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HypoSpace: A Diagnostic Benchmark for Set-Valued Hypothesis Generation under Underdetermination and Sublinear Coverage Bounds

Tingting Chen, Beibei Lin, Zifeng Yuan, Qiran Zou, Hongyu He, Anirudh Goyal, Yew-Soon Ong, Dianbo Liu

arXiv:2510.15614v3cs.CL

TL;DR

Many scientific observations admit multiple consistent explanations, while common benchmarks emphasize single-answer correctness. HypoSpace evaluates LLMs as samplers over exactly enumerated hypothesis sets, and finds that Uniqueness and Recovery decline as those sets grow despite often-high Validity. Its diagnostic scope is controlled set-valued inference, not real-world scientific discovery.

  • Problem

    Many inference problems are underdetermined, but existing benchmarks often reward single-answer correctness rather than testing whether models can explore multiple valid hypotheses.

    Source: §1 · p. 1

  • Method

    HypoSpace treats LLMs as samplers over explicitly enumerated admissible sets and evaluates Validity, Uniqueness, and Recovery with deterministic checks.

    Source: §1 · p. 1, §3 · p. 3, §3.1 · p. 3

  • Results

    As admissible sets grow, models can retain high Validity while Uniqueness and Recovery decline, a pattern also observed on real-world genetic interaction data.

    Source: §1 · p. 2, §9 · p. 9

  • Takeaways & Limitations

    HypoSpace provides a controlled diagnostic for measuring set-valued inference and suggests that reshaping the sampling distribution may be more promising than simply increasing sampling budget.

    Source: §3.2 · p. 3, §9 · p. 9

    The suite abstracts consistency checking and combinatorial hypothesis spaces into controlled settings and does not claim to evaluate real-world scientific discovery.

    Source: §3.2 · p. 3, §3 · p. 3

Abstract

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Many scientific problems are underdetermined: multiple distinct hypotheses are equally consistent with the same observations. In such settings, effective inference requires not only producing valid explanations, but also systematically exploring and covering the admissible hypothesis set. We introduce HypoSpace, a benchmark that treats large language models (LLMs) as samplers over finite hypothesis spaces and evaluates them on three metrics: Validity, Uniqueness, and Recovery. HypoSpace spans three structured domains (causal graph inference, gravity-constrained 3D voxel reconstruction, and Boolean genetic interaction modeling) with deterministic validators and exactly enumerable solution spaces, plus real-world anchored case studies. Empirically, HypoSpace reveals a capability- and scale-dependent coverage failure: models can maintain high Validity while exhibiting reduced Uniqueness and Recovery as admissible hypothesis spaces become larger or more combinatorial. We further show that the analysis on stratified decoding partially mitigates this collapse, demonstrating HypoSpace's utility as a diagnostic benchmark for set-valued inference. Code is available at: https://github.com/CTT-Pavilion/_HypoSpace.

1. Introduction

HypoSpace asks whether LLMs can systematically explore multiple explanations consistent with the same observations, beyond the single-answer correctness emphasized by many existing benchmarks. It evaluates this capacity with exactly enumerable hypothesis sets and finds that coverage degrades as those sets grow, even when validity remains high.

1. Introduction

  • The benchmark targets a gap in single-answer evaluations: whether models can enumerate multiple valid explanations under underdetermination.The motivation is that observations may admit mechanistically distinct explanations, as in EEG source imaging.Source: §1 · p. 1
  • HypoSpace evaluates hypothesis exploration with Validity, Uniqueness, and Recovery, using deterministic validators and enumerated ground truth to avoid rater subjectivity.Validity measures consistency with observations, Uniqueness non-redundancy, and Recovery coverage of the admissible set.Source: §1 · p. 1
  • HypoSpace covers causal graphs, gravity-constrained 3D voxel reconstructions, and Boolean genetic interactions, with task parameters controlling admissible-set size.The suite also includes real-world genetic data with over 100 valid hypotheses consistent with observations.Source: §1 · pp. 1–2
  • Models often maintain high Validity but show declining Uniqueness and Recovery as admissible hypothesis spaces grow, revealing measurable mode collapse.The introduction reports that these failures persist even when traditional accuracy metrics suggest strong performance.Source: §1 · p. 2
  • The benchmark abstracts consistency checking and combinatorial hypothesis spaces into controlled settings; it does not claim to evaluate real-world scientific discovery.This design prioritizes measurement precision and cross-model comparison.Source: §1 · p. 2

