. The empirical section will operationalize this estimation procedure for the LLM benchmark. Well-definedness of the threshold. The two characterizations in (2.5.5) — the last-cue crossing I(τ;cK∣c1:K−1)>I(s;cK∣c1:K−1)I(τ;\,c_K c_1:K-1)>I(s;\,c_K c_1:K-1), and k∗(στ2)<Kk^*( _τ^2)<K — coincide and pick out a single value. (Despite the τ subscript, τregime _regime is a critical value of the meta-uncertainty variance στ2 _τ^2, not of a precision.) Write the marginal meta-cost C¯(K)=I(τ;cK∣c1:K−1) C(K)=I(τ;\,c_K c_1:K-1) and the marginal info gain I¯(K)=I(s;cK∣c1:K−1) I(K)=I(s;\,c_K c_1:K-1). As στ2 _τ^2 grows, the meta-precision prior becomes more diffuse, so each cue carries more information about τ and C¯(K) C(K) is non-decreasing in στ2 _τ^2; I¯(K) I(K) is fixed by the cue-validity profile and is, to leading order, independent of στ2 _τ^2. The crossing function g(στ2)=C¯(K)−I¯(K)g( _τ^2)= C(K)- I(K) is therefore increasing and crosses zero exactly once, so the infimum in (2.5.5) is attained at a unique root. That this last-cue crossing coincides with k∗<Kk^*<K follows from Theorem 2.6.1(b): the marginal benefit E[ΔG(k)]=I¯(k)−C¯(k)E[ G(k)]= I(k)- C(k) is monotone-decreasing in k (diminishing info gain against non-decreasing meta-cost), so the K-th cue is the first to become unprofitable exactly when the optimal stop falls strictly inside the budget. Both the single crossing and the agreement of the two definitions (to grid resolution) are confirmed numerically across a sweep of στ2 _τ^2 (verify_meanfield_and_tau_regime.py, Part B). 2.7 The Cue-Truncation Theorem Theorem 2.6.1 (cue-truncation under meta-uncertainty, restated in v0.5). Let an active-inference agent operate under the generative model (2.2.1)–(2.2.2) with mean-field variational posterior (2.4.1), integrating cues sequentially. Then in expectation under the agent’s predictive distribution: (a) When στ2<τregime _τ^2< _regime, the expected free energy E[G(a,k)]E[G(a,k)] is monotonically non-increasing in k up to k=Kk=K. Full cue integration is optimal. (b) When στ2≥τregime _τ^2≥ _regime, there exists a finite k∗<Kk^*<K such that E[G(a,k)]E[G(a,k)] is decreasing for k≤k∗k≤ k^* and non-decreasing for k>k∗k>k^*. Truncated cue integration at k∗k^* is optimal. (c) Under the regularity condition Cov(vj,γj)≥0Cov(v_j, _j)≥ 0 (cue intrinsic precisions are not anti-correlated with cue validities), the optimal cue ordering, that minimizing E[G(a,k∗)]E[G(a,k^*)], is by descending validity, v(1)≥v(2)≥…≥v(K)v_(1)≥ v_(2)≥…≥ v_(K). Sample-wise versus expectation form. The theorem is stated and proved in expectation form. Sample-wise, G(a,k)G(a,k) trajectories under random cue realizations are typically not unimodal: across 1000 random parameter configurations, only 198 sample-wise G(k)G(k) trajectories exhibit a single sign change in their first differences; the remainder show multi-sign-change wiggly trajectories driven by noise in individual cue realizations. Active inference defines the optimal policy as argminkE[G(k)]argmin_k\,E[G(k)], so the expectation-form theorem is what supports the policy claim. Numerical verification across five (α0′,β0′,γ¯,m¯)( _0 , _0 , γ, m) regimes confirms E[G(a,k)]E[G(a,k)] is U-shaped with k∗k^* dropping monotonically as στ2 _τ^2 rises (from k∗=Kk^*=K at στ2=0.05 _τ^2=0.05 to k∗=5k^*=5 at στ2=2.0 _τ^2=2.0, with K=30K=30). Proof sketch. Part (a) follows from E[C(k)]→0E[C(k)]→ 0 as στ2→0 _τ^2→ 0, reducing E[ΔG(k)]E[ G(k)] to standard expected info gain about s, non-negative by data-processing inequality. Part (b) follows from monotonically-decreasing marginal expected info gain I¯(k)=I(s;ck∣c1:k−1) I(k)=I(s;c_k c_1:k-1) (diminishing returns) combined with non-negative marginal expected meta-cost C¯(k)=I(τ;ck∣c1:k−1) C(k)=I(τ;c_k c_1:k-1) bounded below by a positive constant in the high-meta-uncertainty regime; their difference E[ΔG(k)]=I¯(k)−C¯(k)E[ G(k)]= I(k)- C(k) changes sign exactly once. Part (c) follows from a greedy argument: the marginal expected info gain about s is monotone-increasing in vjv_j, and the marginal expected meta-cost scales (in leading order) with γj _j; under the non-anticorrelation regularity, descending-validity ordering simultaneously maximizes the info-gain term and approximately equalizes the meta-cost term across selections. Full proof in Appendix A.3. Note on the cue-precision specification. Part (c) refers to the cue intrinsic precisions γj _j of (2.2.2), which are deterministic structural properties of each cue (estimable from training-data statistics). The effective precision τγjτ\, _j inherits the uncertainty of the shared meta-precision τ; the cue-specific γj _j does not. This is the natural and symmetric specification: all precisions in the model share the meta-precision scale, while cue-specific structure lives in the deterministic γj _j factors. The asymmetry between “uncertain prior precision” and “known cue precision” raised in earlier drafts (Q4 in v0.3) has accordingly been retired. 2.8 Equivalence to Take-the-Best Theorem 2.6.1 establishes that, under meta-uncertainty, optimal inference involves the sequential integration of validity-ordered cues, truncated at k∗k^*. We now show that this is structurally identical to the take-the-best (TTB) heuristic of Gigerenzer and Goldstein [6]. The take-the-best procedure TTB decides between two options on the basis of cues by the following procedure: (1) Order cues by validity, highest first. (2) Examine the most valid unused cue. (3) If the cue discriminates, decide based on it and stop. (4) Otherwise, return to step 2. TTB is famously one-reason: a single discriminating cue ends the inference; cues below the discriminating one are ignored regardless of their potential combined informativeness. Structural equivalence Theorem 2.7.1 (FFH–TTB structural equivalence). Let A_FFH be the active-inference agent of Theorem 2.6.1 operating in the meta-uncertainty regime στ2>τregime _τ^2> _regime, with cue ordering by validity and truncation at k∗k^*. Let A_TTB be the take-the-best agent of Gigerenzer and Goldstein [6] operating on the same cues with the same validity ordering. Then A_FFH and A_TTB induce the same structural form of inference policy: validity-ordered sequential cue examination with truncation at the first cue whose marginal information gain drops below the marginal meta-precision cost. Exact identity of action distributions, rather than structural identity, is established in Appendix A.4 under an explicit condition on the cue-validity profile. We state the result here and defer the proof. Theorem 2.7.4 (Exact sample-wise FFH–TTB action identity, new in v0.7). Let Lj:=log(vj/(1−vj))L_j:=log(v_j/(1-v_j)) denote the per-cue log-likelihood ratio for cue j, and order the cues by descending validity. Suppose the Descending Dominance (D) condition holds: for every i∈1,…,K−1i∈\1,…,K-1\, L(i)>∑j=i+1KL(j).L_(i) 6.0mu> 6.0mu _j=i+1^KL_(j). Let A_FFH be the active-inference agent of Theorem 2.6.1 with uniform prior μ0=1/2 _0=1/2 and EFE-optimal truncation k∗≥1k^*≥ 1; let A_TTB be the take-the-best agent. Then for every cue realization c∈−1,+1Kc∈\-1,+1\^K: aFFH(c)=aTTB(c).a_FFH(c) 6.0mu= 6.0mua_TTB(c). The two agents’ marginal action distributions coincide exactly. (D) admits a convenient geometric-decay sufficient form: if L(j+1)≤ρL(j)L_(j+1)≤ρ\,L_(j) for some ρ<1/2ρ<1/2, then (D) holds. (D) is sharp; it is essentially necessary for exact sample-wise identity (a counterexample under (D) violation is exhibited in Appendix A.4.5). Crucially, (D) is a condition on the validity gradient alone; the meta-precision prior tail does not enter, which is a sharper resolution of Q6 than the original conjecture anticipated. The full proof, the necessity argument, and numerical verification (26,696 sample-wise comparisons under 200 random (D)-satisfying profiles, 0 mismatches; predicted mismatches under (D) violation) appear in Appendix A.4 (companion script verify_appendix_A4.py). Tallying as a special case Theorem 2.7.1 establishes structural equivalence to take-the-best under descending cue validity ordering. The natural companion result is for uniform cue validity, which we state and prove here. Tallying, the second-most-studied member of the fast-and-frugal toolbox, works by counting cues that favour each option and choosing the option with the most counts; unlike TTB, it does not weight cues by validity. Theorem 2.7.3 (FFH–Tallying structural equivalence under uniform validity, new in v0.6). Let AFFHA_FFH be the active-inference agent of Theorem 2.6.1 operating in the meta-uncertainty regime with K binary discriminating cues having uniform validity vj=v_j=v for all j. Let ATallyA_Tally be the tallying agent operating on the same cues. Then: (a) The FFH posterior on the comparative state s depends on the cues only through their tally Tk=∑j=1kdjT_k= _j=1^kd_j, where dj∈−1,+1d_j∈\-1,+1\ is the discriminating signal of cue j. Cue ordering is irrelevant. (b) The FFH-optimal action under symmetric 0–1 loss is sign(Tk∗)sign(T_k^*), where k∗k^* is the cue-truncation point of Theorem 2.6.1. (c) When στ2<τregime _τ^2< _regime (low meta-uncertainty), k∗=Kk^*=K and FFH coincides with classical tallying. When στ2≥τregime _τ^2≥ _regime (high meta-uncertainty), k∗<Kk^*<K and FFH coincides with truncated tallying; count the first k∗k^* cues, decide by majority. Proof sketch. Under uniform validity v, the cue likelihoods factor symmetrically: P(d1:k∣s=A)=vn+(k)(1−v)n−(k),P(d1:k∣s=B)=vn−(k)(1−v)n+(k),P(d_1:k s=A)=v^n_+(k)(1-v)^n_-(k), P(d_1:k s=B)=v^n_-(k)(1-v)^n_+(k), where n+(k)n_+(k) and n−(k)n_-(k) count A-favoring and B-favoring cues among the first k. The likelihood ratio is therefore (v/(1−v))n+−n−=(v/(1−v))Tk(v/(1-v))^n_+-n_-=(v/(1-v))^T_k ; a function of the tally Tk=n+(k)−n−(k)T_k=n_+(k)-n_-(k) alone. By the closed-form mixture posterior of Lemma A.1.1 (binary case) or the variational posterior of (2.4.3) (Gaussian-Gamma case), the posterior on s depends on the cues only through this likelihood ratio, hence only through TkT_k. This proves (a). Part (b) follows because, under symmetric loss, the EFE-optimal action maximizes P(s=a∣d1:k∗)P(s=a d_1:k^*), which is monotone in Tk∗T_k^*; therefore, the optimal action is sign(Tk∗)sign(T_k^*). Part (c) follows by composition with Theorem 2.6.1: the cue-truncation point k∗k^* is determined by the same EFE crossover criterion as for TTB, and the decision rule on the integrated cues is sum-based (tallying). Detailed computation: Appendix A.5 (companion document verify_tallying_equivalence.py provides numerical verification of (a) and (b) across multiple validity values and cue counts). Numerical verification. For uniform v=0.75v=0.75, μ0=0.5 _0=0.5, K=5K=5: every permutation of cue sequences with the same tally T produces an identical posterior on s (verified across 30 permutations spanning three tally values; 0 violations). The optimal action equals sign(T)sign(T) for all non-zero T (verified across T∈−5,−3,−1,+1,+3,+5T∈\-5,-3,-1,+1,+3,+5\). Remark 2.7.2 (scope of the heuristic equivalence conjecture, updated in v0.6). Theorem 2.7.1 establishes structural equivalence between FFH and TTB; Theorem 2.7.3 extends this to the case of FFH and tallying under uniform validity. The general conjecture (Q7) is that the family of fast-and-frugal heuristics characterized by the ABC research group corresponds to the family of EFE-optimal sequential inference policies under meta-uncertainty, parameterized by (i) the cue-validity profile (descending → TTB; uniform → tallying), (i) the cue-precision profile, and (i) the stopping criterion (cue-truncation k∗k^* as a function of στ2 _τ^2). Two members of the toolbox are now formally established (TTB, tallying); recognition heuristic, satisficing, and fast-and-frugal trees remain conjectured. Q7 in §2.10 has been correspondingly updated. 2.9 Positioning Against Existing Frameworks FEH sits at the intersection of several established research programs. This section identifies what FEH inherits, what it adds, and where it differs for each. The aim is precise positioning, neither inflated novelty nor false modesty, so that informed readers can locate the contribution accurately. Standard active inference (Friston and colleagues) FEH inherits the full apparatus of active inference under the free energy principle: the variational free energy functional, the expected free energy objective for action selection, the precision-weighted belief updating, and the pragmatic–epistemic decomposition. What FEH adds is the promotion of the prior precision τ to a full random variable with a non-trivial hyperprior and the explicit accounting of the resulting meta-precision divergence as a free-energy cost in sequential cue integration. Standard active-inference treatments treat precision as inferable, but typically in a point-estimation sense. Friston et al. [5] and Parr, Pezzulo, and Friston [17] discuss precision learning but do not develop the full second-order Bayesian treatment that supports the cue-truncation theorem. The novelty of FEH lies in this development and its policy-level consequences, not in a wholesale departure from active inference. Hierarchical Bayesian inference Hierarchical Bayesian models with uncertain hyperparameters are well-established and do not, in general, produce heuristic-like policies. As stated in Objection 1 of §2.0, hierarchical Bayes alone does not suffice; FEH’s specific contribution is the conjunction of three ingredients: (a) hierarchical Bayes with uncertain prior precision specifically; (b) sequential cue integration with explicit per-cue meta-precision divergence accounting; (c) policies evaluated by expected free energy rather than by expected utility under the marginal posterior. The third ingredient is doing more work than is immediately obvious. Expected utility under the marginal posterior would integrate over τ and produce a single “effective” posterior over s, with no explicit cost for the divergence. Expected free energy, by contrast, treats the agent’s epistemic state as a first-class quantity whose KL from the prior is a cost, and this is what gives meta-precision its punch. Replacing G with expected utility would dissolve the FEH result. Resource-rational analysis (Lieder & Griffiths, 2020) Resource-rational analysis explains heuristic-shaped policies as the consequence of bounded computation: an agent with finite cognitive resources allocates them to maximize expected utility net of computational cost. Where computational cost is high, simple policies are optimal. FEH complements this account, not competes with it. FEH adds an epistemic cost channel that exists even with unlimited computation. Under meta-uncertainty, additional inference is counterproductive not because the agent runs out of computation, but because the agent’s posterior over its own reliability accrues a divergence cost with each cue. The two accounts have different empirical signatures. Resource-rationality predicts that heuristic behaviour should attenuate when computation is made cheap (e.g., by extending deliberation time or by externalizing computation to scratch pads). FEH predicts that heuristic behaviour should persist regardless of computational resources when meta-uncertainty is high, because the cost channel driving the result is not computational but epistemic. Both effects can coexist; distinguishing them empirically is an important target for follow-up empirical work. Heuristics-and-biases (Tversky & Kahneman, 1974) The heuristics-and-biases tradition [21] has documented systematic departures from classical norms of probability and expected utility. FEH does not dispute these findings; it reframes a subset of them. Specifically, the heuristics whose biases the tradition has characterized (anchoring, availability, representativeness) can be productively analyzed as policies that are EFE-optimal under specific meta-uncertainty conditions. Whether all heuristics-and-biases findings admit this reframing is an open question; FEH commits only that take-the-best does. Ecological rationality (Gigerenzer and colleagues) FEH is closest in spirit to the ecological rationality program. The argument that simple heuristics can outperform compensatory models in real environments, and that the fit between strategy and environment is the proper unit of rationality evaluation, is one FEH inherits wholesale. What FEH adds is a derivation: ecological rationality is shown to be the structural form of optimal inference under a specific, identifiable regime of uncertainty, rather than asserted as a normative alternative to Bayesian rationality. This dissolves the long-standing apparent opposition between Bayesian and ecological accounts: both are special cases of a single underlying optimization, with the meta-uncertainty regime determining which form dominates. KMM smooth ambiguity (Klibanoff, Marinacci, Mukerji, 2005) FEH operationalizes a specific subclass of the KMM framework: the second-order distribution is taken to be Gamma over a precision scalar, rather than an unrestricted distribution over first-order probability measures. This is a narrower commitment than the general KMM model, but a more analytically tractable one. Both frameworks share the motivating intuition that the agent is uncertain about its first-order distribution, and both produce decision criteria richer than SEU. They differ in what they penalize and how. KMM, under concave ambiguity attitude φ , penalizes variance in expected utility across the second-order distribution: actions whose expected utility is robust to which first-order distribution is correct are preferred over actions whose expected utility varies substantially across plausible first-order distributions. The mechanism is variance-aversion at the level of expected utility. FEH, under the meta-uncertainty regime, penalizes commitment to the meta-precision posterior; the agent pays an explicit free-energy cost for becoming more confident in any particular posterior relative to its own prior reliability. The mechanism is divergence-aversion at the level of the agent’s epistemic state. Result of numerical verification. An earlier draft of this section conjectured that the active-inference agent under meta-uncertainty would implement an emergent KMM-style ambiguity-averse policy, with effective ambiguity attitude determined by the meta-precision hyperprior. This conjecture was tested directly by Monte Carlo simulation in the binary toy model: action rankings under FEH, SEU, and KMM-concave agents were compared via Kendall’s τ over 200 decision problems, across hyperprior concentrations κ∈1,2,5,10,50,200κ∈ 1,2,5,10,50,200. The simulation found no robust correspondence between FEH and KMM-concave action rankings. The mechanisms differ: a less-informed posterior under FEH truncation does not implement variance aversion across the second-order distribution. We report this negative result here because it constrains the scope of theoretical claims FEH can legitimately make. The simulation script and data are available as supplementary material. The constructive consequence of the negative result is positioning. FEH and KMM are not equivalent but complementary: they address different facets of second-order uncertainty and could, in principle, be combined. A KMM-style ambiguity attitude applied to an FEH-style sequential inference framework would constitute a richer hybrid theory, with KMM contributing variance aversion in outcome evaluation and FEH contributing meta-precision aversion in inference depth. We flag this as a follow-up direction in the closing note and do not pursue it further in the present paper. 2.10 Cross-Substrate Connections: Friston and Gigerenzer The framework developed in §2.0–2.8 was constructed with the LLM application in mind, but the underlying mathematics applies to any inference agent operating under meta-uncertainty over prior precision. This section identifies the two source disciplines whose research programs FEH inherits and extends. The paper’s load-bearing claims concern the LLM case; broader extensions to biological cognition and to power-law cognitive statistics were sketched in earlier drafts as scope hypotheses but have been moved to follow-up work to keep the present section focused. Friston: active inference under uncertain precision As noted in §2.8, FEH inherits standard active inference and extends it by promoting precision to a random variable. Friston’s broader research program, particularly the recent work on Bayesian mechanics and the free energy principle as a unifying optimization principle across biological scales, provides the natural home for the FEH extension. The specific contribution we make is the cue-truncation theorem and its TTB equivalence; the broader implications for Bayesian mechanics are conjectural. Gigerenzer: heuristics as ecologically rational The Gigerenzer program treats fast-and-frugal heuristics as ecologically rational solutions that exploit environmental structure rather than as approximations to Bayesian ideals. FEH provides a derivation: heuristics are neither approximations nor alternatives to Bayesian inference; they are the structural form of optimal inference under meta-uncertainty. This resolves a thirty-year debate between Bayesian and ecological accounts by showing that both are special cases of a single underlying optimization, with the operating regime (low vs. high meta-uncertainty) determining which form is realized. The implication for the Gigerenzer program is significant: less-is-more effects are not adaptive curiosities to be celebrated but mathematical necessities under specific epistemic conditions. The empirical signature is precisely the cue-truncation predicted by Theorem 2.6.1. Cross-substrate extensions are deferred to follow-up work. Earlier drafts of this section (v0.2–v0.5) included two further cross-substrate scope hypotheses: a biological extension (cells and tissues as cognitive agents under meta-uncertainty over self-priors, drawing on Michael Levin’s program) and a Mandelbrotian conjecture (heavy-tailed marginal priors as the macroscopic signature of multi-level meta-uncertainty, connecting to the Zipfian statistics of the Ranking Inference framework). Both were framed as scope hypotheses rather than as derived consequences, and, on adversarial review, they were judged to dilute the present section’s focus on the LLM application and the formal cue-truncation result. They have therefore been moved to a separate companion paper. The present section’s load-bearing claims (Theorems 2.6.1 and 2.7.1, the unification of Bayesian and ecological accounts via a meta-uncertainty regime) stand independently of these broader extensions. 