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Does Scaling Toward AGI Depend on What System-1 Lets System-2 Execute?

Qi Wang, Yun Qu, Lizhou Cai, Heming Zou, Haoang Chi, Yiqin Lv, Xiang Li, Haiyan Feng, Mengyu Ge, Yize Geng, Yuhang Jiang, Yixiu Mao, Xiangyang Ji

arXiv:2610.39497v1cs.AI

TL;DR

The costs of additional interactions may constrain systems that scale through larger models, more data, and more computation. The paper proposes conditional interaction, coupling a lightweight predictor with a costly executor; under explicit assumptions, it reports screening, memory, regret, and co-scaling results, while presenting informative compute as a scaling hypothesis.

  • Problem

    As systems grow more capable, generating, evaluating, and acting on each additional interaction may become costly, raising the question of which interactions justify their costs.

    Source: THU-IDM · pp. 1–2

  • Method

    Conditional interaction uses a lightweight System-1 to score candidate interactions, a resource-intensive System-2 to execute admitted candidates, and outcomes to refine System-1.

    Source: THU-IDM · p. 2, §2 · p. 6

  • Results

    Under explicit assumptions, the framework yields screening, memory, regret, and co-scaling results, which motivate an informative-compute scaling hypothesis.

    Source: §8 · p. 29

  • Takeaways & Limitations

    The co-design is presented as promising where interactions are costly, unequal in value, and predictable in advance.

    Source: §8 · p. 29

    The informative-compute scaling hypothesis is empirical and depends on constant conditional progress, an assumption that may not hold.

    Source: §7.1 · p. 28

Abstract

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Progress toward artificial general intelligence (AGI) has largely relied on scaling a resource-intensive executive, System-2, through larger models, more data and compute, longer reasoning, and more agents. We ask whether the next stage may come instead from co-designing it with System-1, a lightweight, semantically aware predictor that evaluates candidate interactions before execution, so that System-2 acts only on those worth their cost. Our answer is a qualified yes. We study \textit{conditional interaction}, a predict--gate--execute--update loop: System-1 estimates each candidate's utility, System-2 executes those admitted by an adaptive threshold, and the outcomes update System-1. The loop spans training, test-time reasoning, adaptive decision-making, and agentic tool use, with utility and cost defined by the application. Under ideal screening and equal execution costs, execution compute falls by a factor approaching $1/\rho_u$, where $\rho_u$ is the density of utility-relevant interactions (cost-weighted when costs differ); prediction errors, exploration, and distribution shift reduce the saving. We bound selection regret by predictor error and the utility margin. These results motivate our central hypothesis: $\rho_u$ is a measurable coordinate of scaling efficiency and makes informative compute a better description of scaling than total compute.

1 The Interaction Problem at Scale

As systems scale, the cost of generating, evaluating, and acting on interactions makes their value—not just their number—a central concern. The paper proposes co-designing a lightweight predictor with a costly executor and treats utility-relevant interaction density as a measure of scaling efficiency.

