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arxiv: 2605.00677 · v1 · submitted 2026-05-01 · 💻 cs.LG

Recognition: unknown

Evaluating the Architectural Reasoning Capabilities of LLM Provers via the Obfuscated Natural Number Game

Lixing Li

Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:33 UTC · model grok-4.3

classification 💻 cs.LG
keywords Architectural ReasoningObfuscated Natural Number GameLLM theorem proversLean 4Semantic pattern matchingFormal mathematics benchmarksReasoning modelsAutomated theorem proving
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The pith

Reasoning models maintain proof accuracy on renamed mathematical identifiers while general models lose performance.

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

The paper distinguishes architectural reasoning, which builds proofs from local axioms alone in unfamiliar domains, from semantic pattern matching on pre-trained data. It constructs the Obfuscated Natural Number Game by renaming all identifiers in the Lean 4 Natural Number Game, creating a closed environment without recognizable names. General models such as GPT-4o and Claude-Sonnet-4.5 show accuracy drops under obfuscation, whereas reasoning models including DeepSeek-R1, GPT-5, and DeepSeek-Prover-V2 sustain their original accuracy. All models exhibit longer inference times with the obfuscated version. The benchmark supplies a quantitative test for whether LLM provers can operate without reliance on familiar semantic cues.

Core claim

Architectural Reasoning is defined as the ability to synthesize formal proofs using exclusively local axioms and definitions within an alien math domain. The Obfuscated Natural Number Game renames identifiers in the standard Lean 4 Natural Number Game to produce a zero-knowledge setting. Experiments reveal that reasoning models keep the same accuracy as in the original game, while general models degrade, and a universal latency increase appears across every model tested.

What carries the argument

The Obfuscated Natural Number Game, formed by systematic renaming of all identifiers in the Lean 4 Natural Number Game to strip semantic associations while preserving logical structure.

If this is right

  • Reasoning models can generate proofs in domains lacking any pre-trained semantic anchors.
  • A measurable latency penalty accompanies removal of identifier familiarity for every model type.
  • General-purpose models depend more heavily on surface-level naming patterns than reasoning models do.
  • The obfuscated game supplies a repeatable metric for testing true local reasoning capacity.
  • Automated theorem discovery systems will need architectural reasoning to explore domains outside existing training distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The distinction between model classes suggests that training methods focused on step-by-step verification may produce greater robustness to novel formal languages.
  • Applying the same renaming technique to other Lean libraries or to Coq or Isabelle developments could test whether the observed robustness generalizes across proof assistants.
  • If architectural reasoning scales, it could enable models to propose conjectures in subfields with minimal existing formalization.
  • The latency increase may reflect additional internal search effort once surface cues disappear, pointing to a possible efficiency trade-off for future deployment.

Load-bearing premise

That renaming every identifier fully eliminates semantic pattern matching and that maintained accuracy on this single game measures architectural reasoning ability in arbitrary new domains.

What would settle it

Re-running the evaluation on a second, independently constructed obfuscated game built from different axioms, such as a basic group theory setup, and observing that reasoning models then show accuracy drops would indicate the original result does not reflect general architectural reasoning.

Figures

Figures reproduced from arXiv: 2605.00677 by Lixing Li.

Figure 1
Figure 1. Figure 1: LLM performance metrics across varying noise levels [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

While Large Language Models have achieved notable success on formal mathematics benchmarks such as MiniF2F, it remains unclear whether these results stem from genuine logical reasoning or semantic pattern matching against pre-training data. This paper identifies Architectural Reasoning: the ability to synthesize formal proofs using exclusively local axioms and definitions within an alien math domain, as the necessary ability for future automated theorem discovery AI. We use the Obfuscated Natural Number Game, a benchmark to evaluate Architectural Reasoning. By renaming identifiers in the Natural Number Game in Lean 4, we created a zero-knowledge, closed environment. We evaluate state-of-the-art models, finding a universal latency tax where obfuscation increases inference time. The results also reveal a divergence in robustness: while general models (Claude-Sonnet-4.5, GPT-4o) suffer performance degradation, reasoning models (DeepSeek-R1, GPT-5, DeepSeek-Prover-V2) maintain the same accuracy despite the absence of semantic cues. These findings provide a quantitative metric for assessing the true capacity for mathematical reasoning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that LLM success on formal math benchmarks may reflect semantic pattern matching rather than genuine reasoning. It defines Architectural Reasoning as the capacity to synthesize proofs from local axioms alone in an alien domain. The authors introduce the Obfuscated Natural Number Game by renaming identifiers in the Lean 4 Natural Number Game to create a zero-knowledge setting. Experiments on models including Claude-Sonnet-4.5, GPT-4o, DeepSeek-R1, GPT-5, and DeepSeek-Prover-V2 show that all incur a latency increase under obfuscation, but reasoning models maintain accuracy while general models degrade; this divergence is offered as a quantitative metric of true reasoning ability.

