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arxiv: 2606.01861 · v1 · pith:UFMMEBI6new · submitted 2026-06-01 · 💻 cs.LG

A Theoretical Framework for Self-Play Theorem Proving Algorithms

Thomas Chen , Zhiyuan Li This is my paper

Pith reviewed 2026-06-28 15:31 UTC · model grok-4.3

classification 💻 cs.LG
keywords self-playtheorem provingconjecturerproverrandom walktheorem graphdiversity measurediffusion similarity
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The pith

If a theorem graph is well-connected, a prover paired with a reversible-random-walk conjecturer grows the set of proved theorems exponentially.

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

The paper models the space of theorems as a graph whose nodes are individual theorems and whose edges link theorems that share similar semantics. It states a collection of primitive assumptions that fix the performance guarantees of any trained prover and the manner in which a conjecturer may read the graph’s structure. Under the additional premise that this graph is well-connected, the authors prove that a conjecturer driven by a reversible random walk is enough to make the collection of proved theorems grow exponentially with the number of self-play rounds. The same framework also supplies a diversity objective and a local-maximization procedure, realized by embedding nodes via contrastive learning and measuring diffusion similarity, that discourages the conjecturer from emitting overly complex statements.

Core claim

The central claim is that if the underlying graph of theorems is well-connected, then a prover-conjecturer system, where the conjecturing algorithm is based on a reversible random walk, is sufficient to grow the set of proved theorems exponentially.

What carries the argument

The theorem graph (nodes as theorems, edges as semantic similarity) together with a reversible random walk on that graph that drives conjecture generation.

If this is right

  • Self-play between prover and conjecturer can scale the proved set without any external source of new theorems.
  • A reversible random walk on the semantic graph is a sufficient mechanism for sustained exploration.
  • Maximizing a diffusion-similarity diversity objective locally reduces the generation of artificially complex statements.
  • Contrastive embeddings of graph nodes yield a practical way to compute the required diffusion similarities.

Where Pith is reading between the lines

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

  • The same connectivity-plus-random-walk argument may apply to self-play curricula in other formal domains such as program synthesis.
  • If real-world theorem graphs contain disconnected components, additional bridging mechanisms would be required for the exponential-growth guarantee to hold.
  • The diversity-maximization step suggests a general recipe for preventing mode collapse in any graph-based self-improvement loop.

Load-bearing premise

The primitive assumptions that fix the trained prover’s success guarantees and the conjecturer’s access to the structure of the theorem graph are satisfied.

What would settle it

Simulate or construct a finite, well-connected theorem graph, run the prover-conjecturer loop for many iterations, and check whether the size of the proved set increases exponentially rather than linearly or sub-linearly.

Figures

Figures reproduced from arXiv: 2606.01861 by Thomas Chen, Zhiyuan Li.

Figure 1
Figure 1. Figure 1: Let τ = 1. (Left) Prover f1’s τ -expanded knowledge set is only supported on the left cluster. (Right) Prover f2’s τ -expanded knowledge set is well-supported on both clusters. Their knowledge sets S1, S2 are equal in size (6), but S2 is more diverse since its τ -step expansion is larger. which are statistical-learning-like assumptions which describe how a prover trained on a finite sample of i.i.d. (theor… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of a K-cluster graph: each cluster is internally well-connected (spectral gap γ), but only a sparse set of inter-cluster edges connects clusters, yielding a much smaller global spectral gap. 52 [PITH_FULL_IMAGE:figures/full_fig_p052_2.png] view at source ↗
read the original abstract

Self-play, a type of training algorithm that enables a model to self-improve, has recently shown promising empirical results in the context of formal theorem proving using Large Language Models (LLMs). (Dong & Ma, 2025) instantiate self-play with two cooperating agents: a prover, which proves theorems, and a conjecturer, which generates new theorems as a curriculum to the prover. In this paper, we provide a theoretical framework for understanding the self-improvement capabilities of self-play algorithms for theorem proving. First, we formalize the set of theorems as a graph, with nodes as theorems and edges between pairs of theorems with similar semantics. We introduce a set of primitive assumptions that characterize the guarantees of a trained prover and how a conjecturer can access the structure of the graph. Second, we show that if the underlying graph of theorems is well-connected, then a prover-conjecturer system, where the conjecturing algorithm is based on a reversible random walk, is sufficient to grow the set of proved theorems exponentially. Third, motivated by an issue encountered empirically by self-play algorithms, where the conjecturer tends to generate artificially complex and non-fundamental theorems, we propose a diversity measure for a training distribution of theorems generated by a conjecturer and an improved conjecturing algorithm that locally maximizes this diversity measure, by computing the diffusion similarity between neighboring theorems in the theorem graph. Finally, we describe a method to compute the diffusion similarity by using contrastive learning to embed nodes into Euclidean space and then computing the inner-product between embeddings.

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

2 major / 2 minor

Summary. The paper formalizes self-play theorem proving as a prover-conjecturer pair operating on a graph whose nodes are theorems and edges connect semantically similar theorems. It introduces a set of primitive assumptions characterizing a trained prover's success probabilities and a conjecturer's access to local graph structure. Under these assumptions plus the condition that the theorem graph is well-connected, the central result asserts that a conjecturer implemented via reversible random walk suffices to produce exponential growth in the number of proved theorems. The paper further defines a diversity measure on the conjecturer's output distribution and gives an improved conjecturing procedure that locally maximizes this measure by computing diffusion similarity (via contrastive embeddings of nodes).