1 The Interaction Problem at Scale

  • A costly interaction budget makes selection consequential: systems must decide which interactions justify their cost, not merely how many they can manage.The paper frames this as an alternative scaling challenge to expanding model capacity, data, and computation.Source: THU-IDM · p. 1
  • Conditional interaction couples a lightweight System-1 that predicts candidate utility with a resource-intensive System-2 that executes candidates clearing a resource-aware threshold, then learns from outcomes.The position emphasizes co-designing the systems and their compute, storage, and feedback decisions.Source: THU-IDM · pp. 2, 4
  • In RLVR, prompts whose rollouts all receive the same reward yield no relative policy-gradient signal, so predicting useful prompts can reduce wasted generation.Under GRPO, these prompts have zero group-relative advantages despite consuming rollouts; intermediate-difficulty prompts yield mixed outcomes (Shao et al., 2024; Yu et al., 2026).Source: THU-IDM · p. 3
  • In physical adaptation, rollouts consume real time and cannot be freely parallelized, while the required coverage grows with task-space dimensionality and deployment complexity.Consequently, which tasks are evaluated matters alongside how many are assessed.Source: THU-IDM · p. 3
  • As AI generates more of its own training data and feedback, self-generated interactions can consume compute while teaching little, motivating value judgments before execution.Examples include degenerate RLVR prompts, reasoning tokens that leave answers unchanged, and already-mastered self-proposed tasks (Bender et al., 2021; Wei et al., 2022; Shao et al., 2024; Guo et al., 2025; Silver and Sutton, 2025).Source: THU-IDM · pp. 3–4
  • The utility-relevant interaction density ρu is proposed as an efficiency coordinate, but it depends on the declared utility and is not an additional resource.The paper hypothesizes that gating can direct more of a fixed budget toward relevant interactions and aims for a better cost–latency–success frontier; these are framed as a hypothesis and objective, not an established outcome.Source: THU-IDM · pp. 5–6

2 Conditional Interaction as A Co-Design Strategy: Predict Before You Pay

Conditional interaction co-designs a lightweight System-1 predictor with a resource-intensive System-2 executive: System-1 decides which interactions merit execution, and System-2’s outcomes refine its predictions. The section develops this predict-then-gate approach as a selection policy, an efficiency metric, and a budgeted objective, while specifying when gating is and is not justified.

2 Conditional Interaction as A Co-Design Strategy: Predict Before You Pay

  • Co-design makes System-1’s utility predictions guide System-2’s execution and resource allocation, while System-2’s outcomes recalibrate System-1 as the executive changes.A static cascade routes requests without feedback; the two systems are jointly optimized in co-design.Source: §2 · p. 6, Figure 4 · p. 7

2.1 Core Definition

  • A conditional interaction policy scores candidates, admits those above a utility threshold, and feeds executed outcomes back to the predictor.The threshold controls admissibility, while an inverse-temperature parameter controls selection sharpness among admissible candidates; an empty admissible set triggers abstention.Source: §2.1 · p. 7, Figure 5 · p. 8

2.2 The Interaction Information Efficiency Metric

  • Under perfect screening and equal execution costs, the ideal IIE gain approaches 1/ρ_u; prediction costs and errors reduce efficiency, and false negatives also reduce yield.The section says to report efficiency alongside yield because an aggressive gate can raise IIE while starving the learner of signal.Source: §2.2 · pp. 9–10

2.3 A Master Objective and Its Special Cases

  • A budgeted objective admits candidates by predicted utility relative to cost, providing a shared selection structure across prompt sampling, agentic decisions, and executive routing.The shared structure does not imply interchangeable learning dynamics or guarantees.Source: §2.3 · pp. 11–12

2.4 Scope: When Gating Is and Is Not Justified

  • Gating is suited to individually costly, heterogeneous interactions whose utility can be predicted cheaply; uncertain, exploratory, or safety-critical cases may require relaxing it.The paper identifies audit sampling as a safeguard against off-distribution blind spots and misspecified utility that can cause systematic suppression.Source: §2.4 · p. 12

3 Co-Designing the Two Systems: Architecture and Co-Scaling

The section frames co-design as a coupled learning problem: System-1 selects interactions for System-2, and both systems’ updates depend on one another. It gives a conditional co-scaling result and an architectural realization, while noting that the result is qualitative and assumption-bound.