Significance. If the reported divergence is robust and the benchmark can be extended, the work would supply a practical test for isolating structural reasoning from memorized patterns in LLM provers, with direct relevance to automated theorem discovery. The latency tax observation and model-class split are concrete findings worth documenting, though their scope is currently narrow.

major comments (3)
  1. [Benchmark construction] Benchmark construction (as described in the abstract): the maintained accuracy under renaming is interpreted as evidence of Architectural Reasoning, but the underlying Peano-arithmetic structure, induction rules, and recursion patterns remain identical to the original game. No control experiment with structurally unrelated axioms is reported, so the results may reflect residual structural familiarity rather than domain-agnostic reasoning.
  2. [Experimental results] Experimental results (abstract and evaluation): the paper reports a clear accuracy contrast and latency effect but supplies no sample size, number of trials per model, statistical tests, or error bars. These omissions make it impossible to assess whether the divergence between general and reasoning models is statistically reliable or reproducible.
  3. [Generalization claim] Generalization claim (abstract and conclusion): success on this single renamed instance is used to support a metric for Architectural Reasoning in arbitrary alien domains. Because the logical skeleton is still standard natural-number arithmetic, the observed robustness may be an artifact of this particular structure rather than a general capability; additional domains with unrelated axioms would be required to substantiate the broader interpretation.
minor comments (2)
  1. [Introduction] The definition of Architectural Reasoning is presented without references to prior related concepts in LLM reasoning or formal verification literature.
  2. [Benchmark construction] The precise renaming procedure (which identifiers, how many, preservation of syntax) is not specified, hindering exact reproduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments, which have helped us improve the clarity and rigor of our work. We address each major comment below.

read point-by-point responses

  1. Referee: Benchmark construction (as described in the abstract): the maintained accuracy under renaming is interpreted as evidence of Architectural Reasoning, but the underlying Peano-arithmetic structure, induction rules, and recursion patterns remain identical to the original game. No control experiment with structurally unrelated axioms is reported, so the results may reflect residual structural familiarity rather than domain-agnostic reasoning.

    Authors: We agree that the logical structure is unchanged, as the obfuscation targets only identifier names to eliminate semantic cues from pre-training. Architectural Reasoning, as defined, involves synthesizing proofs from local axioms in an alien domain, where 'alien' here refers to the absence of familiar terminology rather than a completely novel axiom system. This controlled renaming provides a direct test of whether models can operate without relying on memorized semantic patterns. We acknowledge the value of control experiments with unrelated axioms and will include a discussion of this as a limitation, along with suggestions for future extensions, in the revised manuscript. revision: partial

  2. Referee: Experimental results (abstract and evaluation): the paper reports a clear accuracy contrast and latency effect but supplies no sample size, number of trials per model, statistical tests, or error bars. These omissions make it impossible to assess whether the divergence between general and reasoning models is statistically reliable or reproducible.

    Authors: We appreciate this observation. The original manuscript omitted these details for brevity, but we will revise the evaluation section to report the number of trials (multiple runs per problem), sample sizes across the benchmark problems, and include statistical significance tests with error bars to support the reliability of the model-class divergence. revision: yes

  3. Referee: Generalization claim (abstract and conclusion): success on this single renamed instance is used to support a metric for Architectural Reasoning in arbitrary alien domains. Because the logical skeleton is still standard natural-number arithmetic, the observed robustness may be an artifact of this particular structure rather than a general capability; additional domains with unrelated axioms would be required to substantiate the broader interpretation.

    Authors: The Obfuscated Natural Number Game is presented as a specific benchmark to quantify the distinction between semantic pattern matching and architectural reasoning in a zero-knowledge setting. We do not claim it immediately generalizes to all alien domains but rather provides a quantitative metric that can be extended. We will revise the abstract and conclusion to temper the generalization language, explicitly noting that further validation on domains with different logical structures is needed for broader claims. revision: partial

Circularity Check

0 steps flagged

No circularity: empirical benchmark evaluation is self-contained

full rationale

The paper defines Architectural Reasoning as the capacity to synthesize proofs from local axioms in an alien domain, constructs the Obfuscated Natural Number Game by renaming identifiers in the standard Natural Number Game in Lean 4, and reports raw accuracy and latency results from running external LLMs on both versions. No equations, fitted parameters, or derivations are present; the central findings are direct empirical measurements. No self-citations appear in the load-bearing steps, and the evaluation does not rename or refit any input as an output. The analysis therefore contains no reduction of claims to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on standard Lean 4 type theory for the underlying definitions and introduces the new concept of architectural reasoning without external validation beyond the benchmark results.

axioms (1)
  • standard math Lean 4 dependent type theory and the standard natural-number library definitions
    The benchmark is built directly on the existing Lean 4 formalization of natural numbers and its inductive definitions.
invented entities (1)
  • Architectural Reasoning no independent evidence
    purpose: To label the capacity to synthesize formal proofs using only local axioms and definitions in an alien domain
    This term is introduced in the abstract to frame the evaluation; no independent evidence outside the benchmark is provided.

pith-pipeline@v0.9.0 · 5481 in / 1310 out tokens · 40161 ms · 2026-05-09T19:33:06.555712+00:00 · methodology

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Reference graph

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