Completely Feasible Reasoning Boundary

• CFRB is a framework defining the maximal set of instances where reasoning systems deliver reliable, high-accuracy solutions under defined resource or confidence constraints.
• It integrates formal proof complexity, empirical chain-of-thought diagnostics, and reinforcement learning dynamics to precisely delineate solving boundaries.
• Practical optimization strategies, such as black-box boundary detection, tool augmentation, and PoT prompting, effectively reduce computational waste and enhance performance.

A Completely Feasible Reasoning Boundary (CFRB) precisely delineates the maximal set of problem instances or theorems on which a reasoning system (either a formal theory or a machine learning model) achieves reliable, high-accuracy solutions, under well-defined resource or confidence constraints. The CFRB framework integrates perspectives from formal proof complexity, empirical chain-of-thought (CoT) evaluations in LLMs, and reinforcement learning dynamics, yielding an operable theoretical and practical construct for both quantifying and optimizing the attainable limits of algorithmic reasoning.

1. Formal Definitions of the CFRB in Contemporary Frameworks

For LLMs, CFRB can be instantiated as the set of tasks or input questions for which the model's empirical accuracy exceeds a target threshold (typically 90%). If $m$ is a model and $t_1, …, t_n$ are sub-tasks, the mathematical definition is

$$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$

where $Acc(t_1,\ldots,t_n|d, m)$ denotes accuracy at difficulty level $d$ for sub-tasks $t_1,\ldots, t_n$ and model $m$ (Chen et al., 2024, Chen et al., 19 May 2025).

Within formal combinatorial complexity, such as the bounded arithmetic theory $\text{LA}+\Sigma_1^B$-induction (ELA), the CFRB aligns with the class of theorems and problems that admit uniform polynomial-time (extended Frege) proofs. For matrix and combinatorial problems, this boundary encompasses all statements provable in ELA, e.g., König's Min-Max Theorem (KMM), Hall's Marriage Theorem, restricted Menger's Theorem, and Dilworth's Theorem (Fernández et al., 2013).

In reinforcement learning, the CFRB corresponds to the set of problem instances that a model can solve within a fixed sampling or inference budget $k$, as captured empirically by high $Pass@k$ rates (Nguyen et al., 2 Oct 2025).

2. Empirical Diagnostics and Theoretical Characterizations

CFRB identification in LLMs is operationalized via two interlocking approaches:

• Confidence-Trajectory Monitoring: During CoT reasoning, let $t_1, …, t_n$0 (confident expressions) and $t_1, …, t_n$1 (uncertain expressions) represent densities of model-generated confidence markers at normalized step $t_1, …, t_n$2. Solvable problems (within CFRB) show rapidly growing $t_1, …, t_n$3; unsolvable cases plateau with high $t_1, …, t_n$4 (Zhang et al., 29 Sep 2025). This dynamic is quantified by

$t_1, …, t_n$5

with $t_1, …, t_n$6 a threshold yielding high separation early in CoT. Curvature-based tests further refine boundary detection.

• Hidden-State Separability: Before any CoT, the hidden state $t_1, …, t_n$7 of the last input token is extracted and classified via $t_1, …, t_n$8. Solvable and unsolvable problems are linearly separable, achieving 97–99% classification accuracy (Zhang et al., 29 Sep 2025). The minimal score $t_1, …, t_n$9 also correlates with token usage, indicating proximity to the boundary.

This dual characterization enables both black-box (reasoning-expression monitoring) and white-box (internal state probing) CFRB diagnostics, substantially reducing wasted computation and overflows while preserving accuracy.

3. Boundary Combination Laws and Multicomponent Reasoning

Realistic reasoning tasks decompose into sub-tasks with individual boundaries, necessitating a combination law to derive a global CFRB. For $$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$0 independent sub-tasks with boundaries $$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$1, the aggregate CFRB is approximated by the harmonic mean: $$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$2 When unmeasurable sub-boundaries (e.g. visual perception) arise, each is replaced with a fixed reciprocal-difficulty constant $$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$3, yielding

$$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$4

Sub-boundaries can be further partitioned through division laws, e.g., splitting a domain knowledge boundary into separate domain knowledge and multimodal perception, to enhance granularity (Chen et al., 19 May 2025).