3.1 The Two Systems as an Operator Decomposition

  • System-1’s refit and System-2’s gated-gradient update form a coupled operator decomposition, with System-1 adapting faster than System-2.A general convergence guarantee remains open; two-timescale stochastic approximation gives conditions for convergence, and MPTS applies joint training in practice.Source: §3.1 · pp. 12–13
  • With equal costs and a budget of B executions, selecting the B candidates with the largest posterior mean maximizes expected executed utility and hence IIE.The proposition assumes each candidate’s utility depends on predictor outputs only through its own output; the calibrated posterior mean, not an uncalibrated score, is the ranking signal.Source: §3.1 · p. 13

3.2 Co-Scaling: System-1’s Size as System-2 Grows

  • As System-2’s cost scale k grows, optimal System-1 sizes and the maximal IIE ratio are non-decreasing under the proposition’s fixed-error-curve assumptions.The result fixes a direction, not a growth rate; it does not establish how large the resulting loss is.Source: §3.2 · p. 14
  • Scaling System-2 dilutes System-1’s prediction cost relative to execution but does not dilute the cost of prediction mistakes.Thus accuracy becomes relatively cheaper to buy as System-2’s cost scale grows.Source: §3.2 · p. 14
  • The co-scaling result does not establish monotonicity when scaling System-2 changes the utility distribution, interaction separation, or cost structure.The optimal size depends on empirical error and prediction-cost curves, and the proposition does not specify its rate of growth.Source: §3.2 · p. 14

3.3 Example of Architectural Implementation

  • Co-design also sets what knowledge System-2 transfers, which of its representations System-1 sees, and which outcomes supervise System-1.Hierarchical gates can operate at multiple temporal and representational scales, but analyzing their joint dynamics is left open.Source: §3.3 · pp. 14–15
  • The architectural example represents candidates, predicts their utility, gates execution, and learns from outcomes; a threshold controller can target an admission rate.A small audit sample of suppressed candidates can reveal predictor misses, while executed outcomes refit System-1 and train System-2.Source: §3.3 · p. 14, Figure 6 · p. 15

4 What Prediction Buys, and Why a Small System-1 Can Work

This section explains how a cheap predictor can reduce execution cost and, under conditions, memory use, while bounding selection risk by prediction error and utility margins. It examines why a small System-1 can work through information-theoretic, generative-model, and control-theoretic interpretations, and describes the four operations as a closed loop that co-adapts through selective feedback from executed interactions.

4.1 Core Results: What Prediction Buys and What It Risks

  • With a perfect screen, speedup approaches 1/ρ_u as the screening-to-execution cost ratio r approaches zero, provided utility class cannot be known without execution or scoring.This is a ceiling, not a realized speedup: audits, predictor training, screening errors, and re-encoding costs can reduce savings.Source: §4.1 · p. 16
  • Under synchronous evaluation, streamed predictive scoring can make peak rollout memory independent of pool size n, but this advantage is conditional on that evaluation setup.A reactive filter that streams executions can also use n-independent memory, but pays Θ(n c_exec) time; predictive scoring still consumes screening time and score storage.Source: §4.1 · pp. 16–17
  • Selection is exactly oracle-optimal when 2ε_φ < gap_min; otherwise the regret bound degrades linearly with ε_φ and batch size B once the margin is crossed.In continuous interaction spaces, the relevant margin is the one at the B-th order statistic; blind spots can dominate even when the predictor is otherwise accurate.Source: §4.1 · p. 17

4.2 Understanding Through the Lens of Active Inference

  • The information-theoretic view treats expected information gain net of cost as utility, while active-inference epistemic selection is a limiting case under uniform preferences and separately handled cost.A variational posterior approximation does not guarantee better utility rankings, and deployment can price latency, money, context, and safety as separate constraints.Source: §4.2 · pp. 18–19

4.3 Understanding Through the Lens of Decision-Making

  • In the generative view, System-1 cheaply predicts a transition’s scalar value rather than simulating it, forecasting the utility in a verdict that may contain only a single bit of information.The sequence generation bears the cost, while the verdict updates beliefs; in RLVR, the predictor forecasts the declared utility proxy before generation.Source: §4.3 · pp. 19–20

