Compression is all you need: Modeling Mathematics

Abstract: Human mathematics (HM), the mathematics humans discover and value, is a vanishingly small subset of formal mathematics (FM), the totality of all valid deductions. We argue that HM is distinguished by its compressibility through hierarchically nested definitions, lemmas, and theorems. We model this with monoids. A mathematical deduction is a string of primitive symbols; a definition or theorem is a named substring or macro whose use compresses the string. In the free abelian monoid $A_n$, a logarithmically sparse macro set achieves exponential expansion of expressivity. In the free non-abelian monoid $F_n$, even a polynomially-dense macro set only yields linear expansion; superlinear expansion requires near-maximal density. We test these models against MathLib, a large Lean~4 library of mathematics that we take as a proxy for HM. Each element has a depth (layers of definitional nesting), a wrapped length (tokens in its definition), and an unwrapped length (primitive symbols after fully expanding all references). We find unwrapped length grows exponentially with both depth and wrapped length; wrapped length is approximately constant across all depths. These results are consistent with $A_n$ and inconsistent with $F_n$, supporting the thesis that HM occupies a polynomially-growing subset of the exponentially growing space FM. We discuss how compression, measured on the MathLib dependency graph, and a PageRank-style analysis of that graph can quantify mathematical interest and help direct automated reasoning toward the compressible regions where human mathematics lives.

• The paper establishes that hierarchical compression underpins human mathematics by formalizing macro definitions within monoidal models.
• It quantifies exponential expansion using metrics like wrapped and unwrapped lengths, confirming efficient abstraction in MathLib.
• The study reveals that optimal compression guides both automated reasoning and the structural curation of formal mathematical libraries.

Compression as the Core Principle of Human Mathematics: A Technical Analysis

Introduction

The paper "Compression is all you need: Modeling Mathematics" (2603.20396) addresses the structural distinction between the vast space of formal mathematics (FM) and the much narrower set that constitutes human mathematics (HM). The fundamental thesis is that HM is not just a selection of "interesting" theorems and definitions from FM, but is defined by compressibility—HM resides precisely in those regions of FM that admit exponential compression through hierarchical, nested definitions, lemmas, and theorems.

Monoids as Models for Compression in Mathematics

The paper introduces and analyzes monoidal models to formalize the notion of compressibility. Proofs and mathematical objects are modeled as words in either the free abelian monoid $A_n$ or in the free (non-abelian) monoid $F_n$. Here, definitions and theorems act as macros (new generators), whose adoption allows for greater expressivity at reduced syntactic cost.

• Expansion Function: For a macro set $M$, the augmented generator set $G' = G \cup M$ allows expressing more with less. The expansion function $f_{G'}(s)$ quantifies how large a "ball" of primitive words in $G$ can be covered by words of $G'$-length at most $s$.
• Macro Density and Expansion Regimes:
  • In $A_n$, logarithmically sparse macros (e.g., base expansions akin to place notation) yield exponential expansion: a small increase in macro count translates into exponential increases in reach.
  • In $F_n$, only nearly maximal (i.e., exponentially dense) macro sets achieve superlinear expansion, with polynomially dense macro sets supporting merely linear expansion.

Theoretical Results

Key theorems establish the structural limitations on compression:

• $A_n$, Place Notation: With macros at powers of $b$—$M = \{b^j a_i\}$—one obtains $f_{G'}(s) = \Theta(b^{s/(n(b-1))})$ exponential expansion. This is an explicit formalization of the efficiency of numeration systems.
• $A_n$, Waring-Type Macros: Polynomially dense macro sets (e.g., $\{m^k a_i\}$) allow infinite expansion due to additive combinatorial properties (Waring’s problem analogs).
• $F_n$, Polynomial-Dense Macros: Expansion is strictly linear; macro sparsity in the exponentially growing $F_n$ is overwhelmingly outpaced by the volume growth, preventing hierarchical nesting.

Empirical Validation on MathLib

To ground the monoid-theoretic analysis in real mathematical practice, the study analyzes MathLib—a large Lean 4-based library for formalized mathematics.