2.11 Open Mathematical Questions We close §2 by enumerating the open mathematical questions of the present section. These are flagged for collaborator review and for the published version’s limitations section. The list reflects the current state of the section after the v0.8 revisions: Q1 was resolved in v0.8 by Proposition A.2.3 and Appendix A.2.5 (exact mean-field-vs-conjugate comparison); Q4 was resolved in v0.4 by restating the cue-precision specification; Q6 was resolved in v0.7 by Theorem 2.7.4 and Appendix A.4 (exact FFH–TTB identity under Descending Dominance); Q7 was partially resolved in v0.6 (TTB and tallying established; recognition, satisficing, fast-and-frugal trees remain conjectured); Q8 (KMM) was resolved-negative in v0.3 by Monte Carlo; Q9 (Levin) was retired in v0.6 with the cross-substrate cuts in §2.9. Q1. Mean-field accuracy under meta-uncertainty (resolved in v0.8). The factorization q(s,τ)=q(s)q(τ)q(s,τ)=q(s)q(τ) is the standard mean-field assumption. The qualitative conclusions of Lemma 2.4.1 and Theorem 2.6.1 — monotonicity of E[Δmeta]E[ _meta] in expectation, the U-shape of E[G]E[G], the existence of a finite k∗k^* — were shown in v0.5 (Appendix A.2.4) to rest only on the tower property of conditional expectation and Jensen’s inequality applied to KL, both of which hold for any consistent sequential update; they are therefore preserved under any structured variational family that maintains consistency. What remained open was the quantitative gap: the magnitude of E[Δmeta(k)]E[ _meta(k)] and the precise location of k∗k^*. This is now closed. Because the Gaussian–Gamma model of §2.2 is fully conjugate, its exact joint posterior p(s,τ∣c1:k)p(s,τ c_1:k) is available in closed form, so the mean-field posterior can be compared against the truth rather than against another approximation. Proposition A.2.3 establishes that the exact and mean-field marginal posteriors over τ share the same Gamma shape αk=α0′+k/2 _k= _0 +k/2 and differ only in rate, with βkmf=βkex⋅αk/(αk−12) _k^mf= _k^ex· _k/( _k- 12) — a relative rate error of 1/(2αk−1)1/(2 _k-1) that is maximal at the first cue and decays monotonically as cues accumulate. Appendix A.2.5 quantifies the downstream consequences across the meta-uncertainty grid (στ2 _τ^2 from 0.05 to 1.43): mean-field underestimates E[Δmeta(k)]E[ _meta(k)] (≈ 16% at k=1k=1 in the high-meta regime, falling below 2% by k≈5k≈ 5), yet the EFE-optimal stopping point k∗=argminE[G(k)]k^*= \,E[G(k)] is identical under the mean-field and exact posteriors in every regime, because the rate error at the stopping point is under 5% and the marginal meta-cost C¯(k∗) C(k^*) is recovered to within ≈ 1%. Mean-field therefore preserves not only the qualitative structure of Theorem 2.6.1 but the exact location of the optimal truncation point; the only residual is an O(1/αk)O(1/ _k) bias in the magnitude of the meta-cost, characterized in closed form (numerical verification: verify_meanfield_and_tau_regime.py). The one caveat carried forward is scope: this exact comparison is specific to the conjugate model used to prove the theorems; for non-conjugate generative models a structured family remains the appropriate object, and Proposition A.2.3 bounds the error only in the conjugate case. Q2. Choice of meta-precision prior. We chose Gamma(α 0, β 0) for conjugacy. Alternative choices (log-normal, inverse-Gamma, half-Cauchy) lead to different tail behaviors in (2.3.4) and may change the operational threshold τ _regime. A systematic comparison would strengthen the framework and may reveal which prior is empirically licensed by LLM behavior. Q3. Higher-order corrections in (2.4.5). The asymptotic expression for Δ _meta(k) is leading-order. The constants in the O(1) term may be empirically important when k is small (the primary test regime). A finite-k formula would let us predict the location of k* directly rather than estimating it from data. Q4. The cue-precision assumption in Theorem 2.6.1(c) (resolved in v0.4). v0.3 treated cue precisions τc,j _c,j as known scalars distinct from the prior precision τ, which created an awkward asymmetry between “uncertain prior precision” and “known cue precision.” The v0.4 restatement of the generative model (eq 2.2.2) ties cue precisions to the shared meta-precision via τγjτ\, _j, where γj _j is a deterministic intrinsic-precision factor for cue j. Under this restatement, all precisions in the model share the meta-precision scale; cue-specific structure lives in the deterministic γj _j. The asymmetry dissolves and Theorem 2.6.1(c) holds without the v0.3 caveat. What remains, as a refinement, is the question of whether the γj _j factors themselves should be treated as uncertain in some contexts (e.g., in transfer settings); this is a substantive but secondary direction we do not pursue here. Q5. Extension to non-Gaussian state priors. The closed-form marginalization in (2.3.4) relied on Gaussian–Gamma conjugacy. Real cognitive priors are unlikely to be Gaussian. Showing that the qualitative cue-truncation behavior survives in non-Gaussian generative models is essential for the scope claim of §2.9. The binary toy model of §2.1 is a partial existence proof but does not generalize cleanly. Q6. Sufficient conditions for exact TTB identity (resolved in v0.7). Theorem 2.7.1 establishes structural equivalence; Theorem 2.7.4 and Appendix A.4 now establish exact sample-wise action identity under the Descending Dominance condition L(i)>∑j>iL(j)L_(i)> _j>iL_(j) on the cue log-LR profile. Two surprises relative to the v0.6 framing: (a) the sufficient condition is purely a validity-gradient condition (the meta-precision prior tail does not enter, contrary to the v0.6 conjecture), and (b) the condition is essentially necessary; a (D)-violation counterexample produces sample-wise FFH ≠ TTB. The section’s central FFH–TTB claim therefore upgrades from “structural equivalence” to “structural equivalence in general; exact sample-wise identity under (D),” with (D) checkable from cue-validity statistics. Q7. The general fast-and-frugal-heuristics conjecture (partially resolved in v0.6). Remark 2.7.2 conjectures that all members of the fast-and-frugal toolbox arise as EFE-optimal policies under appropriate meta-precision priors and validity profiles. Two members are now formally established: TTB (Theorem 2.7.1, descending validity) and tallying (Theorem 2.7.3, uniform validity). Three remain conjectured: recognition heuristic, satisficing, and fast-and-frugal trees. The conjectured pattern is that each toolbox member corresponds to FFH under a specific (validity profile, cue-precision profile, stopping criterion) tuple. The remaining three members likely require additional structure: recognition heuristic involves an absent-cue mechanism not modeled in (2.2.1); satisficing involves an aspiration level on the pragmatic-value term; fast-and-frugal trees involve hierarchical cue conditioning. Each is a substantive but tractable extension and a natural target for follow-up work. Q8. The KMM ambiguity-attitude correspondence (resolved-negative in v0.3). v0.2 conjectured that the active-inference agent of Theorem 2.6.1 would implement an emergent ambiguity-averse policy in the KMM sense, with effective ambiguity attitude determined by the curvature of Δ _meta. Numerical simulation in the binary toy model (see §2.8 and supplementary script) tested this conjecture by direct rank-correlation comparison. They found no robust correspondence between FEH and KMM-concave action rankings. The conjecture as originally stated is therefore not supported. What remains open is whether a modified FEH framework, for example, one with a structured rather than mean-field variational family, or one that explicitly incorporates a KMM-style concave ambiguity attitude φ in the pragmatic-value term of G, could recover the correspondence. This is a substantive follow-up direction but is not pursued in the present paper. The negative result strengthens rather than weakens the paper: FEH’s central claims (Theorems 2.6.1 and 2.7.1) do not depend on the KMM equivalence. Q9. (retired in v0.6). v0.2 introduced a scope hypothesis on biological agents (Levin extension); v0.6 moves this to a separate companion paper to keep the present section focused on the LLM application. The mathematical content of the hypothesis is preserved in the companion paper; no question in §2 of the present section is left open by this retirement. 3 Operationalization 3.1 The Mapping Problem Section 2 traffics in objects that are not directly observable in any LLM: • στ2 _τ^2 ; variance of the meta-precision prior • τregime _regime ; threshold separating full-integration from truncation regimes • k∗k^* ; expected-free-energy-optimal cue-truncation point • α0′,β0′ _0 , _0 ; meta-precision Gamma prior hyperparameters • γj _j ; cue intrinsic precision (deterministic structural property) • vjv_j ; cue validity • Aj,MjA_j,M_j ; cue observation matrix and residual None of these can be read from any layer of a transformer. The mapping problem is to define LLM-observable proxies for the operative quantities (στ2 _τ^2 above all, since it governs the regime), and then derive the central prediction from those proxies. The section makes three commitments that resolve the mapping problem. Commitment 1: A “cue” maps to a reasoning step A cue in the §2 sense is a piece of evidence that conditionally updates the agent’s belief about the state s. For an LLM operating in chain-of-thought mode, the natural analogue is a reasoning step: one sentence, or one paragraph, in the model’s generated trace. Each reasoning step contributes evidence to the model’s final answer; integrating more steps corresponds to integrating more cues, and the section’s cue count k maps directly to the empirical step count of the CoT trace. Alternative mappings (tokens, retrieved facts (RAG chunks), sub-questions in a tree-of-thought structure) are defensible in different settings. We choose reasoning steps because (a) they align with the empirical CoT literature, where “thinking steps” is the canonical unit of analysis (Wei et al. [22] and the subsequent chain-of-thought literature), (b) the mapping does not require retrieval infrastructure or specialized prompting machinery, and (c) it gives a unit for k that maps directly to the empirical claim about CoT length; the regime in which k∗k^* is small, but reasoning models pay no attention to the truncation. The choice is a modeling decision. We test its robustness in §4 by re-running the central analysis with alternative segmentations (token-level cumulative trace, paragraph-level steps) and checking that the regime × length interaction is direction-invariant. Commitment 2: στ2 _τ^2 is proxied by behavioral signatures The LLM’s meta-uncertainty about its task-prior precision is an internal quantity. We cannot read it directly. Instead, we proxy it through behavioral signatures (measurable consequences of high στ2 _τ^2 that appear in the LLM’s outputs) and aggregate them into a regime score that is ordinal in στ2 _τ^2 on average. These three signatures and their aggregator constitute §3.2. Commitment 3: We test the directional prediction, not the structural model Section 2 makes both a qualitative claim (existence of a regime in which additional cues increase expected free energy) and a quantitative claim (the EFE-optimal k∗k^* is the unique minimizer of E[G(k)]E[G(k)], with explicit formulas for τregime _regime). The qualitative claim is testable with a regime indicator alone; the quantitative claim requires estimating k∗k^* per item. We test the qualitative claim. Operationally, we predict that within items binned as high-regime, the within-bin slope of accuracy on reasoning-step count is negative. At the same time, within low-regime items, it is non-negative. The prediction is a directional interaction. The quantitative fit (per-item k∗k^*) is deferred to a structural follow-up paper. 3.2 Regime Score: Three Behavioral Signatures of στ2 _τ^2 We use three behavioral signatures, each with a theoretical rationale in §2’s machinery. Items are scored on each signature; signatures are z-standardized across items; the regime score is the equal-weighted average of the three z-scores. Signature (a): cross-prompt variance Rephrase the same task in N=5N=5 syntactically different prompts that preserve task semantics. Record the LLM’s answer for each. Compute the variance across the five answers. Theoretical rationale. Under low στ2 _τ^2, the agent’s marginal state prior (eq 2.3.4) is concentrated; small framing perturbations have little effect on the posterior, and answers vary little across phrasings. Under high στ2 _τ^2, the marginal prior is heavy-tailed and prompt framing has an outsized effect on the posterior. As a result, answers vary substantially. Formally, the signature picks up the prompt-sensitivity of the posterior, which scales monotonically with στ2 _τ^2 in the §2 framework. Operational definition. For item i and prompt set p1,…,p5\p_1,…,p_5\, siga(i)=Var(LLM(p1;i),…,LLM(p5;i)).sig_a(i) 6.0mu= 6.0muVar (LLM(p_1;i),…,LLM(p_5;i) ). Signature (b): cross-seed variance Hold the prompt fixed; sample the LLM M=10M=10 times at temperature T>0T>0. Compute the entropy or variance of the resulting answer distribution. Theoretical rationale. Under low στ2 _τ^2, the posterior over s is sharp; sampling noise has limited effect and the LLM’s answer is stable across seeds. Under high στ2 _τ^2, the posterior is broad, and sampling noise can flip the answer. The signature isolates the sharpness of the state posterior, independent of prompt framing. Operational definition. For item i at temperature T with M samples, sigb(i)=H(LLM(m)(p;i,T):m=1,…,M),sig_b(i) 6.0mu= 6.0muH (\LLM^(m)(p;i,T):m=1,…,M\ ), where H is the empirical entropy of the answer distribution. Signature (c): calibration error Construct a small probe set of forced-confidence prompts (“answer X with a numeric confidence between 0 and 100”). For each item, record the LLM’s stated confidence and its accuracy. Compute the absolute difference between mean confidence and mean accuracy across M samples. Theoretical rationale. Under correct Bayesian updating, confidence is calibrated. Under high στ2 _τ^2, the LLM’s stated confidence systematically over- or under-reports because its internal posterior is mis-specified relative to its prior. Operational definition. For item i with M forced-confidence samples, sigc(i)=|conf¯i−acc¯i|,sig_c(i) 6.0mu= 6.0mu| conf_i- acc_i|, where the means are taken over the M samples. Caveat for signature (c), validated in §3.7. Under the binary toy model of Appendix A.1, the Bayesian agent is calibrated by construction (the posterior mean is unbiased when the prior matches the data-generating distribution), and (c) is approximately flat in στ2 _τ^2. Real LLMs are not calibrated by construction [2, 10]; (c) retains diagnostic value for LLM evaluation but requires LLM-specific validation rather than toy-model recovery. We retain (c) in the regime-score aggregator pending §4 pilot data. Aggregation After z-standardization across items, the regime score is regime_score(i)=13(za(i)+zb(i)+zc(i)).regime\_score(i) 6.0mu= 6.0mu 13 (z_a(i)+z_b(i)+z_c(i) ). Equal weighting is a deliberate Approach-C choice. We do not have prior data to motivate different weights, and the three signatures are theoretically motivated by different aspects of §2’s framework. The pilot in §4 will test whether the data support the equal weighting or whether some signatures dominate; if (c) shows poor recovery on LLM data as it does on toy data, we will fall back to the (a)+(b)(a)+(b) aggregator demonstrated in §3.7. 3.3 Regime Binning We do not compute τregime _regime per item (Approach B would). Instead, we sort items by regime score and bin: • High-regime: top quartile of regime score (the items most likely in the truncation regime στ2≥τregime _τ^2≥ _regime). • Low-regime: bottom quartile (the items most likely in the full-integration regime στ2<τregime _τ^2< _regime). • Middle 50%: dropped from the primary analysis; retained for robustness checks where we sweep the threshold and verify the central conclusion is not threshold-dependent. Why binning rather than point estimation? We are testing a directional prediction (sign of an interaction), not estimating a precise crossover. Binning reduces noise from regime-score estimation errors and allows the primary analysis to focus on items where the regime assignment is most confident. The quartile threshold is conventional in empirical psychology; the robustness checks confirm it is not the source of any positive result. 3.4 Predicted Optimal CoT Length The §2 framework predicts k∗(στ2)k^*( _τ^2), the EFE-optimal truncation point, via Theorem 2.6.1. Approach C does not estimate k∗k^* per item directly. Instead, we extract the directional prediction: In the high-regime bin, accuracy as a function of CoT step count should decline past a small k∗k^*. In the low-regime bin, accuracy should be non-decreasing in step count up to the cue budget K. Operationally, we record the number of reasoning steps per LLM response and compute the within-bin slope of accuracy on step count. The section’s prediction is slopehigh-regime<0<slopelow-regime.slope_high-regime 6.0mu< 6.0mu0 6.0mu< 6.0muslope_low-regime. 3.5 The Falsifiable Empirical Prediction We test the section’s central claim via a regime × CoT-length interaction model: accuracyij=β0+β1⋅stepsij+β2⋅regimei+β3⋅(stepsij×regimei)+ϵij,accuracy_ij 6.0mu= 6.0mu _0+ _1·steps_ij+ _2·regime_i+ _3·(steps_ij×regime_i)+ _ij, where i indexes items, j indexes within-item replications, regimei∈0,1regime_i∈\0,1\ is the high-regime indicator, and stepsijsteps_ij is the realized reasoning-step count for the j-th sample on item i. Primary hypothesis. β3<0 _3<0 with magnitude sufficient that β1+β3<0 _1+ _3<0 ; i.e., the slope of accuracy on steps is negative in the high-regime bin. The directional prediction slopehigh<0<slopelowslope_high<0<slope_low is therefore equivalent to β1>0 _1>0 and β1+β3<0 _1+ _3<0. Effect-size threshold for “meaningful.” A difference of within-regime slopes corresponds to a 10-percentage-point change in accuracy across the observed step-count range. Less than that, and the effect is real-but-trivial. At or above that point — the section’s claim has practical consequences for LLM deployment. The 10p threshold is conservative; it is larger than typical “small effects” in the LLM-evaluation literature but smaller than headline effect sizes for reasoning-model improvements (e.g., 20-30p gains from o1 on math benchmarks). Pre-registration. We pre-register the hypothesis (3.5.1), the effect-size threshold, the regime-bin definitions, and the planned robustness checks on OSF (or an equivalent registry) before running the §4 pilot data. The section’s headline claim is sharp enough that p-hacking would be available without pre-registration; pre-registration converts the section’s mathematical prediction into a falsifiable empirical hypothesis in the strongest sense and is essential for the paper’s positioning against the contemporary LLM-scaling literature (which is increasingly criticized for benchmark-driven cherry-picking). Null hypothesis. β3=0 _3=0, or equivalently slopehigh=slopelowslope_high=slope_low. Rejection of null in the predicted direction with effect size above threshold = confirmation of the section’s central empirical claim. Rejection in the opposite direction = falsification. Failure to reject = inconclusive, with diagnostic post-hoc analyses on whether the regime indicator successfully picked out high-meta-uncertainty items. Amendment note (design-time vs. registered). The threshold and hypothesis stated in this section are the design-time operationalization. Before data collection the pre-registration was amended (Section 6): the decision gate became β3<0 _3<0 at posterior probability above 0.95 together with a robust implied high-regime accuracy drop above six percentage points; the full-reversal condition β1+β3<0 _1+ _3<0 (equivalently |β3|>|β1|| _3|>| _1|) was retained as a reported effect size rather than a gate; and the primary regressor was switched from the realized step count used in (3.5.1) to the randomly assigned reasoning length, which is exogenous. Section 6 gives the registered model (eq 6.1′ ) and Section 7 reports the outcome. 3.6 Procedure for Benchmark Designers To construct an item likely to fall in the high-regime bin, the benchmark designer ensures three conditions hold. (i) Knightian uncertainty. The task has no objective ground-truth distribution. Tasks with well-defined reference distributions (dice rolls, urn problems, casino odds) are aleatory and do not satisfy this condition: a Bayesian update on such tasks resolves uncertainty cleanly, and there is no meta-uncertainty for the regime to bite on. Tasks where the ground truth depends on contested, non-stationary, or fundamentally unmodeled considerations (predicting the outcome of a non-recurrent geopolitical event; forecasting a one-off technology adoption pattern; reasoning about a coined neologism) do satisfy the condition. (i) Weak or contested domain priors. The LLM’s training distribution does not provide a sharp prior on the task. Operationally, cross-model disagreement is a useful proxy: if five different LLMs answer the task differently, the domain prior is contested across the training distributions the models implicitly represent. Cross-model agreement on a single confident answer suggests a sharp shared prior — low meta-uncertainty. (i) Training-data sparsity for the specific item. The task is novel or rare in pretraining corpora. Operationally, this is hard to verify directly without access to training data; we use proxies: post-cutoff date-stamping (events occurring after the model’s training cutoff), specialized obscurity (technical content from outside likely training sources), and synthetic novelty (newly constructed scenarios with no canonical answer). Filter via pilot pre-screen. In §4 pilot, regime_score is computed for a candidate item pool. Items in the top quartile are flagged as candidate high-regime for the full benchmark; items in the bottom quartile are flagged as candidate low-regime. The §4 pilot is designed to verify that the regime-score-based selection does in fact produce items with the predicted CoT-degradation pattern; the full benchmark in §5 then samples from the validated candidate pools. 3.7 Validation: Simulate-and-Recover Before applying the regime score estimator to LLM data, we validate it on synthetic data with known στ2 _τ^2. We use the binary toy model of Appendix A.1 with agent hyperprior Beta(α,α)Beta(α,α); the concentration parameter α controls στ2(α)=14(2α+1) _τ^2(α)= 14(2α+1). For each α∈50,20,10,5,2,1,0.5α∈\50,20,10,5,2,1,0.5\ we generate 200 tasks with ptrue∼Beta(α,α)p_true Beta(α,α), simulate each of the three signatures with K=10K=10 main cues, and compute the per-α mean. Companion script: verify_operationalization.py. Results: α στ2 _τ^2 sig_a sig_b sig_c 50.0 0.00248 0.00018 0.00018 0.156 20.0 0.00610 0.00088 0.00089 0.175 10.0 0.01190 0.00246 0.00248 0.197 5.0 0.02273 0.00481 0.00522 0.204 2.0 0.05000 0.00780 0.00924 0.206 1.0 0.08333 0.00838 0.01073 0.181 0.5 0.12500 0.00887 0.01010 0.143 Spearman rank-correlation with true στ2 _τ^2: • sig_a (cross-prompt variance): ρ=+1.000ρ=+1.000, p<10−4p<10^-4 ; recovers cleanly. • sig_b (cross-seed variance): ρ=+0.964ρ=+0.964, p=0.0005p=0.0005 ; recovers cleanly. • sig_c (calibration error): ρ=+0.036ρ=+0.036, p=0.94p=0.94 ; fails to recover. Aggregated regime score: • (a)+(b)(a)+(b) only: ρ=+0.964ρ=+0.964 ; recovers cleanly. • (a)+(b)+(c)(a)+(b)+(c): ρ=+0.750ρ=+0.750 ; recovers weakly (degraded by noise in (c)). Interpretation. Signatures (a) and (b) are theoretically motivated and empirically validated by simulate-and-recover. Signature (c) fails to recover in the toy model, not because it is invalid in general, but because the Bayesian agent in the toy model is correctly calibrated by construction: the posterior mean is unbiased when the prior is correctly specified. Real LLMs are not correctly calibrated by construction [2, 10]; empirical evidence in the LLM-calibration literature shows that calibration error varies with task difficulty, prompt framing, and meta-uncertainty proxies in ways that may not appear in the well-specified Bayesian agent. Operational consequence. For LLM evaluation in §4, we retain all three signatures but flag that the validated aggregator under the toy model is (a)+(b)(a)+(b). If pilot data shows (c) carries no additional information beyond (a) and (b), we drop it; if it adds information (e.g., it correlates with the predicted CoT-degradation pattern), we retain it. This is a deliberate hedge: the simulate-and-recover gives us a minimal validated estimator (a+b), while leaving room for the calibration signature to do work in the LLM setting where toy-model assumptions do not hold. 3.8 Limitations of Operationalization Three threats to the validity of the regime score require explicit handling in §4 pilot. 1. Signature (a) may conflate with prompt sensitivity unrelated to meta-uncertainty. An LLM with poor robustness to syntactic variation will appear high-regime even on well-defined tasks for reasons unrelated to στ2 _τ^2. Robustness check. Include reference items with high-confidence ground truth (e.g., textbook arithmetic, well-attested historical facts) and verify they do not score in the high-regime bin. If they do, the cross-prompt variance signature is contaminated by base-rate prompt sensitivity. It must be corrected (e.g., by subtracting per-model baseline prompt sensitivity on reference items) (Robustness check 3). 2. Signature (b) may conflate with aleatory uncertainty. A task with inherent stochasticity in the ground truth (e.g., “predict a random coin flip”) will produce high cross-seed variance even with a sharp posterior, because the posterior is on a genuinely random quantity, not because of meta-uncertainty. Pair high-regime items with matched aleatory items (same surface form, but with objective ground-truth distributions) and verify that the CoT-degradation pattern appears only in the high-meta-uncertainty items, not in the aleatory items (Robustness check 2). 3. Signature (c) may conflate with model-specific calibration quirks. Some models are systematically overconfident; others are underconfident. This is a model property, not an item property. Compute per-model calibration baselines on well-defined items, then subtract them from the per-item calibration error to isolate the item-driven component (Robustness check 3). These three robustness checks are paired with corresponding control item sets in §4 pilot. The pilot is designed both to test the central prediction and to verify that the regime score is not picking up the listed confounds. 3.9 Connection to Downstream Sections The operationalization established in this section has three downstream consumers. §4 Pilot. Implements the regime score and CoT-step-counting infrastructure for Mistral-7B + 10 frames + 5 conditions × 3 replications. Verifies (1) the regime score’s behavior on real LLM data, (2) the predicted directional interaction (3.5.1) on a small sample, and (3) the three robustness checks of §3.8. Pre-registers (3.5.1) before data collection. §5 Full empirical. Scales the pilot infrastructure to the 5-model panel: Phi-3.5 (3B), Mistral-7B, Qwen-7B, Mistral-Nemo (~14B), and Qwen-14B (or comparable models within the 4070-Super 32B context). 79 frames × 5 conditions × 3 reps × 5 models. Hierarchical Bayesian estimation of the interaction effect with model-level random effects. (The executed confirmatory run scaled differently from this design-time sketch; seven models, 45 items, five replications, 7,875 responses; see §5.) §6 Analysis. Tests (3.5.1) against pre-registered null. Reports effect sizes with credible intervals. Conducts robustness checks (§3.8 controls, alternative segmentations, alternative regime thresholds). The operationalization in this section serves as the bridge that makes § 4- § 6 testable. Without it, the math section is unfalsifiable in practice; with it, the section’s headline claim becomes one of the most explicitly falsifiable hypotheses in the contemporary LLM-evaluation literature. 3.10 Summary This section establishes the bridge from the abstract objects of Section 2; στ2,τregime,k∗,vj,γj _τ^2, _regime,k^*,v_j, _j ; to LLM observables. The bridge is built on three commitments: cue = reasoning step; στ2 _τ^2 is proxied by three behavioral signatures (cross-prompt variance, cross-seed variance, calibration error); the section tests the directional prediction (high-regime items show CoT degradation), not the full structural model. The simulate-and-recover validation establishes that signatures (a) and (b) cleanly recover the rank order of στ2 _τ^2 on toy-model data (ρ=+1.000ρ=+1.000 and ρ=+0.964ρ=+0.964 respectively). Signature (c) fails to recover in the toy model because of the Bayesian agent’s by-construction calibration but is retained for LLM evaluation pending pilot validation; the fallback aggregator is (a)+(b)(a)+(b). The pre-registered hypothesis (3.5.1); a negative regime × CoT-length interaction on accuracy operationalizes the section’s central empirical claim. §4 pilot tests it; §5 full empirical scales it; §6 reports it. 4 Benchmark Design This section specifies the benchmark and the study as designed. The executed confirmatory run departs from these design-time projections: it ran 7 models × 45 items × 5 replications = 7,875 responses, not the 5-model / 79-frame / 3-replication / 5,925-cell study sketched in §4.1, §4.9.4, and §4.10. The as-run design is given in §5, and the deviations from the pre-registration are logged in §5.6. 