4.4 Understanding Through the Lens of Control Theory

  • Conditional interaction resembles receding-horizon control of a learning plant, but is not equivalent to MPC: it selects interaction subsets, executes only the first batch, then replans as outcomes update the model.The plant and controller co-evolve, so stability is not classical and rests on System-1 adapting faster than System-2; greedy selection has a (1 − 1/e) guarantee only for a monotone submodular objective.Source: §4.4 · pp. 20–22
  • An index policy is optimal only under a discounted independent-arm bandit approximation with a retirement option; non-stationarity and dependent interactions make it a motivation for, not a guarantee of, the threshold rule.The control interpretation links to active inference and resource-allocation indices under their respective assumptions.Source: §4.4 · p. 21
  • The four operations form a closed loop in which the predictive subsystem and executive policy co-adapt through shared history.Model predictive task sampling and its in-context extension apply the loop over tasks (Wang et al., 2026; Wen et al., 2026a); agentic RL methods apply it within long-horizon trajectories, allocating rollouts to shared prefixes and propagating terminal rewards through trees (Zou et al., 2026b,a).Source: §4.4 · p. 22
  • Only executed interactions enter H, so the predictor trains on a biased sample of its own choosing, and validity rests on the timescale separation discussed in Section 3.The statistical consequences of this censoring are discussed in Section 7.2.Source: §4.4 · p. 22

5 Conditional Interaction Across Digital and Physical Domains

Digital and physical applications share a predict–gate–execute–update pattern, but use different utilities and conserve different resources. Prior work illustrates predictive selection across both domains without validating the general framework.

5 Conditional Interaction Across Digital and Physical Domains

  • Conditional interaction treats reasoning-model fine-tuning and embodied-agent training as instances of one abstraction, differentiated by state space, cost structure, and verifier.In both, the predictor learns value from past outcomes and gates execution at an adaptive threshold.Source: §5 · p. 22

5.1 Digital and Physical Instantiations in the Literature

  • Predictive prompt selection follows post-generation filtering methods that pay for rollouts before identifying uninformative prompts.GRESO, MoPPS, DPS, BOTS, PCL, and GPS predict prompt value before generation; GPS uses a small predictor to select prompts for a larger reasoning model.Source: §5.1 · pp. 22–23
  • The cited methods illustrate the framework rather than validate it, extending predictive selection to supervised fine-tuning, demonstrations, and agentic rollouts.The broader comparison distinguishes this approach from post-hoc valuation and methods over static candidate pools.Source: §5.1 · p. 23
  • In embodied learning, methods progress from randomization to prioritizing configurations, but many still execute them to assess their value.MPTS instead learns a risk predictor from optimization history; successors add posterior sampling, diversity, or in-context risk prediction.Source: §5.1 · p. 23
  • Across digital and physical settings, System-1 scores candidates from features and history, System-2 executes those clearing a threshold, and outcomes refit System-1.The settings differ in utility and conserved resource: mixed outcomes versus tail adaptation risk, and compute versus actuation.Source: §5.1 · p. 23

6 Conditional Interaction at Test Time

At inference, conditional interaction gates reasoning increments and real-world actions using predicted utility net of application-specific cost. Its efficiency claims depend on accurate prediction and stated conditions, while some test-time premises remain unmeasured.

6 Conditional Interaction at Test Time

  • At inference, the same utility-net-of-cost rule gates interactions, with System-1 rationing System-2’s effort beyond training.The ledger changes to inference compute or real-world actions, and the contribution is to place test-time control alongside training rather than claim to discover it.Source: §6 · pp. 23–24

6.1 Face I: Efficient Test-Time Scaling

  • A next-step utility can halt before delayed gains, so the one-probe policy instead predicts continuation value before stopping or running to the full budget.Sequential halting requires scoring continuation value at each step and charging each prediction cost.Source: §6.1 · p. 24
  • Under a margin condition, a predictor with error at most ε misclassifies at most a fraction mε of candidates when the threshold equals the utility floor.Misclassified candidates lie in the band around that floor.Source: §6.1 · p. 24
  • The test-time cost multiplier depends on a small informative fraction and an accurate halting predictor, but neither premise has yet been measured there.The proposed test measures the fraction of queries made correct by a larger budget and compares gated accuracy and mean compute with unconditional scaling.Source: §6.1 · p. 25