2. Related Work

Related work spans scientific-discovery benchmarks focused on single-answer correctness and newer creativity evaluations that incorporate diversity. HypoSpace differs by measuring set-valued inference against deterministic validators and exactly enumerated admissible sets.

2. Related Work

  • Scientific-discovery benchmarks often assume one ground truth and optimize single-answer correctness, while creativity-aware evaluations have begun incorporating diversity.The passage situates these efforts across equation discovery, AI-scientist loops, and open-ended creativity frameworks.Source: §2 · p. 3
  • Creativity-aware work uses concepts such as novelty, appropriateness, fluency, originality, and flexibility, with several benchmarks and open-ended frameworks proposed.Examples named include HypoBench, IdeaBench, CreativEval, and LiveIdeaBench.Source: §2 · p. 3
  • Unlike these approaches, HypoSpace evaluates set-valued inference with deterministic validators and exactly enumerated admissible sets, without LLM-as-judge.This supports model-agnostic measurement of Validity, Uniqueness, and Recovery.Source: §2 · p. 3

3. Diagnostics for Creative Hypotheses Generation : Details of Formulation and Indicators

The formulation treats inference as sampling from an explicitly enumerated set of hypotheses consistent with revealed observations. Its indicators separate validity, non-redundancy, and coverage, while its intended use is diagnostic rather than leaderboard ranking.

3. Diagnostics for Creative Hypotheses Generation : Details of Formulation and Indicators

  • The admissible set H_O must be sound, complete, and controllable in size through task parameters.Explicit enumeration enables exact validity checks and direct coverage measurement without LLM-as-judge circularity.Source: §3 · p. 3
  • The sampling protocol draws N independent hypotheses from a fixed prompt and decoding setup, logging proposal order for novelty checks.Most experiments set N = |H_O|, though other sampling budgets are possible.Source: §3 · p. 3

3.1. Indicators

  • Validity (VR) measures the share of proposals that satisfy all observations.The formulation denotes the task-specific validity check as val_O(h).Source: §3.1 · p. 3
  • Novelty/Uniqueness (NR) measures non-redundancy within a sampling run, with distinctness assessed relative to earlier samples.The novelty indicator is order-respecting.Source: §3.1 · p. 3
  • Recovery (RR) measures coverage of the enumerated valid set and integrates validity with non-redundancy.Source: §3.1 · p. 3
  • Uniqueness uses task-specific distinctness rules: labeled-edge equality for causal DAGs, voxelwise tensor equality for 3D reconstructions, and a mechanistic canonicalizer for local Boolean equivalences.Source: §3.1 · p. 3

3.2. Intended use and scope

  • HypoSpace is intended as a diagnostic probe for underdetermination, reporting interpretable indicators and coverage curves rather than a single scalar leaderboard score.It supports controlled ablations such as sampling budget and model class, but does not claim real-world scientific discovery.Source: §3.2 · p. 3

4. Why Coverage Collapse Persists Under Peaked Generators

HypoSpace analyzes Recovery by sampling a finite admissible hypothesis set, showing why peaked generators can leave coverage low despite high Validity. Even when all hypotheses have nonzero probability, practical coverage can require very large budgets; zero-probability hypotheses make full Recovery impossible.