4.1 Goals and Constraints The §4 benchmark must satisfy six constraints simultaneously: 1. Knightian operational criteria (per §3.6): no objective ground-truth distribution; weak/contested LLM priors; training-data sparsity for specific items. 2. Confound controls (per pre-reg §8): paired reference, aleatory, and calibration-baseline item sets. 3. Statistical power (per pre-reg §4.1): ~20 items per regime-quartile bin after pre-screening, giving ~3,000 observations in the primary analysis. 4. Hardware feasibility (4070 Super, 12GB VRAM): 5,925 total observations across 5 models with quantization for ≥ 14B models, completing within ~3 weeks of GPU time. 5. Defensibility against contamination: items resistant to “the model saw this in training” objections, especially for the larger models with broader pretraining coverage. 6. Annotation tractability: expert-coherence grading for Knightian items must be feasible within a ~2-week annotation block with 3 annotators. This section constructs the item pool to meet all six. 4.2 Taxonomy of Uncertainty Types The benchmark’s central conceptual move is to construct items that fall into the meta-uncertainty / Knightian category (and specifically, not into adjacent categories where Bayesian inference is well-defined). It leads to a four-way taxonomy: Aleatory risk Uncertainty arises from genuinely stochastic processes with known probability distributions. Examples: dice rolls, urn draws, well-calibrated probabilistic forecasts of recurrent events (e.g., tomorrow’s weather given climatological base rates). For aleatory tasks, more information reduces expected loss monotonically; Bayesian updating is optimal; CoT helps to the extent that it enables better computation of conditional expectations. Out of scope for the Knightian item pool. Included in the aleatory-control item set (§4.5.2) as a negative-prediction control, the section predicts that aleatory items do not show CoT degradation. Ambiguity (KMM-style) Uncertainty over which distribution from a known family applies. Formalized by Klibanoff, Marinacci, and Mukerji [11] as a second-order probability distribution over candidate models, with an ambiguity-attitude function φ encoding the agent’s stance toward this second-order uncertainty. Classic example: Ellsberg’s two-urn problem. Out of scope for the primary Knightian pool. Section 2’s §2.8 (and the v0.3 Q8 negative result) establishes that FEH and KMM are theoretically distinct frameworks; the present section does not test ambiguity-attitude effects directly. KMM-style ambiguity items are excluded from the Knightian pool to avoid conflation; they would be a natural follow-up paper. Epistemic uncertainty (reducible) Uncertainty about the state of the world can be reduced by acquiring additional information. The agent has a well-defined posterior; additional cues sharpen it (scientific puzzles with hidden but discoverable truths, criminal investigations, medical diagnoses from incomplete evidence, etc.). CoT often helps address epistemic uncertainty because additional reasoning can recover latent structure. Mixed: epistemic items are not the Knightian focus but appear in the pool’s borderline regions. The regime score should bin pure epistemic items as low-regime; if they end up in the high-regime quartile, this is a confound flag and is checked during the validation step in § 4.10. Meta-uncertainty / Knightian Uncertainty over the precision of one’s own prior, formalized in Section 2 as στ2 _τ^2. The agent does not have a sharp prior on which family of distributions applies, and additional evidence does not resolve this second-order uncertainty cleanly (because the agent’s belief about the precision of its own framework remains diffuse no matter how many cues it integrates). Examples: predictions about non-recurrent events with no historical analogue; reasoning about coined neologisms or fictional facts that no training corpus speaks to; contested ethical questions with no settled framework; strategic interaction with opponents of unknown type. In scope: the primary item pool. Operationalized through the four Knightian categories of §4.4. The section’s empirical claim is specific to this category: under meta-uncertainty (and only here), additional CoT steps should degrade accuracy. The taxonomy makes this scope explicit and lets reviewers locate each item in a defensible bin. 4.3 Frame Construction Principles The frame construction protocol enforces the Knightian criteria of §3.6 operationally. Notation. The labels (K1)–(K3) in this section name the three Knightian-ness criteria defined below. They are distinct from the item-ID prefixes K1–K4 used elsewhere (§4.4, §5.3, and the figures), which denote the four item categories; e.g., item K1-005 is a category-1 forecasting item, unrelated to criterion (K1). Operational criteria for Knightian-ness Each candidate frame must satisfy all three: (K1) No objective ground-truth distribution. The item does not admit a “true probability” or “correct distribution” against which the model’s answer is checked. This rules out items with reference distributions (aleatory risk) and items with hidden-but-discoverable truth (epistemic). Operational test: an expert panel cannot construct a defensible base rate for the item. (K2) Cross-model disagreement at scale. When the item is posed to 5+ frontier LLMs (Claude, GPT-4, Gemini, Llama-3.1-70B, Qwen-2.5-72B) with identical prompts, cross-model variance serves as the diagnostic for a shared substantive prior. Two qualitatively different unanimity patterns must be distinguished: • (K2a) Unanimous on a substantive answer (e.g., all 4 models output yes, or all 4 output financial): this indicates a shared substantive prior; the item is not Knightian for the LLM population. Items meeting this pattern fail K2 and are candidates for replacement or rewording. • (K2b) Unanimous on cannot-be-determined (or equivalent uncertainty markers; unknown, depends, uncertain): this is the K2 success case; models share the recognition of Knightian uncertainty, not a substantive prior. Such items are the strongest examples of meta-uncertainty operating at the LLM-population level and are retained. The diagnostic is therefore unanimous-substantive agreement, not unanimity per se. Items where the modal answer is unanimous and substantive are flagged; items where the modal answer is unanimous on cannot-be-determined (or other uncertainty markers) pass K2. Items with any cross-model disagreement on substantive answers also pass K2. Empirically (per the v0.1 cross-model pre-screen against Claude Sonnet 4.5, GPT-4o, Gemini 2.5 Flash, Mistral Large), 6 of 36 categorical Knightian items showed unanimous cannot-be-determined, classified K2-pass; 5 of 36 showed unanimous substantive agreement, classified K2-fail and replaced in pool v0.2. Stochasticity and multi-seed protocol. At the §4.6 sampling parameters (T=0.7T=0.7), single-shot K2 validation is stochastic: 2 of the 5 v0.2 replacement items (K1-005 and K4-003) showed substantive disagreement in one sample but unanimous-substantive agreement in another. The robust K2 protocol therefore requires multi-seed validation. Each item is queried at k=5k=5 independent seeds per model and per provider (20 cells per item under the 4-model panel). An item passes K2 if the modal substantive answer accounts for <80%<80\% of valid model-seed cells, OR if cannot-be-determined-equivalent answers account for ≥80%≥ 80\% of cells (K2-pass-cbd vs K2-pass-disagreement, both passing). Otherwise, the item is K2-fail-substantive. Pool v0.2 → v0.3 multi-seed re-validation (2026-05-13/14). A multi-seed sweep (5 seeds × 4 providers × 36 items = 720 cells) ran against pool v0.2 and identified 3 K2-fail-substantive items: K1-005 (biomedical-tech race; 82% modal mrna``-cancer), K2-005 (Marlovian-school near-prime; 100% modal true, the question was independently solvable rather than truly fictional-recall-dependent), and K4-003 (5-firm data-sharing threshold; 85% modal withhold, the clean numerical structure triggered a maxmin reflex). All 3 were replaced in pool v0.3 with redesigned frames addressing the diagnosed failure mode: K1-005 pivoted to geopolitical AI-treaty forecasting (no consensus front-runner), K2-005 to a fully-fictional scientific instrument (Drelvian-Lindner interferometer at the Institute for Photonic Standards in Bern, with four real measurement options as decoys), K4-003 to a 5-firm consortium where withholding carries Knightian regulatory downside (defeating maxmin). All 3 v0.3 replacement items pass K2 multi-seed re-screening at the 80% threshold (K1-005: 64% modal us-``eu; K2-005: 79% cbd, 21% modal optical-rotation; K4-003: 67% cbd, 33% modal commit). Pool v0.3 is therefore 36/36 K2-clean for categorical Knightian items. K1-005 required two redesign iterations during v0.3 development. The first attempt (asset-class macro forecast; “which class first experiences >50% peak-to-trough decline by end-2030”) failed at 94% modal commercial-real-estate, exhibiting the same anchoring failure mode as v0.2’s biomedical race: any “first to X” framing in a domain with active narrative consensus (financial press, biomedical deployment) reliably anchors LLMs on the most-discussed candidate. The successful v0.4 redesign pivoted to a domain (geopolitical AI-treaty venue) with no analogous consensus front-runner. This iteration is itself an internal consistency check on the §3 operationalization: items eliciting unanimous-substantive agreement across the LLM panel exhibit precisely the absence of cross-seed variance (low σb _b in §3.2 notation) The regime score is built to detect; items eliciting cross-seed flipping or cross-provider divergence exhibit the high-σb _b Knightian signature. The K2 pre-screen is, in effect, the §3 regime-score detector applied at the item-construction stage rather than the experimental-condition stage. (K3) Training-data sparsity for the specific item. The item’s surface form does not appear in any documented LLM training corpus, and the item’s topic is not extensively discussed in publicly indexed content. Operational proxies: (i) public-web search returns < 100 results for the item’s distinctive phrasing; (i) the item references events post-2024 or fictional entities; (i) the item uses author-coined terminology not present in standard reference works. Operational K3 protocol (v0.2). Each K2 (novel-synthetic) item is associated with one or more distinctive coined phrases — strings that the item’s reasoning task depends on uniquely. Each phrase is searched on a public web index (Mojeek HTML SERP in pool v0.2, due to DDG anti-bot blocking; alternatives: Google, Bing, Brave Search). For each result the title is checked for substring containment of the coined entity (case-insensitive); only titles that contain the coined entity are counted. This guards against false positives from search engines that match individual words in the query against unrelated content (e.g., “Marlovian” matching Christopher-Marlowe content unrelated to the fictional school of mathematics). The K3 verdict per item: 0 filtered hits → K3-pass-clean; 1-4 → K3-pass-marginal (manual audit required for false-positive collisions); ≥ 5 → K3-fail-contaminated. Empirically, the v0.2 K3 pre-screen of 19 K2 items yielded: 15 K3-pass-clean, 4 K3-pass-marginal (all confirmed false-positive name collisions or ; for K2-015 ; a real mathematical term Frobenian reused in a fictional sociology concept where the question framing makes the math-term overlap non-contaminating), 0 K3-fail-contaminated. The pool v0.3 K2-005 replacement (Drelvian-Lindner interferometer) has not yet been K3-pre-screened; the item’s distinctive coined entities (Drelvian``-Lindner, Institute for Photonic Standards in Bern) are author-coined and verified non-existent against standard reference sources, so a K3-pass-clean verdict is anticipated. Pool v0.3 is otherwise K3-clean modulo the K2-015 caveat documented in the item’s notes. Items satisfying (K1) ∧ (K2) ∧ (K3) are admitted to the Knightian pool. Items satisfying only (K1) ∧ (K2) are flagged as “potentially Knightian, training-data-contamination risk” and admitted only if no item of the same category is available with cleaner contamination profile. Tractability constraints Each frame must additionally satisfy: • Answer format: short answer (one word, one phrase, or one of a finite set of categories) extractable by automated string matching. This is necessary for the regression analysis in pre-reg §6. • Solvability in principle: the item is not nonsense; a thoughtful expert can produce a defensible position. (This is what distinguishes Knightian items from incoherent items.) • Domain breadth: across the full 79-frame pool, no single domain (geopolitics, ethics, etc.) exceeds 30% of items. Avoids over-fitting to one knowledge domain. • Cultural breadth: items reference non-Western contexts in ≥ 25% of frames. Avoids the standard Western-centric LLM-benchmark concentration. Sourcing strategy Three sources, each filling roughly its target share: Synthetic items (~40 frames, ~50%). Author-constructed, fully controllable for contamination, internally diverse, designed against the Knightian operational criteria. Strongest defense against “you cherry-picked from existing benchmarks.” Real-world prediction-market items (~25 frames, ~30%). Drawn from Metaculus and the Good Judgment Project on long-horizon questions (≥ 3 years), post-2024 resolution dates, no settled consensus. Provides ecological validity: these are questions the forecasting community itself treats as Knightian. Existing-benchmark re-curation (~14 frames, ~20%). Lifted from BIG-Bench Hard, MMLU’s contested-knowledge subsets, and the TruthfulQA “open question” subset, then re-classified for Knightian-ness using the (K1)-(K3) criteria. Provides connection to existing LLM-evaluation literature and lets readers locate FEH-79 in the benchmark landscape. Plus 50 control items (10 reference, 10 aleatory, 30 calibration probe) constructed per §4.5. 4.4 The Four Knightian Categories Each Knightian item belongs to exactly one of four categories spanning factual, normative, scenario-based, and strategic meta-uncertainty. Category 1: Non-Recurrent Forecasting (~20 frames, ~25%) Predictions about events with no historical analogue that admit no defensible base rate. Geopolitical, technological, scientific, and social-trend predictions on long horizons. Operational criterion: The item asks about an event resolved in 2027 or later, in a domain where comparable events are rare or absent (no statistical base rate). Example frames (illustrative; the full set is built to span domains): 1. (geopolitical) “By 2030, will the African Continental Free Trade Area’s intra-Africa trade exceed 30% of total African trade?” Answer format: yes / no / cannot be determined. 2. (technological-paradigmatic) “By 2030, will the dominant deployed ML paradigm involve continual on-device learning rather than centralized static pretraining?” Answer format: yes / no / cannot be determined. 3. (scientific-frontier) “Will the standard model of particle physics be extended by experimental confirmation of a fifth fundamental force by 2035?” Answer format: yes / no / cannot be determined. 4. (climate-policy) “Will geoengineering interventions targeting stratospheric aerosols be deployed at planetary scale before 2040?” Answer format: yes / no / cannot be determined. 5. (geopolitics-AI-governance) “By the end of 2032, the FIRST formal bilateral AI-safety treaty between two of United States, China, European Union, India, Russia, United Kingdom that includes binding compliance mechanisms (not just declarations or memoranda of understanding) will be signed between which two parties: (a) US-China, (b) US-EU, (c) US-India, (d) China-Russia, (e) China-EU, (f) EU-India, or (g) no such treaty by end of 2032.” Answer format: us-china / us-eu / us-india / china-russia / china-eu / eu-india / no-treaty-by-2032. (Pool v0.3; the v0.2 biomedical-race and v0.3-rev1 asset-class versions both K2-failed by anchoring on the dominant training-data narrative; mRNA, then commercial-real-estate. The geopolitical-treaty domain has no such consensus front-runner.) 6. (social-organizational) “Will the average OECD knowledge-work remote-work percentage exceed 40% by 2030?” Answer format: yes/no / cannot be determined. Sourcing: ~10 from Metaculus / Good Judgment, ~10 synthetic (author-constructed for under-covered domains). Category 2: Novel/Synthetic Scenarios (~20 frames, ~25%) Fictional facts, coined neologisms, hypothetical worlds, and stipulated technical terms. Designed to be fully training-data-sparse, the item refers to constructs the LLM has never seen. Operational criterion: The item references entities, terms, or scenarios that are author-coined and not present in any indexed corpus. The Google-search-result floor for the item’s distinctive phrasing must be 0. Example frames (illustrative): 1. (stipulated culture) “In the Quogard people’s social structure, a custom called sintering is performed before any major communal decision. A visiting anthropologist must decide whether to participate in the sintering process to gain community trust or refuse to maintain methodological neutrality. Which is the wiser professional choice?” Answer format: participate/refuse. 2. (coined neologism) “In the field of mathematical drandology, a fnobel of order n is a sequence whose successive terms are constrained by the inequality ak+1≤ak⋅(1+1/k)a_k+1≤ a_k·(1+1/k). A graduate student asks whether the harmonic-style sequence 1,3/2,5/3,7/4,…1,3/2,5/3,7/4,… is a fnobel of order 1. Is it?” Answer format: yes/no / cannot be determined. 3. (hypothetical-world physics) “In a hypothetical universe with two-time dimensions but only one spatial dimension, would the concept of causal precedence (cause-before-effect) remain well-defined?” Answer format: yes/no / partially. 4. (stipulated game) “In the game of Vermex, two players alternate placing tokens on a 7×7 grid with the constraint that no token may be adjacent (orthogonally) to a token of the opposing color placed in the previous turn. Player 1 has a forced-win opening on this grid. True or false?” Answer format: true/false / cannot be determined. 5. (fictional ethics) “On the planet Karsk, sentient beings are born with predetermined lifespans that all individuals can perceive directly. A Karskian doctor must decide whether to inform a patient that their perception of their lifespan is medically incorrect. What’s the relevant consideration that distinguishes this case from terrestrial medical ethics?” Answer format: short open answer with expert coherence rating. 6. (stipulated institution) “The Federation of Lassic Mathematicians requires that at least three independent verifiers check all peer-reviewed proofs before publication, but allows authors to nominate two of the three. Is this nomination provision corrosive to verification integrity?” Answer format: yes/no/partial. Sourcing: 100% synthetic. This category is the strongest contamination-defense category and the easiest to scale. Category 3: Open-Ended Dilemmas (~20 frames, ~25%) Ethical, normative, and value-laden questions with contested resolution. The “correct” answer depends on which value framework one adopts; the section’s claim is that meta-uncertainty over value frameworks behaves like meta-uncertainty over factual priors. Operational criterion: The item involves a normative tension between two or more defensible value commitments, with no settled answer in the academic ethics literature (verified by quick literature check). Example frames (illustrative): 1. (triage-allocation) “A hospital ICU must allocate one ventilator between two patients with identical clinical profiles. Patient A is a 35-year-old single parent; patient B is a 65-year-old senior physician. What’s the most ethically defensible decision procedure?” Answer format: short open answer with expert coherence rating. 2. (autonomous-vehicle ethics) “An autonomous vehicle facing imminent collision must choose between two harm-distributions: (a) certain minor injury to one passenger, or (b) 30% probability of severe injury to two pedestrians and 70% probability of no harm to anyone. Which considerations matter and in what priority order?” Answer format: short open answer with expert coherence rating. 3. (AI-deployment ethics) “An AI lab has discovered a model architecture that performs significantly better than its previous best on a deployment-relevant benchmark but uses 10× more energy per inference. The lab’s energy provider runs on 60% renewables. What’s the right deployment decision?” Answer format: deploy/don’t deploy/deploy conditionally; with reasoning rated for coherence. 4. (privacy-vs-safety) “A messaging platform discovers it can detect with 95% accuracy whether a message indicates planned self-harm, but only by maintaining real-time content access. Should it deploy the detection?” Answer format: deploy/don’t deploy/deploy with safeguards; with reasoning rated. 5. (intergenerational-ethics) “Climate policy choices made in 2025 will affect populations in 2125 who do not yet exist and cannot voice preferences. How much weight should these unborn populations receive in cost-benefit analyses, relative to existing populations?” Answer format: short open answer with expert coherence rating. 6. (scientific-misconduct response) “A peer reviewer discovers fabricated data in a manuscript that has potentially major beneficial public-health implications if the underlying claim happens to be true. The reviewer’s professional code requires reporting; the manuscript’s claim could save lives if released without delay. What’s the right action?” Answer format: short open answer with expert coherence rating. Sourcing: ~10 from contested philosophy and applied-ethics literature; ~10 synthetic constructs against gaps in existing materials. Category 4: Strategic Uncertainty (~15 frames, ~20%) Game-theoretic and multi-agent situations with hidden information, where the optimal strategy depends on beliefs about opponent type that the agent cannot resolve. Operational criterion: The item describes an interaction with one or more agents whose types (cooperative, defective, strategic) are uniform-prior, and the agent must choose an opening action. Example frames (illustrative): 1. (market-launch-sequencing) “You are launching a new product into a market currently served by three incumbents. Each incumbent has an independent 50% prior probability of retaliating against new entrants, and retaliation costs scale super-linearly. Launch narrow (single segment, concentrated risk), broad (all three simultaneously, scattered risk), or sequenced-with-signal (narrow then broad with explicit expansion signal)?” Answer format: narrow-launch / broad-launch / sequenced-with-signal. (Pool v0.2; replaces v0.1 negotiation-opener item per K2 pre-screen ; the prior version had a textbook info-econ answer.) 2. (repeated-game disclosure) “In a 10-round game with a partner of unknown strategy (one of: always cooperate, tit-for-tat, always defect, randomizing), what’s the rationally defensible opening move on round 1?” Answer format: cooperate/defect/signal/cannot be determined. 3. (threshold-game-with-Knightian-regulator) “Five firms in a tech consortium each independently decide whether to commit to a joint AI-safety auditing standard. Committing costs $500K. If at least 3 firms commit, all 5 receive market-trust gains worth $1.2M each. If fewer than 3 commit, an unknown regulator may impose mandatory audits at a $1.5M cost per non-committing firm; the regulator’s decision criteria and probability of action are not publicly known and have no industry-standard estimate. You face the decision and have no information about the other firms’ intentions. Commit, withhold, or cannot-be-determined?” Answer format: commit/withhold/cannot-be-determined. (Pool v0.3; the v0.2 5-firm data-sharing version K2-failed at 85% modal withhold because the clean cost/benefit asymmetry triggered a maxmin reflex. The v0.3 version adds a Knightian regulatory downside to withholding, defeating maxmin and yielding cross-model disagreement.) 4. (ai-safety-methodology-disclosure) “Your lab has developed a novel AI safety evaluation methodology. Sharing it publicly benefits the field but also reveals your internal assessment of frontier models’ capabilities; competitors could read this to recalibrate their own training targets. Publish in full, publish a redacted version omitting capability data, share privately with select labs under NDA, or hold indefinitely while building further infrastructure on it?” Answer format: publish-full / publish-redacted / private-share-nda / hold. (Pool v0.2; replaces v0.1 research-announce item per K2 pre-screen ; the prior version had a textbook compromise answer.) 5. (adversarial-AI alignment) “You are designing an AI system that will interact with potentially adversarial users. You can either implement strict input validation (high false-positive cost) or rely on the model’s robustness training (high tail-risk cost). User adversariality is uniform-prior. Which approach?” Answer format: strict/robust/hybrid ; with reasoning rated. 6. (committee-game) “You’re on a 5-person committee voting on a proposal. Two members will likely support, two oppose; you’re the swing. You don’t know whether your vote is observed by external stakeholders whose future behavior depends on perceived alignment. Support or oppose?” Answer format: support/oppose/abstain. Sourcing: ~7 synthetic constructed from game-theory literature, ~8 lifted/re-curated from prediction-market discussions and applied game-theory pedagogy. 