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RBF++: Quantifying and Optimizing Reasoning Boundaries across Measurable and Unmeasurable Capabilities for Chain-of-Thought Reasoning

T0 review · 5 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A weighted harmonic-mean law predicts when chain-of-thought reasoning succeeds and fails.

desk verdict A useful empirical extension wrapped around a combination law that is a restatement of its own definitions, with validation that is curve-fitting rather than a test. read the letter →

arxiv 2505.13307 v1 pith:IKIED77A submitted 2025-05-19 cs.CL cs.AIcs.CV

Qiguang Chen , Libo Qin , Jinhao Liu , Yue Liao , Jiaqi Wang , Jingxuan Zhou , Wanxiang Che This is my paper · ORCID
classification cs.CLcs.AIcs.CV
keywords reasoningboundarychain-of-thoughtlargelanguagemodelsmultimodalpromptoptimizationharmonicmeanlimitsMARPprompting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a large language model's chain-of-thought performance on a combined task is predictable from the model's separate reasoning boundaries on the task's sub-skills: the combined reasoning boundary is approximately the weighted harmonic mean of the sub-boundaries, $B(t_1,\dots,t_n)\approx 1/\sum_i 1/B(t_i)$. A reasoning boundary is defined as the hardest difficulty at which accuracy stays above a chosen threshold, so the law converts the vague idea of a model's reasoning limit into a quantity one can measure and test. The paper reports that this law holds for arithmetic, planning, multi-hop question answering, and multimodal reasoning, and that unmeasurable boundaries can be treated as scenario-specific constants. If the law is right, it explains why program-of-thought and tool use help in text but not multimodal settings, and it justifies prompt designs such as MARP/MARP++ that keep reasoning inside measured feasible regions.

What carries the argument

The load-bearing object is the reasoning boundary, $B^{\text{Acc}=K}(t|m)=\sup\{d:\text{Acc}(t|d,m)\ge K\}$, i.e. the hardest task level a model manages at a fixed accuracy target. The load-bearing identity is the combination law $B(t_1,\dots,t_n)\approx 1/\sum_i 1/B(t_i)$, which the paper derives in Appendix A from additive difficulty and independence. This single reciprocal relation does three jobs: it predicts combined performance from measured sub-boundaries, it lets unmeasurable sub-boundaries be absorbed into a constant $z_i$ in Eq. (4), and it makes boundary division possible because one combined boundary can be split into sub-boundaries that recombine by the same formula. The three boundary categories — completely feasible (accuracy $\ge 90\%$), partially feasible ($10\% <$ accuracy $< 90\%$), and completely infeasible (accuracy $\le 10\%$) — convert the identity into optimization guidance: operate inside the first, use consensus or decomposition in the second, and treat the third as requiring a capability change rather than a prompt change.

What would settle it

Construct a task with two measurable sub-boundaries in which the second sub-task's input is the output of the first, so an error in sub-task 1 changes the difficulty of sub-task 2 (e.g., a multi-step calculation where an early arithmetic slip makes later operands much larger). Measure $B(t_1)$ and $B(t_2)$ separately, compute the Eq. (3) prediction for the combined boundary, then measure the actual combined boundary. If the actual boundary falls systematically below the predicted value as the dependence between sub-tasks is strengthened, the additive-independence assumption is false and the combination law is falsified for that regime.

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Extended reading notes

Core claim

The paper's central claim is that every task has a reasoning boundary $B^{\text{Acc}=K}(t|m)$, the greatest difficulty level $d$ at which accuracy exceeds threshold $K$, and that when a task combines sub-tasks $t_1,\dots,t_n$, the combined boundary satisfies $B(t_1,\dots,t_n)\approx 1/\sum_i 1/B(t_i)$ under a normalization. The derivation starts from an assumption that difficulty is additive and sub-boundaries are independent, then uses the reciprocal relation to justify the harmonic-mean formula. The same formula is extended to unmeasurable capabilities: a constant $z_i$ replaces any sub-boundary that cannot be measured directly, and a unified boundary such as $B(o,p,v)$ can be divided into finer boundaries $B(k)$ and $B(m_m)$ that still combine by the same law. Across 38 models and 13 tasks the paper reports that the predicted 90% and 10% boundary curves align with observed accuracy transitions, and that the three induced regions behave consistently: zero-shot rationales are correct mainly inside the completely feasible region, self-consistency voting helps mainly in the partially feasible region, and the completely infeasible region stays near chance even with voting. Based on this, the paper explains the success or failure of ten chain-of-thought strategies and proposes MARP/MARP++ prompting to stay within the feasible boundary.