Setup

  • For a finite admissible set HO of size M, Recovery is the number of distinct hypotheses observed in N samples divided by M.The analysis considers N independent draws from a model-induced distribution over HO.Source: §4 · pp. 3–4

Asymptotics versus practical budgets

  • With nonzero probability for every hypothesis, Recovery approaches 1 asymptotically, but rare hypotheses can make substantial coverage require exponentially many samples.For a hypothesis with probability ε, achieving a constant chance of observing it requires N = Θ(1/ε).Source: §4 · p. 4

Peaked distributions imply slow coverage

  • When a head of size K carries mass α close to 1, expected discoveries grow at most K + N(1 − α), so Recovery can remain small relative to large M.The slope beyond the head is bounded by the small tail mass 1 − α.Source: §4 · p. 4

Why high Validity can coexist with low Recovery

  • A model can have high Validity but low Recovery when its support lies within HO yet its peaked distribution repeatedly samples the same few admissible hypotheses.Repeatedly rediscovering a small set leaves the long tail uncovered.Source: §4 · p. 4

Zero-probability hypotheses

  • If any admissible hypothesis has zero probability, full Recovery is impossible for every sampling budget.In this case, RR(N) remains bounded away from 1 for all N.Source: §4 · pp. 4–5

Takeaway and Implications for LLMs

  • Coverage collapse reflects sufficiently peaked generator distributions, and addressing it requires reshaping the sampling distribution rather than simply drawing more samples.The analysis notes that near-uniform sampling or reweighting away from prior hypotheses could improve coverage in principle.Source: §4 · pp. 3, 5

5. HypoSpace Construction

HypoSpace constructs three structured diagnostics with exactly enumerable admissible hypothesis sets. Each domain uses its own observations and constraints to define valid candidate explanations and control task difficulty.

5. HypoSpace Construction

  • HypoSpace provides exactly enumerable diagnostics for causal graphs, gravity-constrained voxel worlds, and Boolean genetic interactions.The three domains mirror common scientific inference patterns while permitting exact enumeration of admissible hypotheses.Source: §5 · p. 5

5.1. Causal inference from perturbations

  • In causal inference, models propose labeled DAGs whose descendants reproduce single-node intervention responses; node and intervention counts control difficulty.Two graphs are distinct when their labeled edge sets differ.Source: §5.1 · p. 5

5.2. 3D understanding under gravity

  • In 3D reconstruction, models infer voxel stacks from binary top-down projections, with validity enforcing projection consistency and bottom-contiguous gravity.Grid size, height budget, and projection density control difficulty as valid completions grow rapidly with these parameters.Source: §5.2 · p. 5

5.3. DNA interaction via Boolean programs

  • In Boolean genetic interaction modeling, expression trees over parental phenotypes must match observed outputs, while a restricted canonicalizer merges commutativity, idempotence, and repeated-operator associativity symmetries.Operator set, depth bound, and observation coverage govern the hypothesis-space size and program combinatorics.Source: §5.3 · p. 5

6. Experiments

HypoSpace evaluates LLM hypothesis generation across structured tasks with exactly enumerated admissible sets, separating validity from uniqueness and recovery. Results show that coverage degrades as spaces grow, while complexity-stratified decoding partly improves recovery but has model-dependent trade-offs.

6. Experiments

  • The experiments compare instruction-tuned and reasoning-focused models on three diagnostics, scoring final structured hypotheses rather than rationale text.Difficulty varies task-specific parameters that scale the admissible set; validators check task constraints and canonicalizers assess distinctness.Source: §6 · pp. 5–6, §6.1 · p. 6

6.2. Results by task

  • At larger scales, GPT-5, Gemini-2.5-Pro, and Grok-4 retain 100% Validity on causal sets of 160 and 3D sets of 125, while Uniqueness and Recovery drop substantially.The larger-scale experiments compare these results with the hardest original settings.Source: §6.2 · p. 7, Table 4 · p. 8
  • Across tasks, Uniqueness and Recovery decline as admissible spaces grow, while frontier reasoning models retain higher Validity; Boolean interactions show the sharpest coverage deficits.Reasoning-capable models outperform non-reasoning baselines on Uniqueness and Recovery at medium-to-hard difficulties, but do not eliminate mode collapse.Source: §6.2 · pp. 6–7, Table 1 · p. 6, Table 2 · p. 6

6.4. Where coverage fails: output-level failure modes

  • Coverage loss has multiple causes: larger causal and 3D settings show repeated valid hypotheses, while hard Boolean settings include duplication, invalidity, and constraint violations.Recovery alone does not identify the failure mechanism, so the authors recommend interpreting it alongside Validity, Uniqueness, and output-level analysis.Source: §6.4 · p. 7