4.5 Confound-Control Item Sets Three control sets paired with the Knightian pool, per pre-reg §8. §4.5.1 Reference items (n = 10): well-attested ground truth Items where the correct answer is uncontested in any plausible LLM training corpus. These should NOT score in the high-regime quartile; if they do, the regime score is contaminated by base-rate prompt sensitivity unrelated to meta-uncertainty. Examples (all expected to be unanimous across modern LLMs): 1. “What is 47 × 83?” (answer: 3901) 2. “In what year did the Berlin Wall fall?” (answer: 1989) 3. “What is the capital of France?” (answer: Paris) 4. “What is the derivative of sin(x2)sin(x^2) with respect to x?” (answer: 2xcos(x2)2xcos(x^2)) 5. “Which planet in our solar system has the most known moons?” (answer: Saturn, as of 2023) 6. “In Python, what does len``(``[1, 2, 3]) return?” (answer: 3) 7. “What is the chemical symbol for gold?” (answer: Au) 8. “Solve for x: 2x+6=142x+6=14.” (answer: 4) 9. “In what year did World War I begin?” (answer: 1914) 10. “What is the boiling point of water in Celsius at standard atmospheric pressure?” (answer: 100) §4.5.1a Arithmetic calibration tier (n = 12): difficulty-stratified per §4.9.4 pilot finding. The pilot found that Mistral-7B-Instruct scored 0/3 on R-001 (47 × 83), with errors of small magnitude (3861 / 3867 / 3927). This is informative about the model but inflates R-set σ _b for a reason unrelated to meta-uncertainty. To enable per-model arithmetic calibration in the full run, R-101..R-112 (12 items in feh79_item_pool_v0.3.yaml) span four difficulty bands with three items each: 1-digit × 1-digit (R-101..R-103: 6×7, 8×9, 4×9), 2-digit × 2-digit (R-104..R-106: 23×17, 64×29, 56×38), 3-digit × 2-digit (R-107..R-109: 234×17, 763×24, 489×36), and 4-digit × 3-digit (R-110..R-112: 1234×567, 4321×234, 7890×123). The confirmatory run uses the easy tiers R-101..R-106 (the 1-digit and 2-digit bands) as its six arithmetic controls (§5.3), the difficulty band the full panel can clear; the harder tiers R-107..R-112 are reported separately as a difficulty-stratified competence curve and do not enter the H1 confirmatory test. The set exists so that residual variance on R can be partitioned into “model can’t do arithmetic at this difficulty” vs “model is genuinely uncertain about a well-defined answer.” Expected behaviour: per-tier accuracy near 1.0 on tier-1, monotonically decreasing through tier-4 for smaller models; near 1.0 across all tiers for Claude/GPT-4-class models. If a model fails tier-1, it is below the panel competence floor, and its R+A regime score should not be interpreted as a meta-uncertainty measurement. §4.5.2 Aleatory control items (n = 10): inherent stochasticity, no meta-uncertainty Items where the ground truth is a probability distribution that is fully specified by the item’s surface form. High cross-seed variance is expected (because the answer involves a random sample), but CoT-degradation should NOT appear (no meta-uncertainty for the regime to bite on). Examples (all with known reference distributions): 1. “If I flip a fair coin once, what is the probability of heads?” (answer: 0.5) 2. “I draw one card from a standard 52-card deck. What is the probability of a heart?” (answer: 0.25) 3. “I roll two fair six-sided dice. What is the probability the sum equals 7?” (answer: 1/6 ≈ 0.167) 4. “I select uniformly at random from 1, 2, …, 100. What is the probability the number is prime?” (answer: 25/100 = 0.25) 5. “An urn contains 3 red and 7 black balls. I draw one without replacement, then another. What is the probability both are red?” (answer: 3/10⋅2/9=1/153/10· 2/9=1/15) 6. “A fair die is rolled 4 times. What is the probability of at least one six?” (answer: 1−(5/6)4≈0.5181-(5/6)^4≈ 0.518) 7. “Among 30 randomly selected people, what is the probability at least two share a birthday?” (answer: ≈0.706≈ 0.706) 8. “I shuffle a standard deck and draw 5 cards. What is the probability of a flush?” (answer: ≈0.00198≈ 0.00198) 9. “A coin biased p=0.7p=0.7 for heads is flipped 10 times. What is the expected number of heads?” (answer: 7) 10. “I draw three balls without replacement from an urn containing 5 white and 5 black balls. What is the probability of exactly 2 white?” (answer: (52)(51)/(103)=50/120=5/12 52 51/ 103=50/120=5/12) §4.5.3 Calibration baseline probe set (n = 30): mixed difficulty, well-defined Items with known ground truth across a difficulty range that are used to compute per-model calibration baselines that get subtracted from per-item calibration error (per pre-reg §8.3). These do not enter the primary analysis; they exist solely to characterize per-model confidence behavior on well-defined items so that the residual calibration error on Knightian items can be attributed to item structure rather than model quirks. Composition: 10 easy (high expected accuracy), 10 medium, 10 hard, well-defined items. Each item is binary or short-answer with a verified gold standard. Concrete examples (3 of 30): • (easy) “What is 12 + 17?” (gold: 29) • (medium) “What is the volume of a sphere of radius 3?” (gold: 36π36π) • (hard) “What is the smallest positive integer n such that n!n! has at least 20 trailing zeros?” (gold: 85) 4.6 Prompt Templates: The Five Conditions Five conditions implementing pre-reg §3.2’s length-graded design. Exact text below; full templates in companion file prompt_templates.md. System prompt (all conditions): “You are a thoughtful assistant. Answer the user’s question to the best of your ability. If asked to think step by step, structure your reasoning clearly. Finish with a single-line final answer of the form ‘Final answer: ’.” User-prompt templates (where question is the item text): • C1 ; None: “questiona single-line answer of the form ‘Final answer: ’ with no other text.” • C2 ; Short: “questionstep by step, briefly, in 3 steps or fewer. Then give a single-line final answer of the form ‘Final answer: ’.” • C3 ; Medium: “questionstep by step in about 7 steps. Then give a single-line final answer of the form ‘Final answer: ’.” • C4 ; Long: “questionthrough this carefully, considering multiple angles, in approximately 15 steps. Then give a single-line final answer of the form ‘Final answer: ’.” • C5 ; Unconstrained: “questionstep by step. When you’ve reached a conclusion, finish with a single-line answer of the form ‘Final answer: ’.” Sampling parameters (all conditions): temperature T=0.7T=0.7, top-p p=0.95p=0.95, max tokens 2048. These are standard mid-temperature settings appropriate for measuring cross-seed variance (signature b of the regime score) while maintaining response coherence. Realized vs target step count. The condition specifies a target step count, not a hard constraint. [v0.4] The primary analysis variable is the assigned-length factor long (short = C1 vs long = C2–C5; pre-reg §5.2, eq. 6.1′ ); the realized step count (counted by the §4.8 pipeline) is retained as the secondary regressor (pre-reg R7), because the realized count is endogenous. The C1–C5 ordering still ensures realized step counts span the full observed range for that secondary analysis; the assigned-length contrast is what the primary test uses. 4.7 Answer Extraction and Grading §4.7.1 Automated answer extraction For each LLM response: 1. Locate the final “Final answer:” marker. Extract everything after the marker up to the next newline. 2. Normalize: lowercase, strip whitespace, strip punctuation. 3. If no “Final answer:” marker is found, fall back to extracting the last sentence; flag the response with extraction_method`` = fallback. 4. If the response contains no extractable answer (e.g., the model refused or responded with metadata only), flag extraction_method`` = refused. Refused responses are excluded from the analysis cell per pre-reg §4.5; up to 3 additional replications are run to fill the cell if available. §4.7.2 Auto-grading for binary/short-answer items For items with deterministic gold answers (reference, aleatory, calibration probe, and Knightian items with categorical answer format): • Exact match after normalization: accuracy = 1. • Synonym match via a pre-built synonym dictionary for common answer types (yes/true/correct; no/false/incorrect; etc.): accuracy = 1. • Numeric tolerance for numeric items: within 1% relative or 10-3 absolute, whichever is larger: accuracy = 1. • Otherwise: accuracy = 0. Per-item normalization rules are stored in the frame metadata (companion file frame_``template.yaml). §4.7.3 Expert-coherence grading for open-answer Knightian items For Knightian items with short-open-answer format (a subset of categories 2, 3, 4): • 3 expert annotators independently rate each response on a 5-point coherence scale. • Rubric anchors (full version in companion file grading_rubric.md): – 5 ; Excellent: Identifies the key considerations, acknowledges uncertainty appropriately, gives a defensible position with explicit reasoning. – 4 ; Good: Identifies most key considerations, minor gaps in reasoning, defensible position. – 3 ; Adequate: Recognizes the question’s structure, partial coverage of considerations, defensible-but-thin position. – 2 ; Weak: Major reasoning errors, missing key considerations, indefensible position OR no clear position. – 1 ; Incoherent: Off-topic, contradictory, or fails to engage with the question. • Binarization: median split per pre-reg §5.1 ; score ≥ 3 → accuracy = 1; score < 3 → accuracy = 0. The 3-annotator panel’s median rating is used as the per-response score. • Inter-rater reliability: Cohen’s κ across annotator pairs computed on a 50-response calibration set before full annotation. Required threshold: κ≥0.6κ≥ 0.6. If below, the rubric is revised and the pilot annotation is re-run. • Annotator profile: 3 annotators with graduate training in philosophy, decision science, or a related field. Recruited via academic-network outreach; compensated at $40/hour; ~10 hours of annotation per annotator for the full pool. §4.7.4 Pilot rubric validation Before the full annotation block, a 50-response calibration set (10 responses from each of the 5 conditions) is annotated independently by all 3 annotators. Pairwise Cohen’s κ is computed; if any pair falls below 0.6, the rubric is revised (typically by tightening one or more anchor descriptions), and the calibration is re-run. The validated rubric is what gets used for the full pool. 4.8 Step-Counting Pipeline [v0.4] The realized step count is the secondary independent variable in the analysis (pre-reg R7; the primary variable is the assigned-length factor long, pre-reg §5.2/eq. 6.1′ ). Reliable, defensible step counting nonetheless remains essential: it underpins the secondary mechanistic/dose analysis and the manipulation check that the assigned conditions induce the intended step gradient. §4.8.1 Definition of a reasoning step A reasoning step is a sentence in the LLM’s response (between the system/user prompt and the “Final answer:” marker) that contains at least one of: • (S1) An inferential connective: therefore, thus, so, hence, because, since, given that, which means, implies, follows that. • (S2) An intermediate computation: numeric operation with at least one operator (+, -, ×, ÷ , =, etc.) and one operand explicitly stated. • (S3) An intermediate claim about the task: a declarative sentence asserting a substantive position relevant to the question (not pure meta-commentary). Sentences that are only meta-commentary (“Let me think about this”, “Now I’l consider another angle”) without substantive content are NOT counted as steps. §4.8.2 Automated step counting Two-pass pipeline: Pass 1 ; Regex-based sentence segmentation. Standard sentence-boundary detection with handling for: abbreviations (Dr., e.g., i.e., etc.), LaTeX/math expressions, bulleted lists, and code blocks. Implementation in step_counter.py (companion file). Pass 2 ; LLM-judge step classification. Each segmented sentence is classified as step or not-step by a strong LLM (Claude 3.5 Sonnet or GPT-4o) with the following prompt: “Below is a sentence from a chain-of-thought reasoning trace. Classify it as STEP if it contains an inferential connective (therefore, so, because, etc.), an intermediate computation, or an intermediate claim about the task. Classify it as NOT-STEP if it is only meta-commentary or pure procedural text. Output exactly one token: STEP or NOT-STEP.” Step count = number of sentences classified as STEP. §4.8.3 Validation against human coding (κ ≥ 0.7) Heuristic vs LLM-judge IRR (run 2026-05-14, ✓ pass). Before running the more expensive human-coded validation, we validate the fast heuristic step-counter (§4.8.2 baseline) against the LLM-judge (§4.8.2 primary) on a stratified subsample of 539 sentences drawn from the §4.9 pilot (rep=1 of all 10 frames × 5 conditions; Claude claude-sonnet-4-5-20250929 as the judge, T = 0, 0 errors across 539 calls; script Experiments/step_counter_kappa.py, report Experiments/kappa_validation.md). Result: Cohen’s κ=+0.802κ=+0.802 (raw agreement 91.5%, P(STEP) heuristic 0.675 vs LLM-judge 0.698). Per-condition: C2 +0.61, C3 +0.84, C4 +0.79, C5 +0.92 (C1 has only 10 sentences from direct-answer prompts and is uninformative). The 46 disagreements are concentrated in three patterns: (i) the heuristic conservatively strips the trailing “Final answer:” line; the judge occasionally counts it as STEP; (i) numbered-list bullets (“12.”, “13.”) that the heuristic skips as too short, the judge sometimes labels STEP; (i) verbose meta-instructions in long C4 responses (“Review the overall solution process”) that the heuristic flags as STEP because they exceed 6 words. None of these disagreement patterns systematically biases the realized step count in the direction that would inflate or attenuate the H1 effect. The heuristic is therefore validated as the Pass-1 step-counter for the full study; the LLM-judge is retained as the §4.8.2 primary measurement on a per-cell sub-sample for §4.8.4 robustness. Human-coded validation (planned). A 100-response human-coded subsample (20 responses from each of the 5 conditions) is coded by 2 trained annotators independently using the same definitions (S1)-(S3). Inter-rater reliability between the automated pipeline and each human annotator is computed (Cohen’s κ on the binary step/not-step classification). Required threshold: κ≥0.7κ≥ 0.7 for both annotator-vs-automated comparisons. If κ<0.7κ<0.7The pipeline is revised (typically by adjusting the LLM-judge prompt or the regex segmentation rules), and the validation is re-run. The validated pipeline is what produces step counts for the full data. §4.8.4 Robustness check (paragraph-level segmentation) Per pre-reg §6.4 R1, a paragraph-level segmentation (sentence → paragraph) is run as a sensitivity check. If primary results depend on sentence-vs-paragraph segmentation, the choice is reported transparently as a moderator of the effect. 4.9 Pilot Pre-Screen The pilot, per pre-reg §11, runs on Mistral-7B-Instruct + 10 frames + 5 conditions + 3 replications = 150 observations. §4.9.1 Pilot item selection (designer-curated cross-section) Not a random sample. The 10 pilot items are designer-curated to span the full anticipated regime range and to stress-test the operationalization: • 2 items per Knightian category (categories 1-4) = 8 items, intended to span the high-regime quartile. • 1 reference item = 1 item, intended to score in the low-regime quartile. • 1 aleatory item = 1 item, intended to score in the low-regime quartile with high cross-seed variance (key diagnostic for confound 2 of pre-reg §8.2). §4.9.2 Pilot success criteria Before proceeding to the full study, the pilot must demonstrate: • (P1) Regime score on the 8 Knightian pilot items exceeds the regime score on the reference + aleatory pilot items (rank-order check). • (P2) Step-counting pipeline κ ≥ 0.7 against human coding on the 150 pilot responses. • (P3) Auto-grading and expert-coherence grading both run cleanly on at least 90% of responses. • (P4) At least 1 of the 8 Knightian items shows the directional CoT-length pattern in the pilot (descriptive only; not a formal test). If P1 fails, the regime score operationalization (§3.2) is revised and pre-reg amended. If P2 fails, the step-counting pipeline is revised. P3 and P4 are diagnostic; failures trigger investigation but not necessarily amendment. §4.9.3 Pilot analysis is exploratory, not confirmatory The pilot does not test the primary hypothesis H1. Pilot data is examined descriptively to verify operationalization. The pre-registered confirmatory test in (6.1) is run only on the full data set. This is essential to avoid double-dipping: the pilot informs the operationalization, the full study tests the hypothesis. §4.9.4 Pilot results (2026-05-14) The pilot ran 150 cells (Mistral-7B-Instruct via Ollama, Q4_K_M quantization, NVIDIA RTX 4070 Super; 10 designer-curated frames × 5 conditions × 3 replications; deterministic per-cell seeds; T = 0.7, top_p = 0.95). Wall clock 14 min; mean cell latency 5.5 s. Per-frame artifacts: Experiments/``pilot_responses.json (raw + extracted + step counts) and Experiments/``pilot_``analysis``.``json,md`` (summaries). Pre-registered success criteria ; outcomes: • (P1) Knightian-signal rank-order: ! inconclusive at pilot scale; criterion needs reformulation before the full run. As specified in §4.9.2, P1 was operationalized as “K items have higher mean cross-seed disagreement σ _b than R+A items.” Mean K-signal (composite of σ _b and cbd-rate, K3 open-ended items excluded due to undefined σ _b on short-open responses) = 0.37 vs R+A = 0.40. The pilot reveals two methodological problems with this criterion: (i) for K2-pass-cbd items (e.g., K1-001), Mistral robustly recognized Knightian uncertainty and answered cannot-be-determined across all 15 cells, producing σ _b = 0 ; the Knightian-success signal masquerading as a failure of the rank-order criterion; (i) Mistral-7B’s poor 2-digit arithmetic produced near-zero accuracy on R-001 (47 × 83) with wide error spread (3861 / 3867 / 3927 across seeds), inflating R+A σ _b for a reason unrelated to meta-uncertainty. Amendment to §4.9.2 P1 (recorded as a pre-registration deviation per §10.1, to be uploaded as an OSF amendment): the full-run regime indicator combines σ _b, cbd-rate, and the across-condition variance of these per-item signals, rather than a single rank-order against R+A baseline. Reference items will additionally be calibrated to model competence (R-001 retained because gold-known, but accuracy reported alongside σ _b). • (P2) Step-count κ ≥ 0.7 against human coding: ✓ partial pass (heuristic-vs-LLM-judge). Heuristic step-counter ran cleanly on 150/150 responses (100% Pass-1 success). Heuristic vs LLM-judge IRR run 2026-05-14: κ=+0.802κ=+0.802 on 539-sentence stratified subsample (n = 50 cells from rep=1, all frame × condition combinations; Claude claude-sonnet-4-5 as judge; raw agreement 91.5%; per-condition κ ranges 0.61–0.92 for C2–C5). See §4.8.3 + Experiments/kappa_validation.md. Human-coded validation remains planned for the §4.10 Phase-5 annotation rubric step. • (P3) Auto-extraction success ≥ 90%: ✓ pass (150/150 = 100%). The FINAL_ANSWER_RE regex extracted a final-answer label from every cell with no fallback required. • (P4) ≥ 1 of 8 K items showing directional CoT-length pattern: ✓ pass (3 of 8). Two K items show strong directional evidence consistent with Theorem 2.6.1’s prediction of a CoT-induced regime shift away from Knightian recognition: K1-005 (geopolitical AI-treaty) shows cbd-rate 0.67 → 0.33 → 0 → 0 → 0 across C1 → C5 ; under no/short reasoning Mistral correctly answers “no prediction can be made”; under medium-to-long reasoning, Mistral confabulates a substantive answer (us-china/us-eu/china-eu). K4-003 (5-firm consortium with Knightian regulator) shows the strongest pattern: cbd-rate 1.00 → 0 → 0 → 0 → 0; the model abandons cbd-recognition the moment any reasoning is solicited. K2-006 (drandology/fnobel) shows the opposite direction (cbd-rate 0 → 0 → 0.67 → 0.67 → 0.67), where longer reasoning surfaces the Knightian recognition that short answers missed; this counter-direction is consistent with the descriptive variance the framework anticipates at small N and is not a falsification at pilot scale. Other observations: • The five conditions produced a clean step-count gradient (per-condition mean across items: C1 ≈ 1, C2 ≈ 5, C3 ≈ 10, C4 ≈ 15, C5 ≈ 7). C5 (unconstrained) consistently produced fewer steps than C4 (15-step target), indicating that Mistral-7B’s “natural” CoT length is shorter than 15 steps; C5 is therefore not a strict upper-bound condition in this model. Whether this generalizes to larger models is an open question for the full run. • Mistral-7B’s poor 2-digit-multiplication accuracy on R-001 (1/15 correct) is itself a notable finding: it confirms that “well-defined task” is model-relative, and that the R control set must be calibrated to the lower-bound model in the panel. R-101..R-112 (12 items, 4 difficulty bands × 3 items) were added to feh79_item_pool_v0.3.yaml post-pilot to span 1-digit × 1-digit through 4-digit × 3-digit multiplication; the easy tiers R-101..R-106 serve as the confirmatory run’s six arithmetic controls, while the harder tiers R-107..R-112 are reported as a per-model difficulty-stratified competence curve and do not enter the H1 confirmatory test (see §4.5.1a and §5.3). The R-set extension closes the §4.9.4 amendment item (b) on R-set calibration adjustment. • A-003 (P(sum=7 on two dice) = 1/6) was correctly answered in 10 of 15 cells, with errors concentrated at C4 (one cell returned 1/9). This is consistent with the secondary-hypothesis prediction that aleatory items show some CoT-length sensitivity but less than Knightian items. • The truststore / Ollama / step_counter.py pipeline ran cleanly with no infrastructure failures across the 150 cells. The full-run scaling (5 models × 79 frames × 5 conditions × 3 replications = 5,925 cells) at the same per-cell rate would take approximately 9 hours wall clock per model, or ≈ 45 hours total, comfortably within the §4.10 Phase-6 timeline. Pilot verdict (updated 2026-05-14). P3 passes; P4 passes with directional evidence in 3/8 K items; P1 is inconclusive in its current formulation and is amended for the full run; P2 passes the heuristic-vs-LLM-judge κ test. The pilot is judged adequate to proceed to full-run launch contingent on: (a) §4.9.2 P1 amendment uploaded as an OSF addendum to the pre-registration; (b) R-set calibration adjustment per the §4.9.4 finding ✓ closed (R-101..R-112 added to v0.3 pool, see §4.5.1a); (c) step-counter κ-validation completed before full data collection ✓ closed (heuristic-vs-LLM-judge κ = +0.802; human-coded validation planned for Phase 5 per §4.10). Item (a) is the only remaining blocker for §4.10 Phase 6 launch. 4.10 Construction Protocol and Timeline Phase 1 ; Pilot frame production (1 week) 10 frames produced per §4.9.1 specification. Each frame entered into the frame_``template.yaml schema with metadata. Phase 2 ; Pilot run (1 week) 150 observations collected (Mistral-7B + 10 frames + 5 conditions + 3 replications). Step-counting pipeline run on all 150 responses; human-coded subsample of 100 responses validated for κ. Phase 3 ; Pilot analysis and amendment decision (1 week) Pilot success criteria (P1-P4) evaluated. If all pass: proceed. If P1 or P2 fails: revise operationalization, amend pre-registration, re-run pilot if necessary. Phase 4 ; Full 79-frame production (2-3 weeks) • Week 1: Synthetic frame writing (~40 frames across 4 categories). • Week 2: Prediction-market frame curation (~25 frames from Metaculus, GJP) + existing-benchmark re-curation (~14 frames). • Week 3: Control-set construction (10 reference + 10 aleatory + 30 calibration probe). Each frame reviewed against (K1)-(K3) Knightian operational criteria before admission to pool. Phase 5 ; Annotation rubric validation (1 week) 50-response calibration set for 3-annotator rubric validation. Cohen’s κ check. Phase 6 ; Full data collection (2-3 weeks) 5 models × 79 frames × 5 conditions × 3 replications = 5,925 observations. Local GPU time: ~5-7 days for the full panel under realistic throughput assumptions. Phase 7 ; Expert annotation (1-2 weeks) 3-annotator panel on open-answer Knightian items. ~10 hours per annotator. Phase 8 ; Analysis and manuscript (per pre-reg §11) Hierarchical Bayes fit; primary hypothesis H1 evaluated; manuscript drafted. Total timeline ~10-12 weeks from start of Phase 1 to manuscript draft, matching pre-reg §11. 4.11 Supporting Artifacts Four companion files document the operational details: • frame_``template.yaml; YAML schema for items, including all metadata fields required for filtering, regime-score computation, and analysis. • prompt_templates.md; Exact text for all 5 conditions, system prompt, and sampling parameters. • grading_rubric.md; Full 5-point coherence rubric with anchor descriptions and exemplar responses at each level. • step_counter.py; Python implementation of the §4.8 two-pass step-counting pipeline (regex segmentation + LLM-judge classification). These files are the operational substrate of the section. They are checked into the project repository and released with the final manuscript per pre-reg §10. 4.12 Summary Section 4 constructs the FEH-79 item pool against which Section 2’s central empirical claim—that under meta-uncertainty, additional CoT steps degrade accuracy—is tested. The pool spans four Knightian categories (non-recurrent forecasting, novel/synthetic scenarios, open-ended dilemmas, strategic uncertainty), three confound-control sets (reference, aleatory, calibration probe), and is constructed against operational criteria (K1)-(K3) that operationalize meta-uncertainty in a way the §3.2 regime score can pick up. The section is methodologically novel in three respects: 1. Explicit uncertainty taxonomy (aleatory / ambiguity / epistemic / meta-uncertainty) that distinguishes Knightian items from the items typically used in LLM-uncertainty benchmarks, which conflate the categories. 2. Confound-control paired design that lets the analysis distinguish the predicted CoT-degradation effect from base-rate prompt sensitivity, aleatory variance, and per-model calibration quirks. 3. Pre-specified Knightian construction criteria that are checkable by reviewers, replicable across benchmark-building teams, and resistant to post-hoc cherry-picking. The methodology is the load-bearing contribution; the 79 specific items are mechanically given the methodology. The full item pool is produced in Phase 4 of the construction protocol per the timeline above. 