Load-bearing premise

The load-bearing premise is the additive-difficulty assumption of Definition 1 and the mutual-independence assumption of Assumption 5 in Appendix A: $D(t_1,t_2)=D(t_1)+D(t_2)$ with sub-boundaries independent; if real reasoning difficulty is non-additive — for instance when later steps depend on earlier answers so errors compound superlinearly, or when the model spends one shared effort budget across sub-tasks — then the harmonic-mean law does not follow and the paper's quantitative boundary predictions collapse.

Editorial extensions

If this is right

  • Given separately measured sub-boundaries, the combined boundary can be predicted without running the combined task, so a model's likely performance on a new multi-step task can be estimated in advance.
  • Tool use and program-of-thought improve textual reasoning because they push the calculation sub-boundary toward infinity, making the combined boundary equal to the planning boundary; in multimodal settings the perception constant is not expanded, so the same strategies fail.
  • Decomposing problems with Complex-CoT or Least-to-Most helps only while the smaller steps stay within the planning boundary; beyond that point added planning pressure erases the gain, which explains the observed performance reversal.
  • Self-consistency voting is predicted to improve accuracy only inside the partially feasible region, not in the already-mastered or hopeless regions, so sampling more reasoning paths has a region-dependent payoff.
  • For reasoning-tuned models, a large gain in the completely infeasible region with little gain in the completely feasible region explains why a model like o3-mini can outperform on hard problems yet trail on medium-difficulty benchmarks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the harmonic-mean law holds generally, a model's combined reasoning capacity is limited by its weakest sub-skill, so optimization effort should target the smallest sub-boundary first; the paper does not formulate this budgeting rule explicitly.
  • The constant assumption suggests a direct perception-focused experiment the paper does not run: fix the text reasoning and vary image difficulty systematically, and check whether Eq. (13) predicts the boundary shift with a single fitted constant across all difficulty levels.
  • The same reciprocal law could be applied to multi-model systems where sub-tasks are executed by different agents or models; the prediction would be that the system boundary is dominated by the weakest component, which would be a testable implication for agent orchestrators.
  • Because the law assumes independence, real tasks with correlated errors (e.g., cascading mistakes in a long plan) are where the framework should over-predict; finding a parametric correction for correlated sub-boundaries would be a natural extension.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper introduces RBF++, a framework for quantifying and optimizing the reasoning boundaries (RBs) of LLMs in chain-of-thought reasoning. The main components are: a definition of RB as the maximum task difficulty at which a model maintains a target accuracy; a "combination law" stating that the combined RB is approximately the reciprocal of the sum of reciprocals of sub-RBs (Eq. 3); a constant assumption for unmeasurable boundaries in multimodal settings (Eq. 4); a reasoning-boundary division mechanism (Eqs. 5-7, 12-13); and a three-way categorization of boundaries (CFRB, PFRB, CIRB). The paper also contributes the BigGSM and BigGSM++ benchmarks, the MARP and MARP++ prompting methods, and a broad empirical study across 38 models and 13 tasks. The authors claim the framework explains why strategies such as Program-of-Thought, Tool Usage, Complex-CoT, and Least-to-Most succeed or fail in different modalities.