6.5. A diagnostic intervention: complexity-stratified decoding

  • Complexity-stratified decoding targets a simplicity bias by prompting for hypotheses at successive structural-complexity levels and allocating queries according to their ground-truth distribution.The procedure encourages exploration of complex hypotheses while retaining model-based generation and validity constraints.Source: §6.5 · pp. 7–8
  • Stratified decoding improves overall Recovery for 4/7 models; GPT-4o gains 9.0% and Claude-Opus-4 gains 7.7%, while Grok-4 reaches 17.2% Complex Recovery from a 0% baseline.Trade-offs include lower Simple Recovery for GPT-5 and Gemini and a DeepSeek-R1 Recovery decline from 25.6% to 6.4%.Source: §6.5 · p. 8, Table 5 · p. 8

6.6. Effect of decoding and sampling budget

  • On hard Boolean interactions, temperature, top-p, and larger sampling budgets do not reliably improve Recovery and can produce more duplicates or lower Validity.The authors interpret this pattern as consistent with narrow sampling preferences and motivate explicitly coverage-aware strategies.Source: §6.6 · p. 8, Table 6 · p. 9

6.7. Further analysis

  • The paper defers cost and token usage, model preferences, reasoning-trace analysis, exploration dynamics, and failure-mode decomposition to appendices.These are listed as further analyses rather than findings reported in this section.Source: §6.7 · p. 8

7. Real-World Alignment Study

The study applies HypoSpace to an anonymized yeast vesicle-trafficking module, enumerating Boolean hypotheses consistent with knockout viability observations. Stronger models recover consistent hypotheses, while weaker models can generate diverse but invalid outputs.

7. Real-World Alignment Study

  • Stronger models achieve perfect validity and recover up to 100% of the hypothesis set, while weaker models reach 0% validity despite high uniqueness.The study uses six genes, six single-knockout observations, and seven unique double-knockout outcomes.Source: §7 · p. 9

8. Practical Use of HypoSpace

HypoSpace is intended as a diagnostic layer for LLM-assisted discovery pipelines, supporting model selection, intervention choices, and deployment-risk assessment.

8. Practical Use of HypoSpace

  • High-validity, low-recovery models may suit tasks needing few plausible hypotheses, while higher-recovery models better fit broad exploration.Low recovery also signals risk of missing admissible alternatives that could matter to downstream scientific decisions.Source: §8 · p. 9
  • Metric profiles suggest targeted interventions: constraint prompting or validation for low validity, and duplicate rejection or stratified decoding for low uniqueness.Persistent recovery deficits motivate ensembling or diversity-promoting prompts.Source: §8 · p. 9

9. Conclusion

HypoSpace evaluates set-valued inference using exact validators and enumerated admissible sets. Its results show that high validity can coexist with limited coverage as hypothesis spaces grow, while stratified decoding partially mitigates this pattern.

9. Conclusion

  • HypoSpace is presented as a diagnostic suite for set-valued LLM evaluation, with exact validators and enumerated admissible sets.The conclusion describes its application across three structured domains.Source: §9 · p. 9
  • In the vesicle-trafficking figure, adding double-knockout observations reduces the consistent set most when the synthetic-lethal E–F pair is included.The figure reports the range across different choices of k double-knockout observations.Source: Figure 3 · p. 9
  • As admissible hypothesis spaces grow, frontier models retain high Validity but show limited Uniqueness and Recovery; stratified decoding partially mitigates the collapse.The conclusion reports this pattern on structured tasks and real-world genetic-interaction data.Source: §9 · p. 9
  • Theoretical analysis finds that peaked hypothesis distributions make coverage collapse inevitable under realistic sampling budgets, even when validity remains high.The authors suggest reshaping the sampling distribution as more promising than simply increasing the sampling budget.Source: §9 · p. 9

Impact Statement

These supplementary sections discuss the paper’s impact, model costs, related work, evaluation metadata, and model preferences. They describe HypoSpace as a controlled diagnostic and report cost differences and complexity-related recovery patterns.