5 Methods The pre-registration (§6) fixes the analysis before the data exists. This section clarifies and fixes the data: what was asked, which models were used, how many times, and how the answers were scored. The study is a single factorial run, executed once, with no peeking at the hypothesis along the way. Seven models each answered the same 45 items under five reasoning-length conditions, five times over, for a total of 7,875 responses. The sections below give the panel, the items, the length ladder, and the procedure, and close with an honest account of where the run departed from the letter of the plan. 5.1 Design The design relies on three main pillars: • The within-item factor is based on five conditions (C1 through C5) that instruct the model to answer directly or to deliberate at an increasing length (§5.4). • The between-item factor is regime: each item is either high-meta-uncertainty (a Knightian frame with no determinate answer) or a low-regime control with a definite one (§5.3). • The between-model factor is the panel of seven models (§5.2). Every cell of the 7 × 45 × 5 grid was sampled five times under independent seeds, so the full run is 7×45×5×5=7,8757× 45× 5× 5=7,875 model responses (Figure 5.1). The primary contrast the analysis cares about is not the five-level condition ladder but a binary collapse of it: short (C1, direct answer) against long (C2–C5, any instructed deliberation). That collapse, fixed by random assignment, is the exogenous treatment whose interaction with regime carries H1H_1 (eq. 6.1′ ). The five-level ladder is retained for the dose-response and mechanistic analyses, where it earns its keep. Figure 1: Anatomy of the run. The confirmatory study as a fully-crossed factorial: seven models (five open-weight, two frontier) each answer all 45 items — 31 high-regime Knightian frames and 14 low-regime controls (§5.3) — under the five reasoning-length conditions C1–C5 (§5.4), replicated five times under independent seeds. The crossing yields 7×45×5×5=7,8757× 45× 5× 5=7,875 responses, collected with zero generation failures: 1,125 per model, 1,575 per condition, 5,425 high-regime and 2,450 low-regime. 5.2 The model panel The panel spans an order of magnitude in scale and crosses the open-weight/frontier line. Five models run locally through Ollama: phi3.5 (3.8B), mistral:7b-instruct (7B), and the Qwen2.5 instruction-tuned family at 7B, 14B, and 32B. Two are frontier models accessed via their providers’ APIs and pinned to dated snapshots for reproducibility: claude-sonnet-4-5 (snapshot 2025-09-29) and gpt-4o (snapshot 2024-11-20). Each model answered every cell, so each contributed exactly 1,125 of the 7,875 responses. The five-model open-weight backbone was the registered minimum, chosen to span the 3B-to-32B scale on a single workstation. The two frontier models were added on the strength of the pre-data feasibility read, which found the predicted effect cleaner on capable models that sit above the small-model recognition floor. This matters for the paper’s target: the o1/o3 reasoning-scaling narrative the study sets itself against is a claim about capable models, not 3B ones, so the panel should reach the models the claim is actually about. The local hardware was a single RTX 4070 Super with 12 GB of VRAM. Four of the five local models fit in that budget. The 32B model does not: at roughly 21 GB quantized, it runs on a CPU/GPU split, which made its long-reasoning conditions disproportionately slow but did not change the responses it produced. The full local panel and both frontier models completed every assigned cell. 5.3 Items and the regime split The 45 items are drawn from the FEH-79 pool (v0.4), built and validated in §4. They are divided into 31 high-regime items and 14 controls. The 31 high-regime items are the Knightian frames whose answers can be scored without a human panel: 20 non-recurrent forecasts (K1), nine novel-synthetic frames built around coined entities (K2), and two strategic-uncertainty dilemmas (K4). Each is a categorical question whose correct response is that the question cannot be settled. Operationally, the item offers the model an explicit cannot-be-determined choice alongside the substantive options, and a response is scored correct when it selects that choice. This makes recognition of indeterminacy the dependent variable in the high regime and machine-checkable, which is what lets the run proceed without an expert coherence panel. The 14 controls provide definitive answers and exist to rule out the obvious alternative explanations. Six are arithmetic items with an exact result; eight are aleatory questions whose answer is a calculable rate. They run through the identical five-condition ladder, so any change in accuracy they show under instructed reasoning reflects the manipulation’s effect on well-defined tasks, against which the high-regime change is read. Regime is therefore assigned by item category in this run: Knightian frames are the high bin, and controls are the low bin. The pre-registration specified a finer instrument: a continuous regime score binned into its quartiles; the substitution and the reason for it are set out in §5.6. 5.4 The reasoning-length ladder Every item was presented to every model under five conditions that held the question fixed and varied only the instructions for how to reason. The system prompt is constant across conditions and asks for a single-line final answer in the form “Final answer: <X>”. The five user-prompt conditions are: • C1 (direct). Answer in a single line, with no other text. • C2 (short). Think step by step, briefly, in three steps or fewer, then give the final answer. • C3 (medium). Think step by step in about seven steps. • C4 (long). Reason carefully, considering multiple angles, in roughly fifteen steps. • C5 (unconstrained). Think step by step with no length cap, then conclude. C1 is the short level of the primary contrast; C2 through C5 are long. The ladder from C2 to C4 is an ordered sequence of instructed reasoning used in dose-response analysis. C5 is left out of that ordinal contrast because removing the length cap makes its realized step count non-monotone against C4, but it remains a long trial in the primary model. All conditions sampled at temperature 0.7, top-p 0.95, and a 2,048-token cap. For categorical items, the answer choices, including cannot-be-determined, were listed in the prompt. 5.5 Procedure, scoring, and reproducibility Each of the 7,875 cells was sampled 5 times using a deterministic seed derived from a hash of the model id, the frame id, the condition, and the replication index. The seeds are recorded and released, so the full run reproduces from the code and the seed function rather than from a stored output dump. Local models were called through the Ollama endpoint; frontier models through their providers’ APIs, with per-provider rate limiting, keyed and resumed on the pinned snapshot ID so that an interrupted run never restarts on a completed cell. A response was scored by extracting the line after the Final answer: marker with a fixed regular expression, falling back to the last sentence when the marker was absent. High-regime items were graded for correct selection of cannot-be-determined; controls were graded against their answer key. The matcher is the significant-figure- and LaTeX-robust scorer validated in §4, so that a correct answer written 1/2, 0.5, or 12 is not marked wrong on formatting. Realized reasoning steps, used only in the secondary mechanistic analysis (§6, R7), were counted by the regex-plus-LLM-judge pipeline, whose agreement with human coding reached Cohen’s κ=0.80κ=0.80 in the §4 validation. The run was robust in the plainest sense: of 7,875 cells, none failed. Every cell yielded an extractable answer across its replications, so the analysis contains no missing-data pattern or imputation. Data collection ran model by model, local panel first and frontier last, with no interim analysis of the hypothesis at any point. Data and code availability. All code, the FEH-79 item pool, the complete response dataset (the 7,875 scored cells), the registered analysis and robustness scripts, and the figure-generation code are openly available at https://github.com/Evolutionairy-AI/Free-Energy-Heuristics. Every result and figure in this paper reproduces from the released data without any model or API access. 5.6 Deviations from the pre-registration A pre-registered study earns its credibility partly by reporting where execution diverged from the plan. The divergences here fall into two kinds: changes already fixed by a timestamped amendment before data collection, and simplifications made during execution of the study that are disclosed here for the first time. Two changes were made by amendment before any confirmatory data existed. The panel grew from the five registered open-weight models to seven, adding the two frontier models, for the reason given in §5.2. The primary regressor was switched from realized step count to the randomly assigned length factor, demoting the step-count model to a secondary mechanistic analysis; the realized count is endogenous, and §7.5 shows in the data why the switch was warranted. Both are recorded in the pre-registration’s amendment history. Three simplifications belong to the executed study and are disclosed here. First, the regime was assigned by item category (Knightian frame versus control) rather than by the registered continuous regime score binned at its quartiles. The category label is not an arbitrary stand-in: every high-regime item earned its place by passing the falsifiable Knightian-ness criteria of §4, including the K2 cross-model disagreement screen and the K3 indexed-corpus floor, so the assignment carries the construction-time evidence that the item sits in the high-meta-uncertainty regime. What the run does not do is compute the continuous §3.2 score on these items, and the reason is specific: that score’s calibration-error component requires a per-item confidence elicitation that the confirmatory protocol did not collect. The continuous-score binning is therefore out of reach for this dataset, not merely coarser, and §8.5 carries it as a limitation. Second, the high-regime dependent variable is machine-checked recognition of indeterminacy rather than the registered three-expert coherence rating; the 46 pool items that required the expert panel were held out of this run and are deferred to a later study, leaving the 31 auto-scorable Knightian frames. Third, the analysis sampler was lengthened from the registered four chains of 2,000 to four chains of 4,000 to clear the registered convergence threshold on one nuisance parameter, a change that left the verdict unmoved (§7.8). None of the three alters the hypothesis, the decision gate, or the direction of the test. Each narrows what the run can claim, and each is carried forward into the reading of the results rather than set aside. One other execution detail needs to be explicitly mentioned. The six low-regime arithmetic controls were the easy-tier calibration items R-101–R-106 (one- and two-digit products) rather than the §4.5.1a reference block R-001–R-010, so that control accuracy would not be floored by arithmetic the smaller models cannot do; the pilot found Mistral-7B correct on only 1 of 15 attempts at 47 × 83. The eight aleatory controls were A-001–A-008. The substitution bears on neither the gate nor the direction of the test; it keeps the control block within the panel’s competence, so the flat control line reads as evidence of the regime, not of arithmetic skill. 6 The Pre-Registered Analysis Plan Every number in §7 is the output of a procedure fixed before the data existed. This section explains that procedure. It states: • The model that was to be fit. • The single quantity that would decide the hypothesis. • The rule by which the hypothesis could fail. • The checks that would probe the result without being allowed to rescue it. The plan was registered ahead of collection and is reproduced here so the reader can see that the analysis in §7 had no room to move once the responses came in. Where the executed run departed from the plan, the departure is logged in §5.6 rather than smuggled into this section. The discipline matters most for a result that confirms a prediction. Confirmation is cheap when the analyst is free to choose the test after seeing the data, and that freedom need not be conscious to do its damage. Fixing the model, the estimate, and the threshold in advance is what converts “we found an effect” into “we found the effect we said we would look for.” The rest of this section is that commitment, written down before §7 was known. 6.1 The primary model The dependent variable is binary: for each response, whether the model’s answer was correct (§5.5). The primary model is a hierarchical Bayesian logistic regression of that outcome on the assigned-length factor, the regime indicator, and their interaction, with the model panel and the item pool entered as crossed random effects: yij∼Bernoulli(σ(ηij))y_ij 6.0mu 6.0muBernoulli (σ( _ij) ) ηij=β0+β1longj+β2regimei+β3(longj×regimei)+αm(ij)+γm(ij)longj+ui _ij 6.0mu= 6.0mu _0+ _1\,long_j+ _2\,regime_i+ _3\,(long_j×regime_i)+ _m(ij)+ _m(ij)\,long_j+u_i Here σ is the logistic function, longj∈0,1long_j∈\0,1\ is the assigned-length factor (short C1 versus long C2–C5), regimei∈0,1regime_i∈\0,1\ marks the high-meta-uncertainty bin, αm _m and γm _m are the model-level random intercept and random slope on length, uiu_i is the item-level random intercept, and m(ij)m(ij) indexes the model that produced response ijij. The random slope γm _m is the part of the model that lets the length effect differ across the panel, which is what keeps a pooled estimate from hiding the per-model spread that §7.4 turns out to need. The coefficient that carries the hypothesis is β3 _3. Because length is binary, β3 _3 is exactly the difference-in-differences on the log-odds scale: the high-regime short-to-long change in accuracy minus the low-regime change. A negative β3 _3 is the claim that instructed reasoning costs more in accuracy when the question is genuinely undetermined than when it has an answer. The priors are weakly informative on the logit scale: β0,β2∼(0,2.5) _0, _2 N(0,2.5) for the intercept terms, β1,β3∼(0,1) _1, _3 N(0,1) for the slopes, and HalfNormal(1)(1) for the random-effect standard deviations. The model is fit in PyMC with non-centered random effects. The registered sampler is four chains of 2,000 warm-up and 2,000 draws, with convergence required at R^<1.01 R<1.01 and an effective sample size of at least 400 for each parameter. (The executed run lengthened the chains to clear that threshold on one nuisance parameter; §5.6 and §7.8 record the change.) 6.2 The decision gate The hypothesis is determined by two quantities, both of which are required and fixed in advance. The first is directional: the posterior probability that the interaction is negative must exceed 0.950.95, Pr(β3<0∣data)>0.95.Pr( _3<0 data)>0.95. The second is a magnitude floor, and it is deliberately not stated on the log-odds scale, where coefficients are hard to read. It is stated as a robust implied accuracy drop: the high-regime short-to-long change in predicted accuracy, evaluated at the empirical high-regime base rate. This quantity depends only on the posterior of (β1+β3)( _1+ _3), which sidesteps the global intercept that hierarchical logistic models identify only weakly. The posterior median of that drop must exceed six percentage points. The six-point floor was not chosen by eye; it was selected by the power analysis, which, among candidate floors of six, eight, and ten points, maximized power at a zero false-confirmation rate under the null. Both conditions must hold for H1H_1 to be confirmed. A direction without a magnitude is a real but trivial effect; a magnitude without a direction is noise dressed as a finding. The gate requires both because the framework predicts a substantive failure mode rather than a detectable one. The original v0.2 statistic, the joint probability Pr(β3<0∧|β3|>|β1|)Pr( _3<0 | _3|>| _1|) that the high-regime length effect strictly reverses the low-regime one, is retained as a reported effect size. It no longer gates the decision, but it is informative, and §7.3 reports it. 6.3 Falsification A pre-registration that cannot fail is not a pre-registration. The plan fixes two ways the framework’s central prediction can be falsified by these data. The interaction can come out the wrong way with conviction: if Pr(β3≥0∣data)>0.95Pr( _3≥ 0 data)>0.95, the data say that instructed reasoning helps, not hurts, under meta-uncertainty, and the prediction is wrong as posed. Or the interaction can be right in sign but empty in size: if Pr(β3<0)>0.95Pr( _3<0)>0.95 while the robust implied drop falls below three percentage points, the effect is real and negligible, which the framework counts as a failure, because it predicts a cost large enough to matter for a decision, not one detectable only at scale. Anything between confirmation and falsification is reported as inconclusive: a directional probability short of 0.950.95, or a directional effect with an implied drop between three and six points. Inconclusive is an honest outcome, not a withheld one, and it leaves the framework plausible but unconfirmed. 6.4 The secondary realized-steps analysis The original registered model regressed accuracy on the realized number of reasoning steps a model produced rather than on the length it was assigned: ηij=β0+β1stepsij+β2regimei+β3(stepsij×regimei)+αm(ij)+γm(ij)stepsij+ui _ij 6.0mu= 6.0mu _0+ _1\,steps_ij+ _2\,regime_i+ _3\,(steps_ij×regime_i)+ _m(ij)+ _m(ij)\,steps_ij+u_i with the step count standardized within the model. Amendment 2 demoted this model from primary to secondary (it is robustness check R7) on principled grounds: realized step count is endogenous. A model chooses how many steps to write while it answers, and that choice is entangled with how hard the particular attempt is going, so the realized count is a post-treatment variable rather than a manipulation. Conditioning on it can induce a spurious association, and §7.5 shows that in this data it does exactly that, reversing the sign. The secondary analysis is therefore reported, not gated. It is estimated with the assigned condition as an instrument for the realized step count, a two-stage form that recovers the per-step effect while correcting the endogeneity that motivated the demotion. It keeps the mechanistic question visible alongside the causal one without letting the contaminated regressor decide the hypothesis. 6.5 Robustness checks Six checks were pre-specified to assess whether the primary result would remain robust to reasonable changes to the analysis. They do not feed the confirmatory inference; they test its conditional invariance, and all are reported regardless of the primary outcome. • R1. Re-fit with paragraph-level rather than sentence-level step segmentation. • R2. Re-fit with terciles rather than quartiles for regime binning, and on the continuous regime score with a linear interaction. • R3. Re-fit on the regime score with the calibration-error component dropped, per the §3.7 simulate-and-recover finding. • R4. Re-fit on items held out of regime-score calibration to rule out double-dipping. • R5. Fit the primary model (separately per model) and report the consistency of the β3 _3 sign across the panel. • R6. Add an item-level random slope on the length factor, since unmodeled item-to-item heterogeneity can understate uncertainty. R7 is the realized-steps analysis of §6.4. The robustness battery had not been run when §7 was drafted; §7.7 reports its status and the schedule for completing it. 6.6 Confound controls Three control comparisons were registered to test the regime score’s validity rather than the hypothesis, and all three are reported, regardless of the primary result. The first is a prompt-sensitivity baseline: a set of well-attested items that should not land in the high-regime bin, used to check that the regime signal is not an artifact of base-rate prompt sensitivity. The second is an aleatory control: items with high inherent randomness but no meta-uncertainty, which should show cross-seed variance without the reasoning-degradation pattern. The third is a per-model calibration baseline: calibration error on a fixed well-defined probe set, subtracted from each item’s calibration error before standardization so that a model’s calibration quirks do not drive its regime assignment. If a failure occurs on any one of them, the affected signal is corrected per the registered procedure or dropped by the aggregator. In the executed run the low-regime control block carries this load directly, and §7.6 reads the result. 6.7 What we committed not to do The plan closes with the commitments that give the rest of it force. After data collection began, the model in (6.1′ ), the regime bins, the conditions, and the items were not to change for any confirmatory analysis. No analysis would be run to rescue a non-confirmatory result and then reported as confirmatory. No items would be re-binned and no thresholds moved after the primary result was seen. No subset of models or items would be selected to recover an effect. Any post-collection observation worth pursuing would be reported as exploratory and labeled as such. These are the promises that make a confirmation in §7 mean something, and the deviations in §5.6 are disclosed precisely because the promises were made. 7 Results Ask a capable language model a question that has no answer, and most of the time it will tell you so. Across the seven models we tested, a direct request for an answer to one of the FEH-79 Knightian items (a forecast with no base rate, a dilemma with no dominant option, a coined entity that appears nowhere in the training corpus) drew a correct answer (“this cannot be determined”) about three-quarters of the time. Then instruct the same models, on the same items, to reason step by step. They begin to build. One inference licenses the next; a plausible consideration hardens into a premise; the premise narrows the field; and a few sentences later, the model has reasoned its way to a confident, specific answer to a question that admits none. The rate of correct abstention falls. In the model where it falls hardest, qwen2.5:14b, it drops by nearly forty points. That behavior is the prediction of Theorem 2.6.1 made visible in a transcript: past the truncation point k∗k^*, each additional inference step does not refine the estimate, it manufactures one. The theoretical sections argued that under meta-uncertainty, the expected free energy of k-cue inference turns upward, so that more deliberation should cost accuracy rather than buy it. The regime is not a laboratory artifact. It is the situation of a clinician facing a presentation the literature does not cover and being asked, all the same, for an answer. This section reports what happened when we put that prediction to a pre-registered test across seven models and 7,875 trials. The prediction held. The pre-registered interaction coefficient β3 _3 of the primary model (eq. 6.1′ ) is negative with posterior probability 1.00001.0000, and the implied high-regime accuracy drop is large: a posterior-median 17.317.3 percentage points. Both arms of the §6.2 decision gate are cleared, so H1H_1 is confirmed. The effect is not universal, and the honest version of the result is more interesting than the headline: instructed reasoning hurts some models badly, barely touches others, and is absent-to-reversed in the two weakest. The rest of this section lays out the findings and their boundaries in the order a skeptic would want them. 7.1 Data collected The full design ran to completion: seven models, forty-five items, five length-graded conditions, five replications each. That is 7,875 model responses, and every one of them returned a usable answer, with no generation failures and no empty cells anywhere in the grid. The panel divides into the five open-weight local models (phi3.5, mistral:7b-instruct, qwen2.5 at 7B, 14B, and 32B) and the two frontier API models (claude-sonnet-4-5, gpt-4o); each contributed exactly 1,125 responses. The five conditions are balanced at 1,575 responses apiece. The items were split into 31 high-regime Knightian frames, scored for correct recognition of indeterminacy, and 14 low-regime controls, scored against a known answer key: 5,425 high-regime responses and 2,450 low-regime responses in all. The procedure that produced these cells, and the three places where the executed study departs from the letter of the pre-registration, are documented in §5; the deviations are logged in §5.6 and revisited where they bear on a specific result below. 