Significance. If the exponential-growth claim is rigorously established, the work supplies the first formal account of why self-play can drive unbounded improvement in LLM-based theorem provers, directly addressing the empirical observation that conjecturers tend to produce overly complex statements. The graph-theoretic model and the diffusion-similarity construction are reusable abstractions that could guide both algorithm design and future analyses of exploration in formal mathematics.

major comments (2)
  1. [§3] §3 (exponential-growth theorem): the statement that a reversible random walk on a 'well-connected' graph yields exponential growth of the proved set is load-bearing. Standard random-walk theory shows that path-connectedness alone (finite diameter) produces at most linear coverage per iteration because cover times and hitting times scale polynomially with graph size; exponential growth requires a uniform lower bound on conductance or spectral gap independent of graph size. The manuscript must either (a) strengthen the definition of 'well-connected' to include such a quantitative expansion hypothesis or (b) exhibit an explicit lemma deriving a multiplicative factor from the primitive assumptions and the walk's transition probabilities.
  2. [Primitive assumptions section] Primitive-assumptions section (first part of the paper): the assumptions encode prover success probability and conjecturer access to neighbors, yet the growth analysis must show how these translate into a constant-probability discovery of new provable theorems each round. Without an explicit bound relating the walk's stationary distribution or mixing time to the fraction of newly proved nodes, the sufficiency claim reduces to the modeling choices rather than following from them.
minor comments (2)
  1. [Abstract] The citation 'Dong & Ma, 2025' appears in the abstract but is not expanded in the bibliography; a full reference should be supplied.
  2. [Diversity and improved conjecturer section] Notation for the diffusion similarity (inner product of contrastive embeddings) should be introduced with an equation number when first defined, to aid readability of the improved conjecturing algorithm.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the technical requirements needed to make the exponential-growth claim rigorous. We address each major comment below and will revise the manuscript to incorporate the suggested strengthenings.

read point-by-point responses

  1. Referee: [§3] §3 (exponential-growth theorem): the statement that a reversible random walk on a 'well-connected' graph yields exponential growth of the proved set is load-bearing. Standard random-walk theory shows that path-connectedness alone (finite diameter) produces at most linear coverage per iteration because cover times and hitting times scale polynomially with graph size; exponential growth requires a uniform lower bound on conductance or spectral gap independent of graph size. The manuscript must either (a) strengthen the definition of 'well-connected' to include such a quantitative expansion hypothesis or (b) exhibit an explicit lemma deriving a multiplicative factor from the primitive assumptions and the walk's transition probabilities.

    Authors: We agree that path-connectedness alone is insufficient and that a quantitative expansion condition is required for exponential growth. In the revised manuscript we adopt option (a): we strengthen the definition of 'well-connected' to require a uniform lower bound on conductance (equivalently, spectral gap) that is independent of graph size. We will also insert an explicit lemma in §3 that derives the multiplicative growth factor from this conductance bound, the reversible random-walk transition matrix, and the primitive prover-success probabilities. revision: yes

  2. Referee: [Primitive assumptions section] Primitive-assumptions section (first part of the paper): the assumptions encode prover success probability and conjecturer access to neighbors, yet the growth analysis must show how these translate into a constant-probability discovery of new provable theorems each round. Without an explicit bound relating the walk's stationary distribution or mixing time to the fraction of newly proved nodes, the sufficiency claim reduces to the modeling choices rather than following from them.

    Authors: We acknowledge that the translation from the primitive assumptions to a constant per-round discovery probability is not yet fully explicit. We will add a lemma in the primitive-assumptions section that supplies an explicit lower bound on the probability of discovering new provable theorems. The lemma will relate the stationary distribution and mixing time of the random walk (under the strengthened conductance condition) to the fraction of newly proved nodes, using the prover success probabilities stated in the assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; sufficiency theorem is conditional on explicitly introduced primitive assumptions.

full rationale

The paper introduces a set of primitive assumptions characterizing prover guarantees and conjecturer graph access, then proves a conditional sufficiency result: under those assumptions plus well-connectedness, the reversible random walk conjecturer yields exponential growth of the proved set. This is a standard conditional mathematical derivation, not a reduction of the claimed result to its inputs by construction. No self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work are present in the provided text. The diversity and embedding sections are separate algorithmic proposals without load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on modeling theorems as a graph and on a set of primitive assumptions about prover guarantees and conjecturer access; these are introduced by the paper rather than derived from external benchmarks.

axioms (2)
  • domain assumption The set of theorems forms a graph with edges between pairs of theorems with similar semantics.
    Used to formalize the theorem space for analysis of self-play dynamics.
  • ad hoc to paper Primitive assumptions characterize the guarantees of a trained prover and how a conjecturer can access the structure of the graph.
    These assumptions are introduced to enable the subsequent growth proof.
invented entities (1)
  • Theorem graph no independent evidence
    purpose: To represent semantic relationships among theorems so that random walks and diffusion similarity can be defined.
    New modeling device introduced in the paper; no independent evidence supplied.

pith-pipeline@v0.9.1-grok · 5803 in / 1385 out tokens · 28125 ms · 2026-06-28T15:31:59.289040+00:00 · methodology

discussion (0)

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