Impact Statement

  • The paper presents controlled tasks, explicit validators, and reproducibility details as safeguards against overclaiming model capability or interpreting results as biological conclusions.It says it does not anticipate direct negative impacts on individuals or deployment in high-stakes decision-making.Source: Impact Statement · p. 10

A. Analysis of cost and tokens across LLMs

B. Extended Related Work

  • Prior scientific-discovery benchmarks predominantly emphasize single-answer correctness, while many creativity evaluations lack formally enumerated hypothesis spaces.HypoSpace complements these evaluations with deterministic validators and set-level measurement of Validity, Uniqueness, and Recovery.Source: B. Extended Related Work · p. 12

C. Model Metadata and Prompts

  • The evaluation uses models accessed through OpenRouter, except GPT-5, which was called through the OpenAI API; prompts and decoding settings are available in the repository.Source: C. Model Metadata and Prompts · p. 12

D. Model Preferences

  • The analysis defines task-specific structural complexity, measures generation rate over all valid generated hypotheses (including duplicates, excluding formatting errors), Recovery Rate in simple/complex subsets, and the Recovery Gap RRSimple − RRComplex.Recovery Rate is the fraction of ground-truth hypotheses recovered within each subset. Boolean complexity is simple at ≤3 operators versus complex at 4–5; causal complexity is simple at ≤3 edges versus complex at >3 edges.Source: D. Model Preferences · p. 13
  • Across tasks, many models over-generate simpler hypotheses, and Boolean Hard has 0% RRComplex across models despite validity remaining high.For weaker and mid-tier models, RRSimple exceeds RRComplex, producing a positive recovery gap.Source: D. Model Preferences · p. 13, Table 10 · p. 13

E. Reasoning-Trace Analysis on the Boolean Task

The analyses trace underexploration to repeated outputs and premature stopping: models may retain high validity while missing much of the admissible set. Entropy curves and Boolean reasoning traces reveal distinct patterns of exploration and convergence.

E. Reasoning-Trace Analysis on the Boolean Task

  • The Boolean trace analysis uses one observation admitting 40 hypotheses and prompts GPT-4o, Grok-4, and Gemini-2.5-Pro to explain their reasoning.Automated measures track simplicity cues and stopping patterns, including sufficiency and exhaustion claims.Source: E. Reasoning-Trace Analysis on the… · pp. 13–14
  • In the Boolean instance, models exhaust mostly 1–3-operator hypotheses while under-sampling the 4–5-operator region, and stop on subjective sufficiency or claimed exhaustion despite incompleteness.GPT-4o claims systematic coverage but recovers only 9/40 hypotheses; Grok-4 and Gemini-2.5-Pro also cite sufficiency or unboundedness while missing hypotheses.Source: E. Reasoning-Trace Analysis on the… · p. 14, Table 11 · p. 14

F. Information-theoretic analysis of exploration dynamics

  • Stepwise information gain is the change in entropy of the induced pattern set; positive values indicate expanding diversity, while negative values indicate convergence toward repeated patterns.Sustained positive gain aligns empirically with higher novelty and improved Uniqueness and Recovery.Source: F. Information-theoretic analysis of… · pp. 14–15

G. Failure mode decomposition and coverage analysis

F. Information-theoretic analysis of exploration dynamics

  • Frontier reasoning models show steeper cumulative entropy growth and higher information gain per query, whereas non-reasoning models plateau earlier with flatter curves.The curves link premature entropy saturation with Recovery Rate degradation reported in Tables 1–3.Source: Figure 4 · p. 14, F. Information-theoretic analysis of… · p. 15

G. Failure mode decomposition and coverage analysis

  • GPT-5 and Gemini-2.5-Pro explore only 60–70% of enumerated sets, while duplicates—not validity failures—dominate coverage loss.Parse errors and constraint violations remain relatively low; exact and canonical duplicates are especially common for non-reasoning models.Source: G. Failure mode decomposition and… · p. 15, Figure 5 · p. 15

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