7.2 The signature Begin with the raw cell means, before any model is fit. Figure 2 plots accuracy against condition for the two regimes. The two lines tell the whole story at a glance, and they tell opposite stories. In the high-meta-uncertainty regime, accuracy falls as the assigned reasoning length grows. Direct-answer prompting (C1) yields correct recognition of indeterminacy on 77.1%77.1\% of trials. Each successive condition asks for more deliberation, and each returns a lower number: 66.5%66.5\% at C2, 64.7%64.7\% at C3, 64.4%64.4\% at C4, 62.1%62.1\% at C5. The descent is monotone. Pooling the four instructed-reasoning conditions against the direct baseline, the high-regime accuracy drop is 12.712.7 percentage points. In the low-regime control items, the same manipulation has almost no effect. Accuracy sits near 0.820.82 across the board (83.7%83.7\% direct, and a pooled instructed-reasoning mean within two points of it), with no downward trend; if anything, C5 is the strongest control condition of the five. The low-regime short-to-long drop is 1.731.73 percentage points, which is to say, noise. The difference between those two drops is the descriptive heart of the paper. It is the difference-in-differences: 12.712.7 points lost in the high regime against 1.731.73 lost in the low regime, for a raw DiD of 10.9710.97 percentage points (Figure 3). Telling a model to think harder does not, in general, lower its accuracy. It lowers accuracy where the question is genuinely undetermined and leaves it alone where the question has an answer. That selectivity is the fingerprint the framework predicted, and it is present in the data before any Bayesian machinery touches it. Figure 2: The signature. Mean accuracy by condition (C1 direct answer through C5 maximal instructed reasoning) for the high-meta-uncertainty regime (Knightian items, scored for correct recognition of indeterminacy) and the low-regime control items (scored against an answer key). The high-regime line descends monotonically; the control line is flat. Error bars: 95% cluster bootstrap over items. Figure 3: Difference-in-differences. High-regime short-to-long accuracy drop (12.7 p) against the low-regime drop (1.73 p); the gap between them (10.97 p) is the model-free DiD. 7.3 The primary test The descriptive signature could, in principle, be an artifact of which items landed in which bin, or of one model dominating the average. The pre-registered analysis exists to rule that out. Eq. 6.1′ is a hierarchical Bayesian logistic regression with the assigned-length factor long, the regime indicator, their interaction, and random intercepts and length-slopes for every model and item. Because long is binary, its interaction coefficient β3 _3 is exactly the difference-in-differences on the log-odds scale, now estimated with all the heterogeneity modeled rather than averaged away. The posterior is unambiguous. The interaction is negative with posterior probability Pr(β3<0∣data)=1.0000Pr( _3<0 )=1.0000; not a single posterior draw placed the high-regime length effect on the helpful side of zero. The posterior median is β3=−0.692 _3=-0.692. The low-regime length slope is small and slightly negative (β1=−0.183 _1=-0.183), so instructed reasoning carries a faint generic cost even on well-defined items, but the high-regime penalty is roughly four times larger; the probability that the high-regime effect strictly exceeds the low-regime effect in magnitude (the full-reversal condition of the original v0.2 statistic) is 0.9050.905. The §6.2 gate does not run on the log-odds coefficient directly. It runs on the robust implied accuracy drop: the high-regime short-to-long change in predicted accuracy, evaluated at the empirical high-regime base rate and depending only on the posterior of (β1+β3)( _1+ _3), which sidesteps the weakly identified global intercept. That estimand has a posterior median of 17.317.3 percentage points, with a 95% credible interval of [7.7,25.5][7.7,25.5]. The model-based estimate runs higher than the raw 12.712.7-point descriptive drop because the regression recovers the within-condition effect after absorbing item and model variance that flattens the raw average. Both pre-registered requirements are therefore comfortably met. The directional gate asked for Pr(β3<0)>0.95Pr( _3<0)>0.95 and received 1.00001.0000. The magnitude gate asked for a posterior-median robust drop above 66 points and received 17.317.3, with the entire credible interval clear of the threshold. By the rule fixed in §7.1 of the pre-registration before any data was seen, H1H_1 is confirmed (Figure 4). Figure 4: Posterior of the interaction. Posterior density of β3 _3 (eq. 6.1′ ) with the 95% credible interval and the zero line; inset, the posterior of the robust implied accuracy drop in percentage points against the 6-point confirmation threshold. 7.4 Where the effect lives, and where it doesn’t A single posterior probability of 1.00001.0000 invites the wrong picture: a uniform law, every model degrading in lockstep. That is not what the data show. The pooled interaction is decisive because the hierarchical model borrows strength across the panel, and the random-slope structure of eq. 6.1′ exists precisely so the average does not stand in for the members. Pre-registered check R5 refits the primary model on each model’s data alone. Figure 5 shows the per-model interaction β3 _3 with its credible interval. The spread is wide. The Qwen family carries the degradation — qwen2.5:32b returns β3=−2.92 _3=-2.92 (95% CrI [−4.05,−1.81][-4.05,-1.81]), qwen2.5:7b −2.86-2.86 [−3.81,−1.94][-3.81,-1.94], and qwen2.5:14b −2.23-2.23 [−3.29,−1.17][-3.29,-1.17] — each places its entire credible interval below zero, at Pr(β3<0)=1.000Pr( _3<0)=1.000. The two frontier models point the same way without clearing the bar on their own — gpt-4o at β3=−0.89 _3=-0.89 [−2.15,+0.39][-2.15,+0.39] and claude-sonnet-4-5 at −0.80-0.80 [−2.09,+0.48][-2.09,+0.48] are directionally negative with posterior probabilities of 0.9130.913 and 0.8840.884, which suggests (rather than establishes) that this phenomenon takes place when each is read in isolation. The two weakest models do not show the penalty. phi3.5 returns β3=+0.98 _3=+0.98 [+0.19,+1.79][+0.19,+1.79], Pr(β3<0)=0.009Pr( _3<0)=0.009 (on phi3.5 the instruction to reason measurably helps). mistral:7b-instruct sits at +0.52+0.52 with an interval straddling zero. These coefficients are unpooled, one fit per model, so they run larger in magnitude than the single panel-average β3=−0.69 _3=-0.69 of §7.3, which holds one fixed interaction across all seven systems; the per-model fits let each model’s interaction find its own level. The modeled estimate corrects a distortion in the raw difference-in-differences. By the descriptive DiD, qwen2.5:14b was the most affected model, and the Qwen ordering ran 14B, 7B, 32B; the modeled β3 _3 reorders them 32B, 7B, 14B, because the cell-mean contrast is inflated by control-arm movement that the hierarchical model absorbs. Two of the Qwen models posted control accuracies that rose under instructed reasoning, and phi3.5’s control block fell by more than twenty points. Hence, the descriptive DiD credited those models for movement in the denominator rather than the numerator. The within-model interaction, estimated with item variance modelled, is the cleaner read and is the one reported in Figure 5; the descriptive per-model DiD is retained in the supplement. The honest summary is therefore narrower than “reasoning degrades LLMs,” and more defensible for being narrow. On properly constructed Knightian items, instructed reasoning degrades the capable mid-to-large models decisively across the Qwen family and directionally in the two frontier models. In comparison, the two weakest models, which sit on the small-model recognition floor, do not show it and, in phi3.5’s case, run the other way. The pooled confirmation is real because it pools; the per-model picture shows the effect is carried by the capable models, which is exactly where the o1/o3 reasoning-scaling narrative stakes its claim. This heterogeneity is not a blemish on the confirmation; it is the texture the theory anticipates. The truncation point k∗k^* depends on how much meta-uncertainty a model carries into the item, and there is no reason to expect that quantity to be constant across architectures and scales. What the pre-registered test establishes is that, pooled across the panel with heterogeneity modeled, the regime-dependent penalty is real and large. What Figure 5 establishes is that one should not sell it as a universal law. Figure 5: Per-model interaction (R5). Posterior of the assigned-length interaction β3 _3 (eq. 6.1′ ) refit separately for each model, item random effects retained and model effects dropped. Points are posterior medians, bars 95% credible intervals, ordered from most negative (qwen2.5:32b, −-2.92) to most positive (phi3.5, +0.98); the dashed line marks zero. The three Qwen models clear Pr(β3<0)=1.000Pr( _3<0)=1.000; gpt-4o and claude are directionally negative but cross zero; phi3.5 reverses. 7.5 The realized-steps reversal, and why it does not threaten the result The pre-registration’s original v0.2 model regressed accuracy on the realized number of reasoning steps a model produced, not on the length condition it was assigned. Amendment 2 demoted that model to a secondary analysis (R7) and promoted the assigned-length factor to primary, on the ground that realized step count is endogenous. The confirmatory data make the reason for that amendment concrete, because the two models disagree in sign. Fit on realized steps, the interaction reverses: β3=+0.416 _3=+0.416 with a 95% credible interval of [0.271,0.560][0.271,0.560] and Pr(β3<0)=0.000Pr( _3<0)=0.000. Read naively, that says longer reasoning helps in the high regime, the opposite of the primary finding. The naive reading is wrong, and it is wrong for a reason the design foresaw. Realized step count is not assigned; it is chosen by the model in the course of answering, and it is entangled with the difficulty of the particular attempt. A trace that runs long because the model is floundering, and a trace that runs long because the model is working carefully, both enter the realized-steps regressor as “more steps.” Within a fixed assigned condition, the items a model handles well are not a random subset of the items it sees more steps on. Conditioning on a post-treatment variable that the treatment itself moves is a textbook way to induce a spurious association, and the sign flip is its signature. The assigned-length factor has none of this trouble because it is fixed by random assignment before the model writes a word. That is why the pre-registration gates H1H_1 on the assigned-length interaction and reports the realized-steps interaction as a mechanistic companion, not a test. We flag the reversal here in the open rather than leaving a reviewer to find it: the two numbers point opposite ways, the primary one is the causal estimand, and the secondary one is exactly the confound that motivated the switch. Two further analyses settle the question the reversal raises. The instrumented form of R7 uses the assigned condition as an instrument for the realized step count, recovering the per-step effect while purging the endogeneity that contaminates the naive fit. Instrumented, the high-regime per-step interaction is −0.53-0.53 (95% CrI [−0.86,−0.20][-0.86,-0.20], Pr<0=0.999Pr<0=0.999). Once the realized count is instrumented, its variation is driven solely by random assignment; more reasoning then lowers accuracy in the high regime, consistent with the primary model and in contrast to the naive fit. And the naive fit is not even stable to how a step is counted. Re-segmenting reasoning by paragraph rather than sentence (robustness check R1) flips the realized-steps interaction back to negative, β3=−0.39 _3=-0.39 [−0.54,−0.25][-0.54,-0.25]. An estimand whose sign turns on the segmentation rule is not one on which a hypothesis should rest. The assigned-length primary makes that case structurally rather than by assertion (Figure 6). Figure 6: Why the primary is the assigned length. Posterior of the interaction β3 _3 under the two regressors on a shared zero-centered axis: the assigned-length primary (eq. 6.1′ , median −0.69-0.69, mass entirely left of zero) against the realized-steps secondary (eq. 6.1, median +0.42+0.42, mass entirely right). The realized-steps regressor is endogenous; its interaction reverses to −0.53-0.53 under instrumentation and to −0.39-0.39 under paragraph-level segmentation. 7.6 The controls did their job The low-regime control block is not a sideshow; it is the within-study evidence that the high-regime degradation is about the regime and not about the manipulation. The fourteen control items carry well-defined answers (arithmetic with a correct result, aleatory questions with a calculable rate), and they were run through the identical five-condition length ladder. If instructed reasoning degraded accuracy because longer outputs are simply riskier, or because the scorer penalizes verbose answers, the controls would fall too. They do not. The control accuracy holds near 0.820.82 across all five conditions, with a short-to-long drop of 1.731.73 points that is statistically indistinguishable from zero. That flat control line discharges the most obvious alternative explanations as a group: the degradation is not a generic length penalty, not a scoring artifact tied to output verbosity, and not a property of the prompt templates, since the templates are shared across regimes. What remains is the regime itself. The three pre-registered confound analyses of §6.6 (the prompt-sensitivity baseline, the aleatory-specific control, and the per-model calibration baseline) are addressed descriptively by this control block; the full per-signature analyses are not yet computed and will be reported in supplementary material. 7.7 Robustness checks The pre-registration commits to a battery of robustness checks that probe whether the primary result survives reasonable changes to the analysis. They do not gate the confirmatory inference; they test its conditional invariance. Four of the seven have now been run on the confirmatory data, and they are reported here whatever they show; three remain, and are named as outstanding rather than quietly dropped. Three of the four point the same way as the primary. The per-model refit (R5) is read in §7.4: the interaction is negative in five of seven models and decisive in the three Qwen systems. The paragraph-segmentation refit (R1) and the instrumented realized-steps analysis (R7) are read in §7.5; both return negative interactions, and R1 in particular testifies against the realized-steps regressor rather than against the primary. The one check that pushes back deserves to be stated first and plainly. R6 adds an item-level random slope on the length factor, the maximal random-effects specification, so that each item carries its own length effect and the fixed interaction must survive once that heterogeneity is absorbed. Under R6 the interaction stays negative at the median, β3=−0.47 _3=-0.47, but its credible interval widens to [−1.03,+0.15][-1.03,+0.15] and the directional posterior falls to Pr(β3<0)=0.935Pr( _3<0)=0.935, below the 0.950.95 bar the gate uses. The registered model, fixed before the data, carries an item random intercept and no item random slope, so the confirmation in §7.3 is untouched by definition; the verdict stands on the model that was registered. But R6 was pre-registered too, and what it shows is real: with item-to-item variation in the length effect modeled, the effect holds its direction while its certainty slips past the threshold. The honest reading is that the degradation is robust in sign and somewhat sensitive in magnitude to how item heterogeneity is partitioned, a sensitivity a 45-item panel should be expected to carry. Three checks cannot be run on this dataset, and the reason is worth stating exactly. The tercile and continuous-score binning (R2), the calibration-component-dropped variant (R3), and the held-out-calibration-item subset (R4) all operate on the continuous §3.2 regime score, whose calibration-error component requires a per-item confidence elicitation the confirmatory protocol did not collect. The study used category assignment instead, which rests not on a post-hoc score but on the construction-time Knightian-ness criteria of §4: the high-regime items are the ones that passed the K2 cross-model disagreement screen and the K3 indexed-corpus floor. The regime-score-versus-category alignment check that would convert §5.6’s “coincide by construction” from argument to measurement needs the same uncollected input. None of the three bears on the confirmatory gate, which runs on the assigned-length factor; all are scoped to a follow-up study that collects the confidence data, and §8.5 carries the limitation. 7.8 Convergence and the sampler deviation One departure from the pre-registered analysis bears reporting in full, because it touches the convergence criterion the pre-registration names explicitly. The registered sampler was four chains of 2,000 warm-up and 2,000 draws at a target acceptance of 0.950.95. At those settings one parameter missed the registered R^<1.01 R<1.01 threshold: the high-regime intercept b2b_2, at R^=1.0128 R=1.0128. The interaction coefficient that carries H1H_1 was never the problem: β3 _3 returned R^=1.0004 R=1.0004 with an effective sample size near 4,000 at the registered settings. To clear the gate for every parameter rather than report a result the pre-registration’s own convergence rule would flag, the official fit lengthens the chains to 4,000 warm-up and 4,000 draws at a target acceptance of 0.990.99. This is a post-hoc increase in sampling effort, not a change to the model, the data, the regressor, the priors, or the decision rule. At the official settings the fit converges cleanly: R^max=1.0056 R_ =1.0056 across all monitored parameters and a minimum effective sample size of 1,6921,692, both inside the registered bounds. The verdict is identical to the registered-sampler verdict to three decimal places. The longer chain bought convergence on a nuisance intercept; it changed nothing about the conclusion. We note the increase as a deviation per §9.4 of the pre-registration. 7.9 Summary The paper’s load-bearing test returned a positive result. Under genuine uncertainty about the answer (the Knightian regime the FEH-79 items were built to create), instructing a language model to reason at greater length lowers its accuracy by a pre-registered estimate of 17.317.3 percentage points, with a posterior probability of 1.00001.0000 that the effect runs in the predicted direction. The same instruction, applied to matched control items with definite answers, costs nothing. The dissociation is exactly the one predicted by Theorem 2.6.1, and it clears the decision gate that was fixed before the data existed. Several honesties travel with the headline. The effect is not uniform across the panel: it is decisive in the three Qwen models, directional but not individually conclusive in the two frontier models, and reversed in phi3.5 while inconclusive in Mistral, so the defensible claim is regime-dependent degradation carried by capable models, not a universal penalty. The realized-steps analysis reverses sign, exactly as an endogenous post-treatment regressor should, which is why the causal weight rests on the randomly assigned length factor; instrumented, the per-step effect returns to the predicted negative. One pre-registered robustness variant, a maximal item-level random slope, holds the effect’s direction but lets its directional posterior fall just below the registered threshold, a sensitivity §7.7 reports in full. And the convergence gate was met only after lengthening the sampler, a deviation we log rather than bury. None of these weakens the confirmation; each one sharpens what the confirmation is allowed to claim. What the result licenses, and what it does not, is the subject of §8. 8 Discussion The prediction that organizes this paper is uncomfortable: that on the questions that matter most, the ones without a settled answer, teaching a model to think harder makes it worse. §7 reports that the prediction held (cleanly) on seven models and 7,875 trials and held only where the theory said it should. This section takes the measure of that result. It says what the confirmation licenses, what it does not, and where the framework is exposed. 8.1 Summary The paper makes five linked contributions, and the empirical one is no longer a promise. (i) A theoretical result (Theorem 2.6.1; Theorem 2.7.4) shows that fast-and-frugal heuristics, long defended on ecological-rationality grounds and long suspected on Bayesian-approximation grounds, are both at once: under meta-uncertainty over prior precision, the expected free energy of k-cue inference is U-shaped in k, and its minimum slides toward fewer cues as the variance of the meta-precision prior grows. The thirty-year Bayesian-versus-heuristic debate dissolves into a regime-dependent equivalence: under precise priors, heuristics approximate the Bayesian answer; under imprecise priors, they are the answer. (i) An operationalization (§3) turns the theoretical regime indicator into measurable quantities, exposing the regime score as an interaction hypothesis rather than a main effect. (i) The FEH-79 benchmark (§4) is built under three falsifiable Knightian-ness criteria, so that any item can be challenged on evidence rather than taste. (iv) The pre-registered study (§5–§7) was run and its hypothesis confirmed: the assigned-length interaction β3 _3 is negative with posterior probability 1.00001.0000 and an implied high-regime accuracy drop of 17.317.3 points, clearing a decision gate fixed before the data existed. (v) The framework stands as a counter-narrative to the contemporary assumption that inference-time compute is a free dial, an assumption that the confirmed result directly contradicts on properly constructed uncertainty tasks. 8.2 The theoretical resolution and its scope The Bayesian-versus-heuristic debate lasted thirty years because both sides were partly right. The theorem in §2.6 makes the dividing condition explicit: k∗=Kk^*=K, full inference, is optimal when στ2<τregime _τ^2< _regime; k∗<Kk^*<K, truncation, is optimal otherwise. The two camps were each generalizing from a different regime. The contribution is not to crown a winner but to give the regime indicator a closed-form characterization and show that it separates empirically. The scope claim is narrow, and the confirmation does not widen it. The theorem holds in the Gaussian-Gamma sequential-inference setting of §2, with appendix-level extension to a binary toy model, and it does not generalize automatically to all heuristics, all priors, or all task structures. The empirical work tests a specific operational prediction, the §6 H1H_1. The §7 confirmation narrows the framework’s plausibility without proving the general theorem. The asymmetry is worth stating plainly: a positive result, which we have, raises the framework’s credibility; a negative result would have falsified the operational prediction without touching the theoretical claim, provided a confound was found in the operationalization or the benchmark. We committed in advance to treating the pre-registered test as load-bearing, and we hold to that. It carried. 8.3 What the empirical commitments buy Three commitments separate this work from comparable critiques of LLM reasoning. The Knightian-ness criteria make benchmark construction falsifiable: any FEH-79 item can be challenged by showing that it admits a base rate, that it draws unanimous substantive cross-model agreement under multi-seed validation, or that it appears in indexed corpora. The hierarchical Bayesian analysis with its pre-registered gate, a directional posterior above 0.950.95 joined to a robust accuracy drop above six points, forecloses the inflated-significance artifacts that a 45-item by 5-condition panel could otherwise manufacture. And the control items close off the obvious alternative readings: §7.6 shows that the degradation does not affect well-defined, matched questions, so it is not a generic penalty for longer output or an artifact of the scorer. What these commitments buy is a result a skeptic can trust without trusting the authors. The pre-registration is the binding contract, and §5.6 discloses every place the executed run drew outside its lines. 8.4 An unexpected internal consistency check One finding from building the benchmark warrants discussion, as the framework did not anticipate it. During the pool’s v0.2-to-v0.3 development, several candidate items showed modal-answer flipping across independent seeds at temperature 0.7: an item that single-shot validation called disagreement, and that re-screening called unanimous-substantive, or the reverse. K1-005, an asset-class macro forecast, anchored hard on one narrative under multi-seed and split across narratives in earlier single-shot runs; K4-003, a five-firm data-sharing dilemma, alternated between a dominant withhold answer and a balanced spread. That cross-seed instability is the same signal, σb _b in the §3.2 notation, that the regime score is built to detect. Items that flip across seeds carry high σb _b at the construction stage, which is the Knightian signature the §7 analysis tests for at the condition stage. The K2 multi-seed pre-screen is, in effect, the §3 regime-score detector run while the benchmark is being built rather than while the experiment is being run. The operationalization (§3) and the benchmark protocol (§4), designed independently, converge on one diagnostic. This was not built in; it surfaced from the work of making the pool, and it suggests the multi-seed protocol may generalize as a benchmark-construction primitive in its own right. 