Three key metrics are extracted:

• Depth: Longest path in the dependency DAG from an element to a primitive.
• Wrapped Length: Token count in the defining expression (analogous to $G'$-length).
• Unwrapped Length: Number of primitives after full expansion (analogous to $G$-length).

Findings:

• Exponential Growth of Unwrapped Length with Depth and Wrapped Length: Median $\log_2(\text{unwrapped length})$ grows linearly in both wrapped length and depth, confirming the exponential expansion predicted by $A_n$ with logarithmic macro density.

  Figure 1: Median $\log_2(\text{unwrapped length})$ versus wrapped length, confirming exponential expansion per token.

• Flatness of Wrapped Length Across Depths: Median wrapped length remains essentially constant across large variation in depth, indicating that definitions do not become unwieldy with nesting, analogous to efficient positional notation.

  Figure 2: Median wrapped length versus depth, demonstrating length stability regardless of hierarchical depth.

• Exponential Growth of Unwrapped Length with Depth: The nearly perfect linear relationship (on log scale) between unwrapped length and depth is direct evidence of optimal hierarchical compression.

  Figure 3: Median $\log_2(\text{unwrapped length})$ versus depth, showing exponential increase in expressivity with hierarchical layering.

Regime Discrimination and Macro Set Identification

Comparison of empirical relationships with theoretical models from different monoidal regimes demonstrates that MathLib’s structure aligns with $A_n$ equipped with macros of at most logarithmic density—consistent with efficient expansion and deep reuse of propositions. $F_n$-type behavior is decisively ruled out.

Macro Set Identification Problem: The practical challenge of finding the optimal macro set within a mathematical corpus is formally analogous to optimization over compression strategies under macro cardinality constraints. The empirical approach of taking all non-primitive elements as macros is suboptimal; more nuanced graph-theoretic or information-theoretic macro selection methods remain an open problem with major implications for automating mathematical abstraction.

Implications for Automated Reasoning and "Mathematical Interest"

The focus on compression yields immediate practical consequences for AI in mathematics:

• Automated "Taste": Measures like $T_0$ (reductive compression) and $I_0$ (deductive compression) assign high scores to those elements with maximal compression ratios—either because their succinct definition exploits deep layers of abstraction, or because their proof is highly nontrivial relative to their statement.
• PageRank-Style Centrality: To avoid emphasizing only isolated curiosities, the authors propose a PageRank variant (biased to high-compression nodes) for the dependency graph, quantifying both intrinsic compressibility and network centrality. This approach operationalizes the notion of “interesting” mathematics as a structural property of compressibility and utility.
• Guidance for AI Exploration: It is proposed that AI agents navigating formal mathematics should bias their exploratory strategies toward regions of high compressibility—i.e., the “soft core” where human mathematics resides.

Broader Structural and Algorithmic Consequences

• Structure of HM within FM: The dramatic difference in expansion and compressibility properties between $A_n$ and $F_n$ regimes indicates that HM is statistically thin—parsimony forces HM to inhabit an exponentially small, yet macroscopically accessible subspace of FM.
• FM's Incompressible Regions: Most of FM is not just uninteresting, but algorithmically irreducible; for generic theorems (e.g., random Boolean unsatisfiability claims), no hierarchical compressibility is available—indeed, compressibility would imply breakthroughs in complexity theory (e.g., $P=NP$).
• Compression and Logical Depth: Aligning with concepts like Bennett’s logical depth, only regions of FM with low-depth (fast decompressibility) are mathematically tractable and humanly meaningful.

Conclusion

The paper establishes, theoretically and empirically, that compression is the defining structural property of human mathematics. By formalizing mathematical abstraction as macro selection in monoids and validating the predicted expansion regimes in formalized libraries, the authors provide not just an explanatory, but also a predictive framework for mathematical practice. This compressibility-centric paradigm has immediate consequences for the design of automated theorem provers, the curation and mining of mathematical libraries, and the formalization of mathematical "interest".

Future directions include algorithmic identification of macro sets, efficient approximation of compositional compression measures, and their integration into AI systems for automated theory exploration and conjecture ranking. Understanding the limits and potential of hierarchical compression promises to shape the future path toward machine-assisted mathematical discovery.