Significance. If the harmonic-mean combination law were a validated quantitative law, the paper would provide a practically useful theory for predicting and optimizing CoT performance, and the MARP/MARP++ methods show concrete accuracy gains. The paper has notable strengths: a large experimental scope, detailed benchmark-construction procedures, clear prompting recipes, and a reproducible code/data release. However, the central quantitative claim is not currently established: the derivation in Appendix A restates the model's assumptions, and the empirical validation relies on unreported free parameters and visual inspection. The significance of the framework is therefore conditional on a serious revision that either reframes the combination law as an explicit, testable assumption or derives it from a more substantive model of accuracy and difficulty.

major comments (5)
  1. [Appendix A, Definitions 1-2, Eq. (23)] The harmonic-mean combination law is a restatement of the model's assumptions rather than a derived theorem. Definition 1 stipulates that difficulty is additive, D(t1,t2)=D(t1)+D(t2), and Definition 2 makes RB the reciprocal of difficulty. Consequently, 1/B(t1,...,tn)=Σ 1/B(ti) follows algebraically, and the Taylor expansion in Eqs. (18)-(22) adds no content. The proof also discards k'_i and k0 on the empirical assertion that they vanish, and at Eq. (23) it substitutes B(t_i) for t_i as a "quantized substitute," which reintroduces the reciprocal-of-difficulty identification. As written, Eq. (3) is an identity under the definitions, not a falsifiable prediction. Please either present the combination law as an explicit axiom whose empirical content is stated clearly, or derive it from a model of accuracy as a function of difficulty.
  2. [§IV-B, §VI-A, Figs. 3 and 11] The claimed verification of the combination law is visual only. The manuscript does not report fitted values of N_i, b_i, or z, no confidence intervals, no goodness-of-fit statistics, and no residuals. It also does not compare the harmonic-mean law against alternative combination rules (e.g., arithmetic mean, geometric mean, max/min). Because Eq. (2) contains per-task scaling factors N_i and b_i, overlaying a theoretical line on a scatter of observed samples cannot distinguish the law from an interpolating curve. Please report parameter estimates with uncertainty, perform a quantitative model comparison, and evaluate predictions on held-out data rather than on the same tasks used to estimate parameters.
  3. [§VI-A/B, Eqs. (4), (11), (13)] The constant assumption makes the division-mechanism validation hard to falsify. In Eq. (11), z1 is measured from direct-prompt performance on the same task that is then used to check Eq. (11); in Eq. (13), the multimodal perception RB is absorbed into an unspecified constant z'. A scenario-specific constant fitted to the same data can absorb systematic discrepancies between predicted and observed boundaries. The paper should specify how z is estimated (for example, from a calibration task disjoint from the evaluation task) and test how the conclusions depend on the choice of z.
  4. [§IV-C.4, Fig. 4 caption] The claim that LLMs have self-awareness of their own reasoning boundaries is weakened by the paper's own caption: "all samples in CIRB are special value points, like 25000×1000. In fact, no real CIRB samples are obtained." The finding that generated samples concentrate in CFRB is compatible with the model avoiding implausible or extreme numeric values rather than knowing its own reasoning boundary. This limitation should be acknowledged in the main text, and the self-awareness claim should be softened accordingly.
  5. [§IV-D, §VI-D, Eqs. (9), (13)] The explanations of why Tool Usage and Program-of-Thought help in text but not multimodal settings, and why Complex-CoT and Least-to-Most behave as observed, are all obtained by taking limits of Eq. (3). Since Eq. (3) is not independently validated, these explanations do not provide independent support for the framework; they are conditional consequences of the same combination law. The paper should state this explicitly, rather than presenting these analyses as confirmatory evidence.
minor comments (4)
  1. [Eq. (1)] The formal definition B^{Acc=K1}(t|m) = sup{d | Acc(t|d,m) ≤ K1} conflicts with the accompanying text, which says the boundary is the greatest difficulty at which accuracy exceeds K1. The inequality sign should be reversed or the definition should be reworded.
  2. [Notation throughout] Notation is inconsistent: the paper uses B^{Acc=K1}(t|m), BCoT(c,p), BT(c,p), B(p,o,v), and B(o,p,v) without a single defining list. Please collect all boundary symbols and their meanings in one table or definition block.
  3. [Appendix A, Eq. (18)] The Taylor expansion writes D(x1,...,xn) as a sum of univariate expansions in each xi and omits constant and cross terms without justification. The notation O(x_i) is also undefined for vector arguments; please clarify the expansion and its remainder terms.
  4. [§III, Experimental Setup] The statement that temperature is chosen from the interval [0,1] with temperature serving as the primary error variable is vague. Please specify the exact sampling protocol, number of runs, and how randomness is otherwise controlled.