6.2 Face II: Efficient Adaptive Decision-Making

  • For adaptive decision-making, actions are selected by predicted value net of cost, with task-return improvement or uncertainty reduction as distinct utility choices.Tools, sensing, questions, and replanning can each use the same rule with an application-specific action and utility.Source: §6.2 · p. 25

6.3 Selective Agentic Tool and Computer Use

  • For separable agent actions, the stated bound limits accuracy loss to N mεφ, but prerequisite or jointly useful actions violate the separability assumption.Long interface trajectories may require scoring a sub-trajectory or state transition instead; pure exploration and mandated actions lie outside the gate.Source: §6.3 · p. 26

7 A Testable Scaling Hypothesis, and Open Problems

The paper proposes informative compute as a scaling coordinate and distinguishes a proved horizontal rescaling from a stronger empirical hypothesis about changing interaction density. It also identifies selective supervision as a central open technical challenge.

7 A Testable Scaling Hypothesis, and Open Problems

  • The paper separates proved results, application-specific evidence, and proposed hypotheses, including a co-designed frontier and decay of utility-relevant interaction density.The reported predictive sampling savings support the framework only at the application-specific evidence level.Source: §7 · p. 27

7.1 The Scaling Hypothesis: Informative Compute

  • With constant utility-relevant density, perfect selection, free prediction, equal execution costs, and preserved relevant-interaction order, conditional training shifts the baseline curve left by ln(1/ρu).This rescaling identity holds for any baseline loss-curve shape; if density drifts, the constant factor becomes a time-varying reparameterization.Source: §7.1 · p. 27
  • The hypothesis that ρu declines as capability improves is empirical, can fail, and does not by itself derive a scaling law.The proposed account assumes constant conditional progress; an independently measured density trajectory is needed, and other mechanisms can also produce power-law-like curves.Source: §7.1 · pp. 27–28

7.2 Selective Supervision: The Gate Censors Its Own Training Data

  • Because the gate trains System-1 only on executed interactions, low-scored candidates may never receive outcome labels and their scores may remain uncorrected.Exploration or independent audits can preserve positive observation probability, but required rates and error guarantees under adaptive censored collection remain open.Source: §7.2 · p. 28

7.3 Open Problems and Outlook

  • The cleanest proposed test is to measure the trajectory of ρu; other open questions include optimal admission, predictor cost, System-1 sizing, and multi-agent utility.The paper identifies relational utility in agent societies as a consequential boundary of its single-learner framework.Source: §7.3 · pp. 28–29

8 Conclusion

The conclusion gives a qualified yes: co-designing System-1 and System-2 is promising when interactions are costly, unequal in value, and predictable in advance. The appendix states the scope of its proofs and illustrates a selection result.

8 Conclusion

  • The paper argues that conditional interaction can couple a cheap predictor with a costly executive across training, test-time reasoning, and agentic actions.Its results are conditional on explicit assumptions, and the next step is to measure ρu and compare scaling curves at matched informative compute.Source: §8 · p. 29

A Proofs

  • The appendix treats utility-relevant density as general and informative interaction density as its information-gain special case.Its proofs aim to specify both the benefits and risks of co-design.Source: A Proofs · p. 33

A.1 Proof of Theorem 2

  • With a unique best admissible candidate, the selection policy assigns it probability one as β increases.When there is only one admissible candidate, it is selected with probability one for every β.Source: §A.1 · p. 33

A.2 Proof of Theorem 7

Theorem 7 characterizes how predictor errors and screening costs affect interaction efficiency, while Theorem 14 compares perfect conditional screening with exhaustive execution.