4. CFRB in Combinatorial Proof Complexity

The notion of CFRB is rigorously developed in the bounded arithmetic theory $$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$5-induction, defining the feasible envelope for combinatorial matrix reasoning (Fernández et al., 2013). The CFRB in this context comprises all theorems with both statements and uniform correctness proofs expressible and provable in this system, including:

• König’s Min-Max Theorem (KMM): proven in LA using only $$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$6 induction;
• Restricted versions of Menger’s Theorem, Hall's Marriage, and Dilworth’s Theorem;
• All possess polynomial-time, extended Frege proofs.

The boundary excludes the pigeonhole principle (in hard formulations), general Menger’s Theorem, and certain matrix-algebraic identities, which require more powerful proof-theoretic resources.

5. Boundary Collapse and Expansion under Learning Algorithms

Reinforcement Learning with Verifiable Rewards (RLVR) in LLMs can paradoxically reduce the CFRB. Empirically, standard on-policy RLVR produces negative interference—updating the model to solve some problems decreases correct solution likelihood elsewhere, narrowing solution diversity and collapsing global coverage (Pass@k) (Nguyen et al., 2 Oct 2025). Winner-take-all dynamics further concentrate probability mass on high-likelihood solution modes, excluding low-probability but correct solutions.

The SELF (Selective Low-likelihood RLVR) method counters this effect by focusing updates on problems not yet solved by the current policy and employing a forward-KL objective to preserve known correct modes, thereby maintaining or expanding the CFRB relative to naive RLVR.

6. Practical CFRB Optimization and Monitoring Strategies

Practical interventions for maximizing and maintaining CFRB include:

• Early black-box boundary detection to halt computation on unsolvable queries, yielding up to 93.6% reduction in wasted tokens (Zhang et al., 29 Sep 2025);
• Incorporation of external tools (e.g., calculators, code interpreters) to drive local sub-boundary $$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$7, thus elevating the global CFRB (Chen et al., 2024, Chen et al., 19 May 2025);
• Program-of-Thought (PoT) prompting to optimize global planning boundary, outperforming plain natural language CoT;
• MARP or MARP++ prompt engineering to explicitly cap reasoning step difficulty at the empirically measured CFRB;
• Constant-assumption and boundary-division for systematic treatment of hard-to-measure sub-domains in cross-modal or proprietary domains;
• Continuous measurement of accuracy versus difficulty and decomposition of failures to attribute bottlenecks to specific sub-boundaries.

7. Illustrative Results and Scope of the CFRB Concept

Empirical results validate the CFRB construct across theory and practice. On BigGSM arithmetic benchmarks, tool-enriched and PoT strategies yield accuracies up to 80.55%, aligning closely with CFRB theoretical predictions (Chen et al., 2024). In cross-modal science reasoning, precise boundary location and targeted MARP++ intervention provide absolute accuracy gains (Chen et al., 19 May 2025).

In formal proof contexts, ELA identifies the maximal class of combinatorial-theoretic results provable within feasible (polytime) boundaries. Open questions pertain to further extensions: equivalence to full Menger or pigeonhole principles in $$CFRB := \mathcal{B}_{Acc\geq90\%}(t_1,\ldots,t_n|m) = \sup\{ d\ |\ Acc(t_1,\ldots, t_n|d, m)\geq 0.9 \}$$8, reduction to quasi-polynomial extended-Frege, and the full cataloging of theorems inside or outside the CFRB (Fernández et al., 2013).

CFRB provides an essential analytical and operational framework for delimiting, monitoring, and algorithmically optimizing the “fully attainable” frontiers of reasoning in both symbolic and learning-based systems.