Circularity Check

4 steps flagged · score 7.0 of 10

The harmonic-mean combination law is a restatement of the additive-difficulty and reciprocal-RB definitions, and its validation uses free scaling parameters and scenario-fitted constants, so the central quantitative claim is not independently tested.

  1. self definitional [Appendix A, Definitions 1 and 2, Assumption 5; Section II-B, Eqs. (2)-(3)]
    "Definition 1: The difficulty of solving a certain problem during model reasoning is an independent constant. That is, the difficulty D(t1,t2) satisfies: D(t1,t2|m)=D(t1|m)+D(t2|m)=K1+K2 ... Definition 2: The RB is defined as the reciprocal of the difficulty of solving the problem. The greater the difficulty of solving the problem, the lower the RB and the smaller the feasible area."

    With difficulty declared additive in Definition 1 and RB declared to be the reciprocal of difficulty in Definition 2, the combined boundary B(t1,...,tn) ≈ 1 / Σ_i (1/B(t_i)) in Eq. (3) is an algebraic restatement: 1/B_comb ∝ D_comb = Σ_i D_i ∝ Σ_i 1/B_i. The 'combination law' is therefore not an independent empirical discovery; it is the definitional pair rewritten. The experiments in Section IV-B can only check whether real tasks happen to satisfy the assumed additivity and independence, and no independent measure of D is supplied to break the definitional identity.

  2. fitted input called prediction [Appendix A, Eqs. (18)-(23); Section II-B, Eq. (2)]
    "We set ti = 1/xi + bi and 1/B(t1,t2,...,tn|m) ∝ D(x1,x2,...,xn|m). Then the original formula is expressed as: B(...)≈ N0/(Σ_i [N_i1/(t_i−b_i) − N_i1 k_i] + N_i1 k0) ... Given the minimal change in the derivative within the observable range, N′i1 = Ni1/N0 is treated as a constant Ni in this task for simplicity. Experimental results show that, if sub-RBs are separated independently, k′i = ki/N0 and k0 is typically 0."

    The derivation turns the derivative N_i1 and the offset b_i into free task-specific parameters of Eq. (2), and it disposes of the constants k'_i and k0 by the paper's own empirical assertion ('Experimental results show') rather than by derivation or external measurement. Because N_i and b_i are not reported in the verification figures, the 'theoretical' boundary lines in Section IV-B are flexible curve fits to the observed boundaries, not predictions from independently measured sub-boundaries. The fitted form is then reused as the 'combination law' that the experiments are said to confirm.

2 more flagged steps
  1. fitted input called prediction [Section VI-A, Eq. (11) and Fig. 11(a)]
    "We evaluate this by assessing the model's performance using direct prompts without CoT outputs. The resulting performance is used to measure B(v) as z1, which aids in determining the corresponding constant. Thus, the combined RB is defined as: B(p,o,v) = 1/(1/B(p)+1/B(o)+z1)."

    The scenario-specific constant z1 is measured from the same model and task whose combined boundary is then displayed as satisfying Eq. (11). Substituting a per-scenario fitted constant into the formula and then showing that the data lie on the resulting curve is not an independent validation of the constant assumption; any systematic discrepancy can be absorbed by z1. The same pattern is repeated in Eq. (13), where the multimodal perception RB is set to a free constant z' before the 'validation' in Fig. 11(b).

  2. other [Fig. 4 caption, Section IV-C (self-awareness analysis)]
    "For Fig. (c), all samples in CIRB are special value points, like 25000×1000. In fact, no real CIRB samples are obtained."