A.2 Proof of Theorem 7

  • With a perfect, free predictor, the efficiency ratio equals 1/ρ_u when dead and relevant candidates have equal execution costs, and exceeds it when dead candidates cost more.Theorem 7’s clean case assumes dead candidates carry no utility.Source: §A.2 · p. 34
  • False-positive rate, false-negative rate, and predictor cost each make the efficiency ratio non-increasing; it strictly decreases when the corresponding coefficient is positive.False negatives also reduce expected yield by a factor of 1 − fn, increasing the pool needed to reach a target.Source: §A.2 · pp. 34–35

A.3 Proof of Theorem 14

  • With screening cost ratio r, conditional interaction’s speedup is 1/(ρ_u + r), approaching 1/ρ_u as r approaches zero.The comparison assumes the utility class cannot be known without executing or scoring a candidate; exhaustive execute-then-filter must execute all candidates to guarantee the full relevant subset.Source: §A.3 · pp. 35–36

A.4 Proof of Theorem 15 and Theorem 16

Predictive selection keeps peak rollout memory independent of pool size under the stated synchronous-evaluation assumptions. Increasing the pool can raise its best available utility, provided the predictor ranks the top candidates correctly.

A.4 Proof of Theorem 15 and Theorem 16

  • Predictive selection’s peak memory is Θ(b d_e + B + B G L d_model), independent of n, under the stated embedding and scoring assumptions.The terms are micro-batch embeddings, the running top-B structure, and the B executed survivors.Source: §A.4 · p. 36
  • Screening all candidates still takes Θ(n c_pred) time; storing all scores takes Θ(n) memory, or Θ(B) when only the running top-B is kept.For streaming reactive evaluation, the memory advantage applies to synchronous pipelines; otherwise it reduces to Theorem 14’s time advantage.Source: §A.4 · pp. 36–37
  • The best utility among n candidates is non-decreasing and can grow as n^ξ in probability for the stated heavy upper tail, if the predictor ranks the pool’s top correctly.For bounded utility it converges to the essential supremum; predictive sampling adds no rollout memory as the pool grows.Source: §A.4 · p. 37

A.5 Proof of Theorem 20

Theorem 20 relates expected free energy to variational information gain under uniform outcome preferences. With non-uniform preferences, the selection criterion also reflects preference satisfaction.

A.5 Proof of Theorem 20

  • Under uniform preferences and separately accounted costs, minimizing expected free energy is equivalent to maximizing variational information gain.The variational information gain equals the utility information gain when the variational posterior and predictive are exact.Source: §A.5 · pp. 37–38
  • With non-uniform outcome preferences, the pragmatic term depends on the interaction, so minimizing expected free energy trades information gain against preference satisfaction.The variational surrogate’s maximizer need not maximize true information gain.Source: §A.5 · p. 38

A.6 Proof of Theorem 18 and Theorem 19

Theorems 18 and 19 bound the regret from selecting candidates by predicted utility. Their guarantees depend on predictor error, utility gaps, and restrictions on blind-spot errors.

A.6 Proof of Theorem 18 and Theorem 19

  • When 2ε_φ is below the minimum utility gap, deterministic top-B selection by predicted utility matches the oracle selection exactly.More generally, regret is bounded using predictor error and utility gaps, in the units of the declared utility.Source: §A.6 · pp. 38–39
  • With zero-mean sub-Gaussian errors outside blind spots, the stated high-probability regret bound combines clean-pair error with a blind-spot count term.The theorem bounds blind spots by their count, not pool fraction; tied utilities at the B-th position leave the error indicators active.Source: §A.6 · pp. 38–39

A.7 Proof of Theorem 21

Theorem 21 bounds gate misclassification by the mass of candidates near the utility threshold; under a margin condition, this yields a linear error bound.

A.7 Proof of Theorem 21

  • δFP + δFN ≤ Pr(Bε): false positives and false negatives are confined to a band around the utility threshold.For a general gate threshold κ, Bε includes candidates whose true utility lies within ε of either the relevance boundary or κ.Source: §A.7 · p. 40
  • With κ = umin and the stated margin condition, the gate misclassifies at most a fraction mε of candidates.The margin condition bounds the probability mass within ε of umin by mε.Source: §A.7 · p. 40
  • A false negative lies just above the relevance threshold, while a false positive lies just below it, placing both in Bε.The proof obtains these two cases from the predictor’s uniform error bound.Source: §A.7 · p. 40

A.8 Proof of Theorem 22 and Theorem 23

These proofs bound the accuracy and compute effects of gating, characterize optimal selection under a fixed execution budget, and show when the best System-1 size rises with System-2 scale.