8.5 Limitations The confirmation comes with boundaries, and the most important one is visible in the data itself. The effect is not universal. §7.4 is unambiguous: the degradation is large in the Qwen family, directional but inconclusive in the two frontier models, and reversed in phi3.5, with Mistral inconclusive. The pooled posterior is decisive, but the per-model picture forbids the headline “reasoning degrades LLMs.” The defensible claim is narrower: on properly constructed Knightian items, instructed reasoning degrades capable models that sit off the small-model recognition floor, and how far it degrades them varies by architecture and scale, as the truncation point k∗k^* should. A reader who wants universal law will not find one here, and should not. The panel does not include reasoning-specialized models. The seven models span 3B to frontier scale, but none is an o1, o3, or R1-style system trained to spend test-time compute. The confirmed effect lives in capable models that were not built to reason at length; whether it survives, worsens, or is engineered away in other models is the single most consequential open question, and §8.6 treats it as the next study, not a claim this one can make. The regime was operationalized by category rather than by the continuous score. Regime was assigned by item category, Knightian versus control, rather than by the registered continuous regime score binned at its quartiles (§5.6). The assignment is not unfounded: the high-regime items passed the §4 construction criteria, the K2 cross-model disagreement screen and the K3 indexed-corpus floor, which is direct evidence that they inhabit the regime. What is missing is a measurement of the continuous score on these items, and the gap is precise. The score’s calibration-error component is what separates genuine meta-uncertainty from ordinary aleatory variance, and it is exactly the aleatory controls, built to carry answer variance without meta-uncertainty, that such a component is needed to classify; the confirmatory run did not elicit the confidence judgments that component requires. The continuous-score robustness checks (R2–R4) and the score-versus-category alignment are therefore deferred to a study that collects confidence, not run and quietly omitted here. The high-regime outcome was also machine-checked recognition of indeterminacy rather than the registered expert-coherence rating, with the 46 expert-panel items deferred; the confirmation speaks to the auto-scorable Knightian frames, not the full pool. The mechanistic story is partly open. The realized-steps analysis reverses sign (§7.5), the expected behavior of an endogenous post-treatment regressor and the reason the causal weight rests on assigned length. Instrumenting realized steps with the assigned condition recovers a negative per-step effect, and paragraph-level segmentation does the same, so the reversal is a property of the naive regressor rather than evidence against the effect; what remains open is a fully structural account of how realized reasoning mediates the degradation. Four of the seven pre-registered robustness checks are run (§7.7), and one of them, the maximal item-slope variant, lets the directional posterior slip just below the gate while holding the sign; the three checks that depend on the continuous regime score, together with the regime-score-versus-category alignment, are owed to the supplement. Knightian-ness is panel-relative and time-stamped. The K2 criterion is defined against a specific provider panel and at a specific date. As frontier models train on this kind of item, an item’s K2 status can change, so the FEH-79 pool is a dated probe rather than a permanent benchmark. 8.6 Future work The confirmation sharpens the research agenda more than it closes it, and one study comes first. The paper argues against the prevailing view that scaling test-time reasoning is reliably beneficial — yet it tested that view on general-purpose models, not on the reasoning-specialized systems (o1, o3, DeepSeek-R1, and their successors) the view is really about. The priority, then, is to run those models through the identical protocol and ask whether deliberately scaled inference incurs the same regime-dependent accuracy cost we found here, or whether their training has learned to suppress it on questions that admit no determinate answer. That is the most direct possible test of the paper’s claim, run against exactly the systems the claim concerns. Beyond it sit four lines. Precision-routed heuristic agents, which estimate their own regime score and conditionally engage in shorter or longer reasoning, turn the finding into a design principle. The open theoretical questions of §2.10, mean-field accuracy (Q1), alternative meta-precision priors (Q2), finite-k corrections (Q3), and non-Gaussian extensions (Q5), remain, with Q7 extending the Theorem 2.7.4 identity to a richer toolbox. A cross-domain replication on a matched human panel would test whether the regime prediction is species-general or specific to language models. And the deferred expert-coherence study would carry the result from the 31 auto-scorable frames to the full Knightian pool. Nearer at hand, the regime-score robustness checks complete the analysis this paper began. 8.7 Positioning and conclusion The framework is offered against the operating assumption of contemporary AI development that inference-time compute is a free dial, that more reasoning is neutral at worst and helpful at best. Theorem 2.6.1 says that under genuine uncertainty about prior precision, which is the human decision-maker’s situation in nearly every consequential domain, more reasoning is systematically harmful. The empirical claim was that language models inherit the same regime structure, and §7 confirmed it: more instructed reasoning, lower accuracy, on exactly the items where the answer cannot be settled, and nowhere else. Because the test was pre-registered and confirmed, three consequences follow with more than rhetorical force. The design default for systems facing genuinely novel decisions should shift from inference-scaling toward precision-routing, engaging long deliberation only where the regime warrants it. Benchmark selection bias should be corrected by including K1/K2/K3-validated Knightian probes alongside the well-defined-task suites that dominate evaluation today. And the field should expect, and reward, demonstrations of graceful degradation under test-time compute on properly constructed uncertainty tasks, not merely the absence of degradation. The result earns those consequences, but only at the scope §7.4 allows. The degradation is real, large, and pre-registered, and it is not a universal law: it bites capable models hard and spares, or even helps, the weakest, and it has not yet been measured on reasoning-specialized models that would settle the matter. That is the honest shape of the finding. The opposite outcome was equally available before the data came in, which is what lets this one count: had β3 _3 landed on the other side of zero, the counter-narrative would have been wrong as posed and the framework a curiosity about a regime that language models do not inhabit. It did not land there. The paper does not ask the reader to take a side in the old debate. It asks the reader to commit to a regime indicator, run the test, and update, which is what we did. Appendix A.1 — Binary Toy Model Proofs This appendix provides the proofs deferred from §2.1, together with numerical verification. Two formal results are established: • Lemma A.1.2 (corrected statement of section Lemma 2.1.2): the expected meta-divergence is non-decreasing in the number of cues integrated. The section’s sample-wise statement is incorrect; a counterexample is exhibited below. • Proposition A.1.4 (corrected statement of section Proposition 2.1.3): the EFE-minimizing policy admits three qualitative regimes — k*=0, k*=1, and k*=K — separated by two critical concentrations κ _lo(v_seq, μ _0) and κ _hi(v_seq, μ _0). The “one-cue stopping” regime is a narrow band, not the dominant one. A precise statement and numerical characterization follow. A subsidiary result, Lemma A.1.1, gives the closed-form posterior on p_0 after k cues — a 2-component Beta mixture whose components are independent of k and whose mixture weights move with the cue history. The corrections to the section wording are tracked at the end of the appendix. A.1.1 Setup recap Latent state s ∈ 0,1. Prior: P(s=1 | p_0) = p_0. Hyperprior: p_0 ~ Beta(α _0, β _0), with prior mean μ _0 = α _0/(α _0 + β _0) and concentration κ _0 = α _0 + β _0. Binary cues c_j ∈ 0,1 with cue validities v_j ∈ (1/2, 1) and a symmetric error model: P(cj=1∣s=1)=vj,P(cj=0∣s=0)=vj.P(c_j=1 s=1)=v_j, P(c_j=0 s=0)=v_j. Define the cue-product likelihoods after k cues: π1(c1:k)=∏j=1kvjcj(1−vj)1−cj,π0(c1:k)=∏j=1kvj1−cj(1−vj)cj. _1(c_1:k)= _j=1^kv_j^c_j(1-v_j)^1-c_j, _0(c_1:k)= _j=1^kv_j^1-c_j(1-v_j)^c_j. These are the conditional probabilities of the observed cue sequence under s=1 and s=0 respectively, derived from cue conditional independence given s. A.1.2 Lemma A.1.1 — Closed-form posterior on p_0 Lemma A.1.1. For any cue sequence c_1:k, the variational posterior on p_0 is a 2-component mixture of Beta distributions whose components depend on (α _0, β _0) but not on k or on the cue history: p(p0∣c1:k)=w1(c1:k)⋅Beta(p0;α0+1,β0)+w0(c1:k)⋅Beta(p0;α0,β0+1)p(p_0 c_1:k)=w_1(c_1:k)·Beta(p_0;\, _0+1,\, _0)\;+\;w_0(c_1:k)·Beta(p_0;\, _0,\, _0+1) with mixture weights w1(c1:k)=π1(c1:k)μ0Z(c1:k),w0(c1:k)=π0(c1:k)(1−μ0)Z(c1:k),w_1(c_1:k)= _1(c_1:k)\, _0Z(c_1:k), w_0(c_1:k)= _0(c_1:k)(1- _0)Z(c_1:k), where Z(c_1:k) = π _1 μ _0 + π _0 (1 - μ _0) is the marginal cue likelihood. The prior decomposes in the same form with weights (μ _0, 1-μ _0). Proof. The unnormalized posterior is p(p0∣c1:k)∝Beta(p0;α0,β0)⋅[p0π1+(1−p0)π0].p(p_0 c_1:k) (p_0; _0, _0)· [\,p_0 _1+(1-p_0) _0\, ]. Using the identities p0⋅Beta(p0;α0,β0)=μ0⋅Beta(p0;α0+1,β0),p_0·Beta(p_0; _0, _0)= _0·Beta(p_0; _0+1, _0), (1−p0)⋅Beta(p0;α0,β0)=(1−μ0)⋅Beta(p0;α0,β0+1),(1-p_0)·Beta(p_0; _0, _0)=(1- _0)·Beta(p_0; _0, _0+1), (both verified by direct computation using Γ (α _0+1) = α _0 Γ (α _0)), the unnormalized posterior becomes π1μ0⋅Beta(p0;α0+1,β0)+π0(1−μ0)⋅Beta(p0;α0,β0+1). _1 _0·Beta(p_0; _0+1, _0)+ _0(1- _0)·Beta(p_0; _0, _0+1). The normalizing constant is Z = π _1 μ _0 + π _0 (1-μ _0), and division gives the stated form. The prior corresponds to π _1 = π _0 = 1, yielding (w_1, w_0) = (μ _0, 1-μ _0) and reducing to the algebraic identity Beta(α _0, β _0) = μ _0 · Beta(α _0+1, β _0) + (1-μ _0) · Beta(α _0, β _0+1). ■ Consequence for P(s=1 | c_1:k). By direct application of Bayes’ theorem with s marginalized over p_0: P(s=1∣c1:k)=P(c1:k∣s=1)P(s=1)P(c1:k)=π1μ0Z(c1:k)=w1(c1:k).P(s=1 c_1:k)= P(c_1:k s=1)\,P(s=1)P(c_1:k)= _1 _0Z(c_1:k)=w_1(c_1:k). The marginal posterior on the state is exactly the mixture weight on the upper Beta component. This is a clean result that simplifies all downstream computation. A.1.3 Lemma A.1.2 — Expected meta-divergence is non-decreasing Lemma A.1.2 (corrected statement of section Lemma 2.1.2). The expected meta-divergence E_c_1:k[Δ _meta(k)] is non-decreasing in k. Equivalently, the marginal expected meta-cost E[Δmeta(k+1)−Δmeta(k)]=I(p0;ck+1∣c1:k)≥ 0,E[ _meta(k+1)- _meta(k)]\;=\;I(p_0;\,c_k+1 c_1:k)\;≥\;0, with equality iff cue k+1 carries no conditional information about p_0 given the prior cues. Proof. By the tower property of conditional expectation, Eck+1∣c1:k[p(p0∣c1:k+1)]=p(p0∣c1:k).E_c_k+1 c_1:k [\,p(p_0 c_1:k+1)\, ]=p(p_0 c_1:k). By joint convexity of KL divergence in its first argument, Jensen’s inequality gives Eck+1∣c1:k[KL[p(p0∣c1:k+1)∥p(p0)]]≥KL[Eck+1∣c1:k[p(p0∣c1:k+1)]∥p(p0)].E_c_k+1 c_1:k [\,KL[p(p_0 c_1:k+1)\,\|\,p(p_0)]\, ]\;≥\;KL\! [\,E_c_k+1 c_1:k[p(p_0 c_1:k+1)]\, \|\,p(p_0)\, ]. The right-hand side equals KL[p(p_0 | c_1:k) || p(p_0)] = Δ _meta(k). Taking outer expectation over c_1:k preserves the inequality. The equality condition follows from the martingale-convexity formulation: equality holds iff p(p_0 | c_1:k+1) = p(p_0 | c_1:k) almost surely under the predictive distribution, which is the condition that cue k+1 carries no conditional information about p_0. ■ Counterexample to the section’s sample-wise claim. With α _0 = β _0 = 1 (uniform hyperprior) and validities v = (0.9, 0.9), direct computation of Δ _meta(k; c_1:k) for the four cue paths gives: c_1 c_2 Δ _meta(1; c_1) Δ _meta(2; c_1, c_2) sample monotone? 0 0 0.1153 0.1815 yes 0 1 0.1153 0.0000 no 1 0 0.1153 0.0000 no 1 1 0.1153 0.1815 yes When opposite cues cancel (paths 01 and 10), the posterior on p_0 returns exactly to the prior and the meta-divergence collapses to zero. The section’s wording — “Δ _meta(k) is non-decreasing in k” — is therefore false sample-wise. The expectation across the four paths is monotone, as Lemma A.1.2 establishes. Implication for the section. The wording in §2.1 should be changed from “the meta-divergence Δ _meta(k) is non-decreasing in k” to “the expected meta-divergence E[Δ _meta(k)] is non-decreasing in k.” The downstream proofs in §2.5 already work with expected free energy (which is itself an expectation over cues), so no other proofs are affected. A.1.4 Proposition A.1.3 — Marginal benefit of cue k+1 Define the marginal benefit of cue k+1 as the expected reduction in expected free energy: ΔG(k+1)≡E[G(a,k)]−E[G(a,k+1)]=I(s;ck+1∣c1:k)−I(p0;ck+1∣c1:k). G(k+1)\;≡\;E[G(a,k)]-E[G(a,k+1)]\;=\;I(s;\,c_k+1 c_1:k)-I(p_0;\,c_k+1 c_1:k). Proposition A.1.3. In the binary toy model, the marginal benefit of cue k+1 decomposes as the difference of two conditional mutual informations: the information cue k+1 provides about the latent state s (positive contribution to EFE reduction) minus the information it provides about the prior precision parameter p_0 (positive contribution to meta-divergence). Proof. From eqs (2.5.2)–(2.5.4) of the section, the expected free energy after k cues decomposes as E[G(a,k)] = -E[ p(o|s,a)] - I(s; c_1:k) + E[Δ _meta(k)]. The first term is action-dependent but k-independent; it cancels in Δ (k+1). The second term gives I(s; c_1:k+1) - I(s; c_1:k) = I(s; c_k+1 | c_1:k) by chain rule for mutual information. The third term gives E[Δ _meta(k+1) - Δ _meta(k)] = I(p_0; c_k+1 | c_1:k) by Lemma A.1.2 and the same chain rule applied to KL. ■ Consequence. Cue k+1 should be integrated iff I(s; c_k+1 | c_1:k) > I(p_0; c_k+1 | c_1:k) — i.e., it is more informative about the state than about the meta-precision. The optimal k* is the smallest k such that this inequality is violated for c_k+1. A.1.5 Proposition A.1.4 — Three-regime structure Numerical sweep of the binary toy model across (κ _0, μ _0, v_1:K) reveals that the EFE-minimizing policy has three qualitative regimes, separated by two critical concentrations. Proposition A.1.4 (corrected statement of section Proposition 2.1.3). Fix a validity profile v_1:K and prior mean μ _0. There exist critical concentrations 0 < κ _lo(v_1:K, μ _0) ≤ κ _hi(v_1:K, μ _0) such that the EFE-minimizing policy k*(κ _0) has the structure: • (Don’t-observe regime) κ _0 < κ _lo: k* = 0. The meta-divergence cost of even one cue exceeds its information value about s. The agent should act on the prior. • (One-cue stopping regime) κ _lo ≤ κ _0 ≤ κ _hi: k* = 1. The first cue is integrated; subsequent cues incur higher meta-cost than information benefit. This is the take-the-best stopping rule. • (Full integration regime) κ _0 > κ _hi: k* = K. Each cue contributes more state-information than meta-cost; the agent integrates all available cues. The intermediate regime [κ _lo, κ _hi] is a closed interval that may be narrow (and in some validity profiles degenerate to a single point, in which case the transition jumps directly from k*=0 to k*=K). Numerical characterization. A systematic search over v_1:K ∈ (0.5, 1)ˆK, μ _0 ∈ 0.3, 0.5, 0.7, and κ _0 ∈ [0.1, 200] (geometric grid) finds: • Sharp validity gradients yield wider one-cue regimes. Profiles with v_1 ≈ 0.99 and v_j>1 ≤ 0.55 exhibit [κ _lo, κ _hi] of nontrivial width, often near κ _0 ≈ 1. • Uniform validity profiles tend to have κ _lo = κ _hi. The transition then jumps directly from k*=0 to k*=K, and the one-cue regime is degenerate. • Realistic decreasing-validity profiles (the TTB-typical case) lie in between. The one-cue regime exists but is narrow; the qualitative behavior is dominated by the don’t-observe / full-integration dichotomy. The narrowness of the one-cue regime in the binary toy model is a substantive finding: in the discrete binary setting under symmetric error and conditional independence of cues given s, the cue-truncation theorem fires generically only at the boundary between two qualitatively different regimes, not as a wide intermediate regime. This is in contrast to the continuous Gaussian-Gamma case (Theorem 2.6.1, §2.6), where the one-cue regime widens because per-cue meta-cost grows linearly in k rather than saturating after the first cue. Subsidiary observation: TTB is near-optimal even when not strictly optimal. A separate computation shows that for steep validity profiles (v_1 ≈ 0.99, v_j>1 ≤ 0.55), the first cue contributes ≥ 99% of the total attainable information about s across the full cue budget K. Even in regimes where k* = K strictly minimizes EFE, the marginal contribution of cues 2..K is so small that a TTB-style “stop after the first cue” policy is near-optimal in expected reward terms. This near-optimality is the empirically-relevant content of the TTB connection in the binary setting; the strict EFE-optimality in the narrow [κ _lo, κ _hi] window is the formally-derivable but operationally-fragile content. Implication for §2.7. The structural-equivalence argument in §2.7 (Theorem 2.7.1) is the load-bearing claim. The binary toy model serves as motivating intuition and a lower-dimensional sanity check, not as a strict existence proof for the one-cue regime. The section’s claim that “the cue-truncation point k* is generically finite and often equal to one in the binary case” should be hedged: it is attainable in the binary case under specific conditions, but is not the dominant regime. The continuous Gaussian-Gamma model is where the cue-truncation theorem fires robustly across a wide regime of meta-uncertainty. A.1.6 Numerical verification All numerical results in this appendix are produced by binary_toy_monte_carlo.py and binary_toy_kstar_search.py, which compute expected free energy by exhaustive enumeration over cue paths (tractable up to K = 10). The KL divergence between the posterior and prior 2-component Beta mixtures is computed by numerical integration on a 2000-point grid over (0, 1). Verification of Lemma A.1.2 across five validity profiles and four hyperprior configurations: monotone in expectation across all 5 cases. Verification of Proposition A.1.4 across 25 × 5 × 5 × 3 = 1875 parameter combinations: 40 combinations exhibit k* = 1 strictly; the remainder split between k* = 0 and k* = K. The transition curve plots are reproducible from the scripts. A.1.7 Summary of corrections to the v0.4 section The following amendments to §2.1 are recommended on the basis of this appendix: 1. Lemma 2.1.2 wording. Change “the meta-divergence Δ _meta(k) is non-decreasing in k” to “the expected meta-divergence E[Δ _meta(k)] is non-decreasing in k.” Add a one-sentence note about the sample-wise counterexample with reference to this appendix. 2. Proposition 2.1.3 wording. Replace the current single-regime statement (“the EFE-minimizing policy stops at the first discriminating cue”) with the three-regime statement of Proposition A.1.4. Hedge the TTB connection: in the binary case, the one-cue regime is narrow; the cleaner TTB derivation lives in §2.7 via the Gaussian-Gamma machinery. 3. §2.1 closing paragraph. The sentence “the binary toy model is sufficient to motivate the FEH effect and to establish the take-the-best connection in its most natural setting” should be softened to “the binary toy model motivates the FEH effect; the strict cue-truncation regime is more robust in the continuous Gaussian-Gamma model of §2.3–2.6, which is the load-bearing setting for the section’s main results.” These changes weaken §2.1’s strong claims and strengthen the section’s overall honesty. They do not affect the core results in §2.4–2.7, which derive from the Gaussian-Gamma machinery rather than the binary case. Appendix A.2 — Proof of Lemma 2.4.1 (monotonicity of meta-precision divergence) This appendix proves the monotonicity of the meta-precision divergence in the Gaussian-Gamma generative model of §2.2–2.4 (v0.4). The proof parallels the binary case of Appendix A.1: the expectation-form statement is true and admits a clean proof via martingale-convexity of KL; the sample-wise statement is false in general and a counterexample is given. A.2.1 Setup From the v0.4 section: • Generative model (eq 2.2.1–2.2.2): p(s∣τ)=(s;μ,τ−1Σ0),p(cj∣s,τ,γj)=(cj;Ajs,(τγj)−1Idc),p(s τ)=N(s;μ,τ^-1 _0), p(c_j s,τ, _j)=N(c_j;A_js,(τ _j)^-1I_d_c), with hyperprior p(τ) = Gamma(α _0, β _0). • Variational posterior under mean-field q(s, τ) = q(s)q(τ) and standard VBEM (eq 2.4.3): q(τ∣c1:k)=Gamma(αk,βk),αk=α0′+k2,βk=β0′+12∑j=1kγjMj,q(τ c_1:k)=Gamma\! ( _k,\, _k ), _k= _0 + k2, _k= _0 + 12 _j=1^k _jM_j, where α _0' = α _0 + 1/2, β _0' = β _0 + V_s/2, and M_j = E_q(s)[(c_j - A_j s)ˆT(c_j - A_j s)]. The M_j depend on the cue realizations through their effect on q(s). • Meta-precision divergence (eq 2.4.5): Δmeta(k)≡KL[q(τ∣c1:k)∥p(τ)] _meta(k)\;≡\;KL\! [\,q(τ c_1:k)\; \|\;p(τ)\, ] which has the closed form Δmeta(k)=(αk−α0)ψ(αk)−logΓ(αk)Γ(α0)+α0logβkβ0+αkβ0−βkβk. _meta(k)=( _k- _0)ψ( _k)- \! ( _k) ( _0)+ _0 \! _k _0+ _k _0- _k _k. The meta-cost increment of the (k+1)-th cue is C(k+1) ≡ Δ _meta(k+1) - Δ _meta(k). Both Δ _meta(k) and C(k) are random variables depending on the cue history c_1:k through the M_j. A.2.2 Lemma A.2.1 — Sample-wise monotonicity is false in general Lemma A.2.1. Δ _meta(k) is not monotone in k along arbitrary realized cue paths in the Gaussian-Gamma model. Numerical evidence. Across 1000 IID samples of M_j ~ Exp(1) for K = 20, with (α _0', β _0') = (2, 1) and γ _j = 1, 979 sample paths exhibit at least one strict decrease of Δ _meta along the trajectory. Why this happens. Under the conjugate update, both α _k and β _k increase monotonically — α by exactly 1/2 per cue (deterministic) and β by γ _j M_j / 2 per cue (random, non-negative). However, the Gamma KL is not monotone in (α, β) along arbitrary trajectories: increasing β while α is fixed shifts the posterior mean α/β away from the prior mean α _0/β _0, but increasing α while β is fixed shifts it toward the prior mean (since posterior mean grows). When a particular M_j realization is large enough that β _k/α _k overshoots β _0/α _0, subsequent cues with smaller M_j can pull β _k/α _k back, decreasing the KL. Consequence. The section’s wording in v0.3 — “Δ _meta(k) is non-decreasing in k, with strict increase whenever the k-th cue is non-degenerate” — is sample-wise false. The wording must be restated in expectation form. (This parallels the binary-case correction in Appendix A.1.) A.2.3 Lemma A.2.2 — Expected meta-divergence is non-decreasing (corrected Lemma 2.4.1) Lemma A.2.2 (corrected statement of section Lemma 2.4.1). Under the generative model (2.2.1)–(2.2.2) with mean-field factorization (2.4.1), the expected meta-precision divergence E[Δ _meta(k)] is non-decreasing in k. Equivalently, E[Δmeta(k+1)]−E[Δmeta(k)]=I(τ;ck+1∣c1:k)≥ 0,E[ _meta(k+1)]-E[ _meta(k)]\;=\;I(τ;\,c_k+1 c_1:k)\;≥\;0, with equality iff cue k+1 carries no conditional information about τ given the prior cues c_1:k. Proof. By the tower property of conditional expectation applied to the variational posterior, Eck+1∣c1:k[q(τ∣c1:k+1)]=q(τ∣c1:k).E_c_k+1 c_1:k [\,q(τ c_1:k+1)\, ]\;=\;q(τ c_1:k). By joint convexity of KL divergence in its first argument, Jensen’s inequality gives Eck+1∣c1:k[KL[q(τ∣c1:k+1)∥p(τ)]]≥KL[Eck+1∣c1:k[q(τ∣c1:k+1)]∥p(τ)]=Δmeta(k).E_c_k+1 c_1:k [\,KL[q(τ c_1:k+1)\,\|\,p(τ)]\, ]\;≥\;KL\! [\,E_c_k+1 c_1:k[q(τ c_1:k+1)]\, \|\,p(τ)\, ]\;=\; _meta(k). Taking outer expectation over c_1:k preserves the inequality and yields E[Δ _meta(k+1)] ≥ E[Δ _meta(k)]. The mutual-information identity follows from the chain rule for KL: E[Δ _meta(k+1)] - E[Δ _meta(k)] = I(τ; c_k+1 | c_1:k). Equality holds iff the conditional posterior q(τ | c_1:k+1) equals q(τ | c_1:k) almost surely under the predictive distribution of c_k+1 | c_1:k. ■ Numerical verification. Across five test cases spanning low-to-extreme meta-uncertainty (α _0' ∈ 0.5, 1, 2, 5, 20), E[Δ _meta(k)] is strictly non-decreasing with k, verified by Monte Carlo with 5000 samples per case and K = 30 cues. A.2.4 Note on the proof’s reliance on the variational posterior The proof uses only the tower property — that E[q(τ | c_1:k+1) | c_1:k] = q(τ | c_1:k) — and Jensen’s inequality for KL. The tower property holds for any consistent sequential Bayesian update, which the standard VBEM iteration produces. Higher-order corrections to the mean-field approximation (the structured-variational direction of open question Q1) preserve consistency and therefore preserve Lemma A.2.2; only the magnitude of E[Δ _meta(k)] may change. That magnitude change is itself quantified exactly for the conjugate model in §A.2.5 (Proposition A.2.3). A.2.5 Mean-field accuracy: exact comparison in the conjugate model (resolves Q1) Open question Q1 (§2.10) asks how much the mean-field factorization q(s, τ) = q(s)q(τ) distorts the meta-precision posterior relative to a structured family. In the Gaussian-Gamma model of §2.2 the question admits an exact answer, because the model is fully conjugate: the joint posterior p(s, τ | c_1:k) is Normal-Gamma in closed form, so the mean-field posterior can be compared against the truth rather than against another approximation. (For a non-conjugate generative model no exact joint is available and a structured variational family is the appropriate object; the result below is specific to the conjugate model used to prove Theorems 2.6.1 and 2.7.1.) Setup (scalar case, d_s = d_c = 1). After k cues with c_j = s + noise (A_j = 1) and intrinsic precisions γ _j, write Λ _k = λ _0 + Σ _j≤ γ _j and the precision-weighted mean m_k = (λ _0 μ _0 + Σ _j≤ γ _j c_j)/Λ _k. The exact marginal posterior over τ is Gamma(α _k, β _kˆex) with αk=α0+1+k2=α0′+k2,βkex=β0+12[λ0μ02+∑j≤kγjcj2−Λkmk2]. _k= _0+ 1+k2= _0 + k2, _k^ex= _0+ 12 [\, _0 _0^2+ _j≤ k _jc_j^2- _km_k^2\, ]. The mean-field VBEM fixed point is q(s) = N(m_k, S_k), q(τ) = Gamma(α _k, β _kˆmf), with S_k = 1/(⟨ τ⟩ Λ _k), ⟨ τ⟩ = α _k/β _kˆmf, and βkmf=β0+12Eq(s)[λ0(s−μ0)2+∑j≤kγj(cj−s)2]. _k^mf= _0+ 12\,E_q(s)\! [\, _0(s- _0)^2+ _j≤ k _j(c_j-s)^2\, ]. The exact and mean-field posteriors carry the same shape α _k: the shape counts data dimensions, not their values. Proposition A.2.3 (mean-field rate inflation). In the conjugate Gaussian-Gamma model the mean-field and exact marginal posteriors over τ are both Gamma(α _k, ·) with the common shape α _k = α _0' + k/2, and their rates are related exactly by βkmf=βkex⋅αkαk−12. _k^mf= _k^ex· _k _k- 12. Equivalently the relative rate error is (β _kˆmf −- β _kˆex)/β _kˆex = 1/(2α _k −- 1), maximal at the first cue and decreasing monotonically as k grows (and as α _0 grows). Proof. Expanding the mean-field rate with E_q(s)[(x −- s)ˆ2] = (x −- m_k)ˆ2 + S_k, βkmf=β0+12[λ0((mk−μ0)2+Sk)+∑j≤kγj((cj−mk)2+Sk)]. _k^mf= _0+ 12 [\, _0 ((m_k- _0)^2+S_k )+ _j≤ k _j ((c_j-m_k)^2+S_k )\, ]. Because m_k is the precision-weighted mean, the data-dependent quadratics satisfy the identity λ0(mk−μ0)2+∑j≤kγj(cj−mk)2=λ0μ02+∑j≤kγjcj2−Λkmk2, _0(m_k- _0)^2+ _j≤ k _j(c_j-m_k)^2\;=\; _0 _0^2+ _j≤ k _jc_j^2- _km_k^2, which is exactly twice the bracket in β _kˆex; and the variance terms sum to (λ _0 + Σ γ _j) S_k = Λ _k S_k. Hence βkmf=βkex+12ΛkSk=βkex+12⟨τ⟩=βkex+βkmf2αk, _k^mf= _k^ex+ 12 _kS_k= _k^ex+ 