    This is not itself a circular reduction, but it is an explicit missing-support flag for the claim that 'LLMs possess an inherent understanding of their RBs': the CIRB column of the supporting figure contains no representative samples by the authors' own caption, so the asserted boundary-awareness evidence for the CIRB region is absent. This limitation is weighed in the overall assessment even though it is not a circularity.

full rationale

The core quantitative claim of RBF++ is the harmonic-mean combination law, Eq. (3). The paper's own Appendix A defines task difficulty to be additive (Definition 1) and defines RB as the reciprocal of difficulty (Definition 2), with mutual independence assumed (Assumption 5). Under those definitions, Eq. (3) follows by substitution: 1/B_comb is proportional to the sum of the inverse sub-boundaries. That makes the combination law a definitional restatement rather than a testable prediction, so the 'verification' plots in Sections IV-B, VI-A, and VI-B confirm the equation the authors built into the definitions. The Taylor-expansion derivation in Eqs. (18)-(23) does not rescue the empirical content: it drops k'_i and k0 by asserting 'Experimental results show' that they vanish, and it leaves N_i and b_i unreported free parameters, so the plotted 'theoretical intervals' can be fit to the observed boundary. In the multimodal setting, the constant assumption is implemented by measuring z1 from the same task (Section VI-A) and by assuming a free constant z' in the division mechanism, which again makes the subsequent agreement with Eqs. (11) and (13) partly tautological. The paper does contain substantive independent measurements: the existence of RB regions, the 100x CIRB improvement for DeepSeek-R1, correlations between RB values and benchmark accuracy, and qualitative comparisons of CoT strategies. These parts are not circular. Self-citations to the authors' prior RBF paper [1] are present but are not the main load-bearing mechanism, because the derivation is reproduced in this manuscript. The admitted absence of real CIRB samples in Fig. 4(c) is a further limitation, but it is a missing-support issue rather than a circular step. Overall, because the central law is definitional and its validation relies on hidden fitted parameters and scenario-measured constants, the score is 7 rather than 0-2; the framework's auxiliary empirical findings keep it from being a fully vacuous renaming.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The framework's central formula is carried by three families of assumptions: (1) the additivity and independence of difficulty, stated as Definition 1 and Assumption 5 in Appendix A, which make the harmonic-mean law true by construction; (2) the Taylor-expansion smoothness assumption used to introduce N_i and b_i, which are effectively free parameters; and (3) the constant assumption for multimodal boundaries, where an "unmeasurable" quantity is set to a measured value from the same model and then folded into the formula being validated. No independent anchors, such as held-out predictions with reported parameters or formal proofs, are provided.

free parameters (3)
  • N_i, b_i (task-specific scaling factors) = Not reported; estimated by eye to align curves.
    Introduced in Eq (2) as task-specific scaling factors and in Appendix A as Taylor coefficients; the paper never reports their values, and the "theoretical" curves in Figs 3 and 11 are drawn with them, making the claimed agreement a fit.
  • z (multimodal perception constant) = Measured as no-CoT direct-prompt accuracy on the same task (e.g., z1 in Eq 11).
    The constant assumption in Section VI.A replaces an "unmeasurable" vertical-domain boundary with a constant z measured from the same model and data, then uses Eq (11) with that same constant to claim validation.
  • MARP/MARP++ operation limits (5 operations, 1.5e5 multiplication cap) = 5 basic operations per step; multiplication less than 1.5e5.
    Hand-chosen in Appendix D based on the authors' measured boundaries; the reported MARP gains depend on these numbers, which are not derived.
assumptions (4)
  • domain assumption Task difficulty is additive: D(t1,t2|m) = D(t1|m) + D(t2|m).
    Stated as Definition 1 in Appendix A; the combination law is a direct consequence, so the paper's central formula is assumed rather than derived.
  • domain assumption All basic reasoning boundaries are mutually independent.
    Assumption 5 in Appendix A; no test of independence is provided, and it is load-bearing for the harmonic-mean combination.
  • standard math The combination-law function is continuously differentiable everywhere.
    Assumption 4 in Appendix A; used for the Taylor expansion, but the expansion is first-order with remainder dropped.
  • ad hoc to paper Unmeasurable boundaries can be replaced by scenario-specific constants z.
    Constant assumption in Section VI.A; the boundary is called unmeasurable, yet it is measured as direct-prompt accuracy, and the constant is then folded into the formula being validated.
invented entities (1)
  • Reasoning boundary (B)
    purpose: Quantifies the maximum problem difficulty at which a model maintains accuracy above a threshold (Eq 1).
    Operational definition based on accuracy thresholds; it is not observable independently of the accuracy measurements that define it, and it is not shown to be a stable model property across prompts or datasets.