A.8 Proof of Theorem 22 and Theorem 23

  • At test time, a one-probe gate loses at most δFN accuracy, while its compute fraction approaches ρtt as C grows and false positives vanish.The corresponding compute multiplier approaches 1/ρtt; the accuracy bound concerns false negatives only.Source: §A.8 · pp. 40–41
  • For tasks with N candidate actions, accuracy is within N m εφ of call-everything accuracy under the theorem’s assumptions, including that omitting decision-irrelevant actions leaves the answer unchanged.Separability is essential: jointly relevant actions can invalidate the accuracy guarantee even when each action is individually irrelevant.Source: §A.8 · p. 41

A.9 Proof of Theorem 12

  • With equal costs and candidate-local conditional utility, executing the B candidates with the largest posterior mean maximizes expected executed utility and IIE.The result characterizes how to use a given predictor; it does not establish which predictor is best.Source: §A.9 · p. 42

A.10 Proof of Theorem 13

  • Under the stated error-rate and scoring-cost assumptions, the optimal System-1 size and maximal IIE ratio are non-decreasing as System-2 scale k increases.The error rates’ independence from k is the substantive assumption; the monotonicity conclusion can fail if the scoring-cost condition fails.Source: §A.10 · pp. 42–43

A.11 Proof of Theorem 24

Under ideal screening, conditional training matches the unconditional loss curve at 1/ρu of its compute; the appendix also gives digital and physical acquisition functions for gating.

A.11 Proof of Theorem 24

  • Under perfect selection, free prediction, equal execution costs, and the stated interaction assumptions, the conditional loss curve equals the unconditional curve shifted left by ln(1/ρu) on a log-compute axis.The identity holds for any shape of the unconditional loss curve.Source: §A.11 · pp. 43–44

B Acquisition Functions for the Digital and Physical Instantiations

  • The digital acquisition score estimates how much a prompt’s outcome distribution falls in the informative, mixed-outcome band.The prompt selector uses prompt context and recent model history to estimate that distribution.Source: B Acquisition Functions for the… · p. 44
  • The physical acquisition score targets predicted tail adaptation risk; MPTS uses an upper-confidence score, while PDTS samples risk and selects a diverse batch.The batch objective is NP-hard; a pairwise-distance diversity term is supermodular, so the stated greedy guarantee does not apply.Source: B Acquisition Functions for the… · p. 44
  • Both acquisition functions are local approximations of cost-normalized expected utility, but they use different scalars: mixed-outcome informativeness digitally and predicted tail risk physically.The diversity term represents diminishing returns from similar interactions.Source: B Acquisition Functions for the… · p. 45

B.1 Derivation for Theorem 26

The derivation uses a local Gaussian approximation to connect information gain with interaction utility, while emphasizing that its broader acquisition-function link is heuristic.

B.1 Derivation for Theorem 26

  • The acquisition-function connection is a local analogy, not a theorem; a large gradient may reflect noise or a harmful direction rather than useful learning.Only Steps 2 and 7 are exact, and only within their stated Gaussian models.Source: §B.1 · p. 45
  • The derivation assumes an approximately Gaussian belief and a small interaction update proportional to the observed learning-signal score.The local model uses accumulated Fisher information as precision.Source: §B.1 · p. 45
  • In the linear-Gaussian model, information gain from adding an interaction decreases as it aligns with already probed directions, making information gain submodular in the executed set.The diversity term is a cheap proxy for this diminishing return, but a pairwise-distance sum is not itself submodular.Source: §B.1 · p. 45
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