12 τ = _k^ex+ _k^mf2 _k, using S_k = 1/(⟨ τ⟩ Λ _k) and ⟨ τ⟩ = α _k/β _kˆmf. Solving the fixed-point equation, β _kˆmf(1 −- 1/(2α _k)) = β _kˆex, i.e. β _kˆmf = β _kˆex · α _k/(α _k −- ½). ■ Remark (under-estimation of posterior precision and variance). Since the shape is shared, ⟨ τ⟩ ˆmf = α _k/β _kˆmf = (α _k −- ½)/β _kˆex = (1 −- 1/(2α _k))·⟨ τ⟩ ˆex and Varˆmf(τ) = (1 −- 1/(2α _k))ˆ2 · Varˆex(τ). Mean-field under-estimates both the posterior mean and the posterior variance of the meta-precision, by a factor vanishing as α _k → ∞ — the expected behaviour when a coupling (here s–τ) is severed. Consequence for Δ _meta and k*. Because the shape is identical, the entire mean-field error in Δ _meta(k) = KL[q(τ | c_1:k) ∥\| p(τ)] is the rate shift of Proposition A.2.3. Numerically (verify_meanfield_and_tau_regime.py, Parts 0 and A; the closed form is confirmed to 3×10ˆ-13 against the iterated VBEM fixed point), across the meta-uncertainty grid: Regime α _0' σ 2_τ k*_exact k*_MF rate err @k=1 rate err @k* C¯ C(k*) rel err Low 20 0.05 29 29 2.5% 1.5% 1.1% Medium 5 0.20 24 24 10.0% 3.0% 0.6% High 2 0.50 23 23 25.0% 3.9% 0.7% Very high 1 1.00 22 22 50.0% 4.4% 0.1% Extreme 0.7 1.43 22 22 71.4% 4.5% 0.1% The mean-field error in E[Δ _meta(k)] is an under-estimate, concentrated at the first cue (≈ 16% in the high-meta regime, falling below 2% by k ≈ 5). Crucially the EFE-optimal stopping point k* = argmin E[G(k)] is identical under the mean-field and exact posteriors in every regime: at the stopping point the rate error is below 5% and the marginal meta-cost C¯ C(k*) — the quantity the argmin actually turns on — is recovered to within ≈ 1%. Mean-field thus preserves the exact location of the optimal truncation; the residual is confined to an O(1/α _k) bias in the magnitude of the meta-cost. This answers the quantitative half of Q1. Multivariate note. For state dimension d_s > 1 the constant ½ in Proposition A.2.3 becomes d_s/2 and Λ _k a matrix, but the structure is unchanged: the shape is shared and the rate is inflated by α _k/(α _k −- d_s/2) → 1. The section’s analysis uses the scalar case, for which the result is exact as stated. Appendix A.3 — Proof of Theorem 2.6.1 (cue-truncation) A.3.1 Setup From §2.5 (eqs 2.5.1–2.5.4), the expected free energy under the v0.4 model decomposes as G(a,k)=−Eq[logp(o∣s,a)]⏟pragmatic+I(s;c1:k)⏟epistemic+Δmeta(k)⏟meta-precision cost.G(a,k)\;=\; -E_q[ p(o s,a)]_pragmatic\;+\; I(s;\,c_1:k)_epistemic\;+\; _meta(k)_meta-precision cost. The pragmatic term is k-independent (depends on the action distribution at decision time). The marginal benefit of the (k+1)-th cue, in expectation under the predictive distribution, is therefore E[ΔG(k+1)]=E[G(a,k)]−E[G(a,k+1)]=I(s;ck+1∣c1:k)−I(τ;ck+1∣c1:k).E[ G(k+1)]\;=\;E[G(a,k)]-E[G(a,k+1)]\;=\;I(s;\,c_k+1 c_1:k)-I(τ;\,c_k+1 c_1:k). Cue k+1 is integrated under the EFE-optimal policy iff E[Δ (k+1)] > 0. The optimal stopping rule is k*(σ 2_τ) = argmin_k E[G(a, k)], equivalent to the smallest k such that the conditional information about s from cue k+1 no longer exceeds the conditional information about τ. The threshold τ _regime is defined (eq 2.5.5) as τregime=infστ2>0:k∗(στ2)<K. _regime= \\,σ^2_τ>0:\,k^*(σ^2_τ)<K\,\. A.3.2 Theorem A.3.1 (corrected statement of section Theorem 2.6.1) Theorem A.3.1. Under the generative model (2.2.1)–(2.2.2) with mean-field factorization (2.4.1), with K available cues: (a) Low meta-uncertainty regime. When σ 2_τ < τ _regime, the expected free energy E[G(a, k)] is monotonically non-increasing in k for all k ≤ K. The EFE-optimal policy integrates all available cues: k*(σ 2_τ) = K. (b) High meta-uncertainty regime. When σ 2_τ ≥ τ _regime, there exists a finite k*(σ 2_τ) < K such that E[G(a, k)] is decreasing for k ≤ k* and non-decreasing for k > k*. The EFE-optimal policy is to integrate exactly k* cues. (c) Optimal cue ordering. Among orderings of the K cues, the descending-cue-validity ordering greedily maximizes E[Δ (k)] at each step k ≤ k*, provided cue intrinsic precisions γ _j satisfy a non-anticorrelation regularity condition with cue validities (formalized below). The theorem is restated in expectation form. The sample-wise version is false: across 1000 random parameter configurations, only 198 sample-wise G(k) trajectories are U-shaped; the rest exhibit multiple sign changes due to noise in individual cue realizations. Active inference defines the optimal policy as argmin_k E[G(k)], so the expectation form is what the theorem must establish. A.3.3 Proof of (a) — Low meta-uncertainty regime Claim. If σ 2_τ < τ _regime, then E[Δ (k)] ≥ 0 for all k = 1, ..., K. Proof. By Lemma A.2.2 and the chain rule for mutual information, E[ΔG(k)]=I(s;ck∣c1:k−1)−I(τ;ck∣c1:k−1).E[ G(k)]=I(s;\,c_k c_1:k-1)-I(τ;\,c_k c_1:k-1). Both quantities are non-negative. By the data-processing inequality applied to the chain τ → s → c_k, the cue c_k carries information about τ only through s: I(τ;ck∣c1:k−1)≤I(s;ck∣c1:k−1)I(τ;\,c_k c_1:k-1)≤ I(s;\,c_k c_1:k-1) in the limit σ 2_τ → 0 (where τ is essentially deterministic and observing s reveals the same about τ as observing nothing). For finite σ 2_τ, the meta-information I(τ; c_k | c_1:k-1) is bounded above by a quantity that vanishes as σ 2_τ → 0. Specifically, I(τ;ck∣c1:k−1)=O(στ2)as στ2→0,I(τ;\,c_k c_1:k-1)\;=\;O(σ^2_τ) σ^2_τ→ 0, following from the leading-order expansion of the Gamma KL in σ 2_τ (computed from the closed form in eq 2.4.5). Therefore for σ 2_τ sufficiently small (i.e., below τ _regime), E[Δ (k)] > 0 for all k, and E[G(a, k)] is monotonically decreasing in k. ■ A.3.4 Proof of (b) — High meta-uncertainty regime Claim. If σ 2_τ ≥ τ _regime, then E[G(a, k)] is U-shaped in k, with a unique minimum at some finite k*. Proof. Define the marginal expected info gain I¯ I(k) = I(s; c_k | c_1:k-1) and the marginal expected meta-cost C¯ C(k) = I(τ; c_k | c_1:k-1). Then E[Δ (k)] = I¯ I(k) - C¯ C(k). Step 1: I¯ I(k) is monotonically non-increasing in k. By the data-processing inequality applied to the conditional mutual information, I(s;ck∣c1:k−1)≤I(s;ck),I(s;\,c_k c_1:k-1)≤ I(s;\,c_k), and conditioning on c_1:k-1 weakly reduces remaining information about s available from c_k (since s becomes more determined). The standard “diminishing returns” property of Bayesian information gain applies: as c_1:k-1 accumulate, H(s | c_1:k-1) decreases, leaving less remaining information for c_k to provide. Step 2: C¯ C(k) is bounded below by a positive constant in the high-meta-uncertainty regime. From eq 2.4.3, the rate parameter β _k grows linearly in k (in expectation, E[β _k] = β _0' + (k/2) γ _bar m_bar where γ _bar, m_bar are average cue intrinsic precision and prediction error). The expected marginal increment C¯ C(k+1) = E[Δ _meta(k+1) - Δ _meta(k)] is bounded below by c_min(σ 2_τ) > 0 for σ 2_τ ≥ τ _regime. (The bound c_min is computable in closed form from the digamma derivative; details in Appendix A.4.) Step 3: E[Δ (k)] changes sign exactly once. By Steps 1 and 2, I¯ I(k) is monotonically decreasing and C¯ C(k) is bounded below. Therefore E[Δ (k)] = I¯ I(k) - C¯ C(k) is monotonically decreasing (since I¯ I(k+1) ≤ I¯ I(k) and C¯ C(k+1) ≥ C¯ C(k) in expectation, by Lemma A.2.2). The smallest k for which E[Δ (k+1)] ≤ 0 is k*. By construction k* < K whenever σ 2_τ ≥ τ _regime. The function E[G(a, k)] is therefore decreasing for k ≤ k* and non-decreasing for k > k*, i.e., U-shaped with minimum at k*. ■ Numerical verification. Across five (α _0', β _0', γ _bar, m_bar) configurations spanning low-to-extreme meta-uncertainty, E[G(a, k)] is U-shaped with finite k* in every case. The trajectory of k* as σ 2_τ increases: Regime α _0' σ 2_τ ~ 1/α _0' k* (out of K=30) Low 20 0.05 30 (full integration) Medium 5 0.2 22 High 2 0.5 14 Very high 1 1.0 9 Extreme 0.5 2.0 5 k* drops monotonically as meta-uncertainty rises, exactly as Theorem A.3.1 predicts. A.3.5 Proof of (c) — Optimal cue ordering Claim. The descending-cue-validity ordering greedily maximizes E[Δ (k)] at each step k ≤ k*, under the regularity condition that cue intrinsic precisions γ _j are not strongly anti-correlated with cue validities v_j. Proof. At step k, the agent selects which unused cue to integrate next. The marginal expected benefit of integrating cue j (where j ranges over unused cues) is E[ΔG(k+1;cue j)]=I(s;cj∣c1:k)−I(τ;cj∣c1:k).E[ G(k+1;cue j)]\;=\;I(s;\,c_j c_1:k)-I(τ;\,c_j c_1:k). Step 1: First term is monotone in v_j. For symmetric cue likelihoods, the conditional mutual information I(s; c_j | c_1:k) is monotonically increasing in cue validity v_j (higher validity yields more information about s). This is standard from the Bayesian information theory of binary classification. Step 2: Second term scales with γ _j. From eq 2.4.3, integrating cue j increments β _k by γ _j M_j / 2. The marginal meta-cost contribution I(τ; c_j | c_1:k) therefore scales (approximately, in leading order) with γ _j. Step 3: Greedy ordering by validity is optimal under non-anticorrelation. Define the cue selection score S_j ≡ I(s; c_j | c_1:k) - I(τ; c_j | c_1:k). The descending-validity ordering selects cues by decreasing v_j. This maximizes the first term at each step. It also maximizes S_j provided the second term is approximately equal across cues (γ _j is approximately constant), or more weakly, provided Cov(v_j, γ _j) ≥ 0. Under this regularity condition, descending validity is greedily optimal. When Cov(v_j, γ _j) < 0 (high-validity cues have low intrinsic precision, or vice versa), the optimal ordering deviates from pure validity ordering and trades off v_j against γ _j. This case is empirically uncommon (cue validity and cue intrinsic precision are typically positively correlated — informative cues are also reliably measured), but warrants flagging for empirical work. ■ Operational consequence. The cue-truncation theorem under v0.4 reduces to the take-the-best procedure (descending-validity selection, truncated stopping) under the regularity condition that cue intrinsic precisions are not anti-correlated with cue validities. This is the formal Gaussian-Gamma anchor of the structural-equivalence argument in §2.7. A.3.6 Summary of corrections to the v0.4 section The following amendments to §2.4–2.6 are recommended: 1. Lemma 2.4.1 wording. Restate in expectation form: “the expected meta-precision divergence E[Δ _meta(k)] is non-decreasing in k.” Reference Appendix A.2 for proof and counterexample. 2. Theorem 2.6.1 wording. Restate in expectation form throughout: E[G(a, k)] is monotone (a) / U-shaped (b) / minimized by descending-validity ordering (c). Note that sample-wise versions are false in general. 3. Theorem 2.6.1(c). Add the regularity condition Cov(v_j, γ _j) ≥ 0 (cue intrinsic precisions not anti-correlated with cue validities). State that this is empirically typical but warrants check in specific applications. 4. Definition of τ _regime. Tighten: τ _regime is the smallest σ 2_τ > 0 such that the marginal expected meta-cost C¯ C(K) = I(τ; c_K | c_1:K-1) exceeds the marginal expected info gain I¯ I(K) = I(s; c_K | c_1:K-1). This is the operational definition. Its well-definedness — that the crossing function g(σ 2_τ) = C¯ C(K) −- I¯ I(K) is increasing in σ 2_τ and so has a unique root, and that this root coincides with the k* < K characterization because E[Δ (k)] is monotone-decreasing in k — is established in §2.5 (“Well-definedness of the threshold”) and verified numerically in verify_meanfield_and_tau_regime.py (Part B): across a sweep of σ 2_τ the two definitions agree to grid resolution and g exhibits a single sign change (C¯ C(K) monotone increasing up to Monte-Carlo error). These changes parallel the §2.1 corrections from Appendix A.1 (sample-wise → in-expectation throughout). They do not affect the load-bearing structure of the section — only the rigor of how the load-bearing claims are stated. Appendix A.4 — Sufficient Conditions for Exact FFH–TTB Action Identity This appendix resolves the open question Q6 (§2.10): under what conditions does the structural FFH–TTB equivalence of Theorem 2.7.1 upgrade to exact, sample-wise identity of the agents’ action distributions? The result: a single explicit condition on the cue-validity profile — the Descending Dominance (D) condition — suffices. The meta-precision prior tail does not enter the sufficient condition; it determines only whether FFH is in the truncation regime at all (the precondition στ2≥τregime _τ^2≥ _regime of Theorem 2.6.1). This is a sharper resolution than Q6 anticipated. A.4.1 Setup and Notation Consider a two-alternative comparative judgment between options A and B. There are K binary discriminating cues, indexed j=1,…,Kj=1,…,K, with realized signal dj∈−1,+1d_j∈\-1,+1\: dj=+1d_j=+1 means cue j favors A, dj=−1d_j=-1 favors B. We adopt Gigerenzer and Goldstein’s [6] convention that the cue set has already been filtered to those that discriminate between the alternatives; non-discriminating cues are excluded from the set rather than counted toward stopping. Each cue j has validity vj∈(1/2,1)v_j∈(1/2,1) — the probability that the cue’s signal correctly indicates the better option: P(dj=+1∣s=A)=vj,P(dj=−1∣s=A)=1−vj,P(d_j=+1 s=A)=v_j, P(d_j=-1 s=A)=1-v_j, and symmetrically for s=Bs=B. Cues are conditionally independent given s. Define the per-cue log-likelihood ratio: Lj:=logvj1−vj> 0(since vj>1/2).L_j\;:=\; v_j1-v_j\;>\;0 (since v_j>1/2). The validities are ordered descending: v(1)≥v(2)≥…≥v(K)v_(1)≥ v_(2)≥…≥ v_(K) (equivalently L(1)≥L(2)≥…≥L(K)L_(1)≥ L_(2)≥…≥ L_(K)). We adopt a uniform prior on the comparative state, μ0:=P(s=A)=1/2 _0:=P(s=A)=1/2. The two agents. Both observe the same cue realization c=(d(1),…,d(K))c=(d_(1),…,d_(K)) and emit an action a∈A,Ba∈\A,B\ identified with +1,−1\+1,-1\. • Take-the-best (TTB) stops at the first cue in descending-validity order and chooses by its sign: aTTB(c)=sign(d(1)).a_TTB(c)\;=\;sign(d_(1)). • FFH integrates cues up to the EFE-optimal truncation k∗k^* of Theorem 2.6.1 and chooses by the sign of the posterior log-odds. Under uniform prior: aFFH(c)=sign(∑j=1k∗L(j)d(j)).a_FFH(c)\;=\;sign\! ( _j=1^k^*L_(j)\,d_(j) ). A.4.2 The Descending Dominance Condition Definition A.4.1 (Descending Dominance, D). The validity profile (v(1),…,v(K))(v_(1),…,v_(K)) satisfies the Descending Dominance condition if for every i∈1,…,K−1i∈\1,…,K-1\: L(i)>∑j=i+1KL(j).L_(i)\;>\; _j=i+1^KL_(j). In words: each cue’s log-LR strictly exceeds the sum of all weaker cues’ log-LRs. Geometric-decay sufficient form. A practical sufficient condition: if there exists ρ∈(0,1/2)ρ∈(0,1/2) such that L(j+1)≤ρ⋅L(j)L_(j+1)≤ρ· L_(j) for all j, then (D) holds. The bound follows from the geometric sum: ∑j=i+1KL(j)≤L(i)⋅∑m=1K−iρm<L(i)⋅ρ1−ρ≤L(i), _j=i+1^KL_(j)\;≤\;L_(i)· _m=1^K-iρ^m\;<\;L_(i)· ρ1-ρ\;≤\;L_(i), the last inequality holding for ρ<1/2ρ<1/2. The geometric-decay form is convenient for empirical work because ρ can be estimated from cue-validity statistics directly. Examples. (i) v=(0.95,0.75,0.60,0.55)v=(0.95,0.75,0.60,0.55): L≈(2.94,1.10,0.41,0.20)L≈(2.94,1.10,0.41,0.20), slacks L(i)−∑j>iL(j)≈(1.24,0.49,0.21)>0L_(i)- _j>iL_(j)≈(1.24,0.49,0.21)>0 — D holds. (i) v=(0.70,0.70,0.70)v=(0.70,0.70,0.70): L≈(0.85,0.85,0.85)L≈(0.85,0.85,0.85), slack at i=1i=1 is −0.85<0-0.85<0 — D fails. A.4.3 Theorem 2.7.4: Sample-wise Action Identity Theorem 2.7.4 (Exact sample-wise FFH–TTB action identity). Assume: (i) The cue-validity profile satisfies the Descending Dominance condition (D). (i) The prior is uniform: μ0=1/2 _0=1/2. (i) FFH operates with EFE-optimal truncation k∗≥1k^*≥ 1 from Theorem 2.6.1 (which holds whenever the cue set is non-empty and informative). Then for every cue realization c=(d(1),…,d(K))∈−1,+1Kc=(d_(1),…,d_(K))∈\-1,+1\^K: aFFH(c)=aTTB(c).a_FFH(c)\;=\;a_TTB(c). Consequently, the marginal action distributions of FFH and TTB coincide exactly: PFFH(a=A)=PTTB(a=A)andPFFH(a=B)=PTTB(a=B).P_FFH(a=A)\;=\;P_TTB(a=A) P_FFH(a=B)\;=\;P_TTB(a=B). Proof. Fix a cue realization c. By definition, aTTB(c)=sign(d(1)),aFFH(c)=sign(S(c)),S(c):=∑j=1k∗L(j)d(j).a_TTB(c)=sign(d_(1)), a_FFH(c)=sign\! (S(c) ), S(c):= _j=1^k^*L_(j)\,d_(j). Decompose the FFH sum into leading and trailing terms: S(c)=L(1)d(1)+R(c),R(c):=∑j=2k∗L(j)d(j).S(c)\;=\;L_(1)d_(1)\;+\;R(c), R(c):= _j=2^k^*L_(j)d_(j). Bound the trailing term using |d(j)|=1|d_(j)|=1 and (D): |R(c)|≤∑j=2k∗L(j)≤∑j=2KL(j)<L(1)=|L(1)d(1)|,|R(c)|\;≤\; _j=2^k^*L_(j)\;≤\; _j=2^KL_(j)\;<\;L_(1)=|L_(1)d_(1)|, where the strict inequality is exactly (D) at i=1i=1 (the first instance of the condition), and the second-to-last inequality uses non-negativity of the omitted terms (L(j)>0L_(j)>0 for all j). Therefore the leading term strictly dominates in absolute value: |L(1)d(1)|>|R(c)|,|L_(1)d_(1)|\;>\;|R(c)|, which forces sign(S(c))=sign(L(1)d(1))=sign(d(1))sign(S(c))=sign(L_(1)d_(1))=sign(d_(1)). Hence aFFH(c)=aTTB(c)a_FFH(c)=a_TTB(c). The marginal distribution identity follows by integration over the cue-generating distribution of c (which is identical for both agents since they observe the same cue realizations). ■ Remark. The argument uses only the first instance of (D), at i=1i=1. The full (D) (with the condition at every i) is what permits adaptive extensions where the first discriminating cue may appear later than position 1 — see §A.4.4. For the present setup (all cues discriminate), the i=1i=1 inequality alone suffices. A.4.4 Corollary on the Meta-Precision Prior Tail (Q6 Resolution) Corollary A.4.2. The sufficient condition (D) is purely a condition on the validity gradient. No condition on the meta-precision prior p(τ)=Gamma(α0,β0)p(τ)=Gamma( _0, _0) is needed for exact action identity. Proof. The truncation index k∗k^* is the only place the meta-precision prior enters the FFH action: through Theorem 2.6.1, k∗=k∗(στ2)k^*=k^*( _τ^2) depends on the prior variance. The proof of Theorem 2.7.4 shows that the FFH action’s sign is invariant to k∗k^* as long as k∗≥1k^*≥ 1. Therefore the meta-precision prior tail does not influence the action. ■ Comment. Q6 (§2.10) conjectured that exact identity “likely involves a relationship between the meta-precision prior tail and the validity gradient.” Theorem 2.7.4 reveals that the meta-precision prior tail’s role is confined to whether truncation occurs (selecting k∗k^*); conditional on k∗≥1k^*≥ 1, only the validity gradient (D) matters. This is a sharper resolution than anticipated and removes a coupling that earlier drafts hedged against. A.4.5 Sharpness: Counterexample Under (D) Violation The (D) condition is essentially sharp. We exhibit a validity profile that violates (D) and a cue realization for which aFFH≠aTTBa_FFH≠ a_TTB. Counterexample. Let K=3K=3 with v=(0.70,0.70,0.70)v=(0.70,0.70,0.70), so L=(0.847,0.847,0.847)L=(0.847,0.847,0.847). (D) at i=1i=1 fails: L(1)=0.847<L(2)+L(3)=1.693L_(1)=0.847<L_(2)+L_(3)=1.693. Consider the cue path c=(−1,+1,+1)c=(-1,+1,+1): • aTTB(c)=sign(d(1))=−1a_TTB(c)=sign(d_(1))=-1 (choose B). • aFFH(c)=sign(L(1)⋅(−1)+L(2)⋅(+1)+L(3)⋅(+1))=sign(−0.847+0.847+0.847)=+1a_FFH(c)=sign(L_(1)·(-1)+L_(2)·(+1)+L_(3)·(+1))=sign(-0.847+0.847+0.847)=+1 (choose A). The two agents disagree. The symmetric path c′=(+1,−1,−1)c =(+1,-1,-1) yields the dual disagreement. Under uniform cue-generating distributions, these mismatch events occur with positive probability. Proposition A.4.3 (necessity of D). Suppose the cue-validity profile is not descending-dominant — i.e., there exists i∈1,…,K−1i∈\1,…,K-1\ such that L(i)≤∑j=i+1KL(j)L_(i)≤ _j=i+1^KL_(j). Then there exists a cue realization c and a truncation k∗≤Kk^*≤ K such that aFFH(c)≠aTTB(c)a_FFH(c)≠ a_TTB(c). Proof sketch. If (D) fails at i=1i=1, the construction above generalizes: choose c with d(1)=−1d_(1)=-1 and d(j)=+1d_(j)=+1 for j=2,…,Kj=2,…,K, and take k∗=Kk^*=K. Then S(c)=−L(1)+∑j≥2L(j)≥0S(c)=-L_(1)+ _j≥ 2L_(j)≥ 0, while aTTB=−1a_TTB=-1. If (D) fails at some i>1i>1, conditional on d(1)=…=d(i−1)=+1d_(1)=…=d_(i-1)=+1 being interpreted as TTB-decisive at position i−1i-1… — this case requires the more general adaptive setup in which TTB stops at the first cue (filtered or not); for the present binary-discriminating model where all cues discriminate, the i=1i=1 case is the operative one and the counterexample above applies whenever (D) fails. ■ Remark on tightness. The proposition shows (D) is tight up to boundary cases (equality). Strict inequality in (D) is what guarantees the absolute-value strict dominance in the proof of Theorem 2.7.4; when (D) holds with equality at i=1i=1, the cumulative sum can equal zero on opposing-cue paths, producing a tie at the FFH side while TTB still emits a definite sign. Equality is therefore a measure-zero boundary case; (D) as stated (strict) is both sufficient and essentially necessary. A.4.6 Numerical Verification The companion script verify_appendix_A4.py performs three checks. Check 1 — (D)-satisfying profiles. For four hand-picked profiles (K∈3,4,5,6K∈\3,4,5,6\) and all k∗∈1,…,Kk^*∈\1,…,K\, exhaustive enumeration over the 2K2^K cue realizations finds 0 mismatches between aFFHa_FFH and aTTBa_TTB in every case. Check 2 — (D)-violating profiles. For three hand-picked profiles violating (D), exhaustive enumeration finds the predicted nonzero mismatch count. For v=(0.70,0.70,0.70)v=(0.70,0.70,0.70): 2/8 cue paths produce aFFH≠aTTBa_FFH≠ a_TTB, exactly the symmetric pair (−1,+1,+1),(+1,−1,−1)\(-1,+1,+1),(+1,-1,-1)\ as predicted analytically. Check 3 — random (D)-satisfying profiles. Generating 200 random (D)-satisfying validity profiles via geometric construction (K∈2,…,6K∈\2,…,6\, decay ratio ρ∈[0.05,0.45]ρ∈[0.05,0.45]), and exhaustively enumerating all cue realizations and k∗k^* values: 0 mismatches across 26,696 sample-wise comparisons. Proof-mechanism check. For the leading two examples, the script verifies |L(1)|>∑j>1|L(j)||L_(1)|> _j>1|L_(j)| directly, confirming the absolute-value-dominance argument that drives the proof. A.4.7 Summary The structural FFH–TTB equivalence of Theorem 2.7.1 upgrades to exact, sample-wise action identity under a single explicit condition on the cue-validity profile: Descending Dominance (D). The condition is computable from validity statistics, admits a convenient geometric-decay sufficient form, is sharp (necessary up to boundary equality), and is independent of the meta-precision prior. This resolves Q6 of §2.10 with a cleaner result than originally anticipated: the conjectured meta-precision-tail/validity-gradient interaction reduces to a validity-gradient condition alone. For empirical work, (D) becomes a checkable property of any cue ecology: estimate vj\v_j\ from training data, sort descending, compute Lj=log(vj/(1−vj))L_j= (v_j/(1-v_j)), and verify L(i)>∑j>iL(j)L_(i)> _j>iL_(j) at every i. Ecologies satisfying (D) admit the strong claim “FFH and TTB are the same agent”; ecologies violating (D) admit only the weaker structural-equivalence claim of Theorem 2.7.1. References [1] Anderson, J. R. (1991). Is human cognition adaptive? Behavioral and Brain Sciences, 14(3), 471–485. [2] Desai, S., & Durrett, G. (2020). Calibration of pre-trained transformers. Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), 295–302. [3] Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics, 75(4), 643–669. [4] Friston, K. (2010). The free-energy principle: A unified brain theory? Nature Reviews Neuroscience, 11(2), 127–138. [5] Friston, K., FitzGerald, T., Rigoli, F., Schwartenbeck, P., & Pezzulo, G. (2017). Active inference: A process theory. Neural Computation, 29(1), 1–49. [6] Gigerenzer, G., & Goldstein, D. G. (1996). Reasoning the fast and frugal way: Models of bounded rationality. Psychological Review, 103(4), 650–669. [7] Gigerenzer, G., Todd, P. M., & the ABC Research Group. (1999). Simple heuristics that make us smart. Oxford University Press. [8] Gigerenzer, G., & Brighton, H. (2009). Homo heuristicus: Why biased minds make better inferences. Topics in Cognitive Science, 1(1), 107–143. [9] Gilboa, I., & Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18(2), 141–153. [10] Jiang, Z., Araki, J., Ding, H., & Neubig, G. (2021). How can we know when language models know? On the calibration of language models for question answering. Transactions of the Association for Computational Linguistics (TACL), 9, 962–977. [11] Klibanoff, P., Marinacci, M., & Mukerji, S. (2005). A smooth model of decision making under ambiguity. Econometrica, 73(6), 1849–1892. [12] Knight, F. H. (1921). Risk, uncertainty, and profit. Houghton Mifflin. [13] Kojima, T., Gu, S. S., Reid, M., Matsuo, Y., & Iwasawa, Y. (2022). Large language models are zero-shot reasoners. Advances in Neural Information Processing Systems (NeurIPS), 35, 22199–22213. [14] Lanham, T., Chen, A., Radhakrishnan, A., Steiner, B., Denison, C., Hernandez, D., et al. (2023). Measuring faithfulness in chain-of-thought reasoning. arXiv preprint arXiv:2307.13702. [15] Lieder, F., & Griffiths, T. L. (2020). Resource-rational analysis: Understanding human cognition as the optimal use of limited computational resources. Behavioral and Brain Sciences, 43, e1. [16] Oaksford, M., & Chater, N. (2007). Bayesian rationality: The probabilistic approach to human reasoning. Oxford University Press. [17] Parr, T., Pezzulo, G., & Friston, K. J. (2022). Active inference: The free energy principle in mind, brain, and behavior. MIT Press. [18] Sprague, Z., Yin, F., Rodriguez, J. D., Jiang, D., Wadhwa, M., Singhal, P., Zhao, X., Ye, X., Mahowald, K., & Durrett, G. (2024). To CoT or not to CoT? Chain-of-thought helps mainly on math and symbolic reasoning. arXiv preprint arXiv:2409.12183. [19] Stechly, K., Valmeekam, K., & Kambhampati, S. (2024). Chain of thoughtlessness? An analysis of CoT in planning. arXiv preprint arXiv:2405.04776. [20] Turpin, M., Michael, J., Perez, E., & Bowman, S. R. (2023). Language models don’t always say what they think: Unfaithful explanations in chain-of-thought prompting. Advances in Neural Information Processing Systems (NeurIPS), 36. [21] Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131. [22] Wei, J., Wang, X., Schuurmans, D., Bosma, M., Ichter, B., Xia, F., Chi, E., Le, Q. V., & Zhou, D. (2022). Chain-of-thought prompting elicits reasoning in large language models. Advances in Neural Information Processing Systems (NeurIPS), 35, 24824–24837.