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Cite this review

Pith. "Pith review of RBF++: Quantifying and Optimizing Reasoning Boundaries across Measurable and Unmeasurable Capabilities for Chain-of-Thought Reasoning." pith.science (2026). https://pith.science/paper/IKIED77A

@misc{pith2026250513307,
  author       = {Pith},
  title        = {Pith review of: RBF++: Quantifying and Optimizing Reasoning Boundaries across Measurable and Unmeasurable Capabilities for Chain-of-Thought Reasoning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IKIED77A}},
  note         = {Machine review of arXiv:2505.13307}
}
read the original abstract

Chain-of-Thought (CoT) reasoning has proven effective in enhancing large language models (LLMs) on complex tasks, spurring research into its underlying mechanisms. However, two primary challenges remain for real-world applications: (1) the lack of quantitative metrics and actionable guidelines for evaluating and optimizing measurable boundaries of CoT capability, and (2) the absence of methods to assess boundaries of unmeasurable CoT capability, such as multimodal perception. To address these gaps, we introduce the Reasoning Boundary Framework++ (RBF++). To tackle the first challenge, we define the reasoning boundary (RB) as the maximum limit of CoT performance. We also propose a combination law for RBs, enabling quantitative analysis and offering actionable guidance across various CoT tasks. For the second challenge, particularly in multimodal scenarios, we introduce a constant assumption, which replaces unmeasurable RBs with scenario-specific constants. Additionally, we propose the reasoning boundary division mechanism, which divides unmeasurable RBs into two sub-boundaries, facilitating the quantification and optimization of both unmeasurable domain knowledge and multimodal perception capabilities. Extensive experiments involving 38 models across 13 tasks validate the feasibility of our framework in cross-modal settings. Additionally, we evaluate 10 CoT strategies, offer insights into optimization and decay from two complementary perspectives, and expand evaluation benchmarks for measuring RBs in LLM reasoning. We hope this work advances the understanding of RBs and optimization strategies in LLMs. Code and data are available at https://github.com/LightChen233/reasoning-boundary.

Figures

Figures reproduced from arXiv: 2505.13307 by the authors.

Figure 1
Figure 1. Overview of the introduced concepts: (a) reasoning boundary (RB), (b) combination law for quantifying the upper [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Existence Verification for Reasoning Boundary. Figures (b, c) present evaluations performed on BigGSM, where the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Combination law verification of RB on different tasks for RBF in textual modalities. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 7
Figure 7. Figure 7: Correlation between the number of steps and perfor [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 9
Figure 9. Figure 9: Correlation between the values of RB for different [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]
Figure 10
Figure 10. Figure 10: Nature analysis at different reasoning boundaries for unmeasurable boundaries on M3CoT with multi-modal scenarios. [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: Combination law verification of RB on different tasks [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]
Figure 12
Figure 12. Figure 12: The performance analysis of Least-to-Most prompting in multimodal scenarios. [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 13
Figure 13. Figure 13: Analysis of the impact of PoT in multimodal scenarios. [PITH_FULL_IMAGE:figures/full_fig_p009_13.png]
Figure 15
Figure 15. Figure 15: Correlation between RB values for different models [PITH_FULL_IMAGE:figures/full_fig_p010_15.png]
Figure 16
Figure 16. Figure 16: Existence verification for reasoning boundaries on basic arithmetic calculation tasks, including division, addition, and [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]

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Forward citations

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.