Compression Model of Mathematics

• Compression Model of Mathematics is a framework where mathematical notations and reasoning serve as compressed forms of larger informational structures via pattern matching and unification.
• It illustrates how equations, names, and constructs like SP-multiple-alignment act as efficient, compressed representations that preserve computational and inferential content.
• The model bridges mathematics, logic, and computing, extending to empirical tests on formal libraries and offering insights into compressibility criteria in human versus formal mathematics.

The compression model of mathematics denotes a family of research programs in which mathematical notation, reasoning, and structure are treated as compressed representations of larger bodies of information, or as computations performed directly on compressed objects. In one foundational formulation, mathematics is understood as information compression via the matching and unification of patterns (ICMUP), with “mathematics as ICMUP” abbreviated MICMUP; in parallel, other work studies matrices, combinatorial objects, stochastic trajectories, and formal libraries as mathematical entities whose algebraic or inferential structure remains available under compression (Wolff, 2013, Wolff, 2018, Paixão et al., 2013, Aksenov et al., 20 Mar 2026).

1. Foundational formulations

The foundational thesis is stated most explicitly in the SP literature. There, mathematics is not treated as a separate formal field built on numbers first and foremost; instead, many mathematical practices are reinterpreted as compressed representations of information or as computations driven by pattern matching. The central primitive is matching and unification of patterns, where “unification” means “a simple merging of two or more matching patterns to make one,” and knowledge is represented as patterns—arrays of atomic symbols in one or two dimensions. On this view, the paper argues not that mathematics provides a basis for understanding compression, but that ICMUP may provide a basis for mathematics (Wolff, 2013).

This formulation is extended by the claim that mathematics can be understood, at a deep level, as a system of information compression via the matching and unification of patterns, and that this is not merely the familiar use of mathematics for compression in the Shannon/Kolmogorov sense. The relevant mechanism in the SP framework is SP-multiple-alignment, described as a generalized version of ICMUP and as the “bedrock” of the SP system. The same framework is then used to connect mathematics with logic, computing, and human learning, perception, and cognition (Wolff, 2018).

A later formulation recasts the thesis in terms of the distinction between formal mathematics (FM) and human mathematics (HM). FM is the full space of all valid deductions in a formal system, whereas HM is the small, “interesting,” and human-discoverable subset characterized by compressibility through hierarchically nested definitions, lemmas, and theorems. This suggests a sharper operational criterion for mathematical significance: HM occupies the compressible regions of FM, where naming and reuse create leverage (Aksenov et al., 20 Mar 2026).

2. Compression mechanisms inside mathematical notation and reasoning

The SP-based account identifies several recurring compression mechanisms in ordinary mathematics. The most direct case is the equation. A compact law such as $s = (gt^2)/2$ is presented as a compressed representation of a potentially large table of distance–time values, and the same point is made with $E = mc^2$, $a^2 + b^2 = c^2$, and $P = k / V$ or $PV = k$. The claim is not merely that equations summarize data; they are said to function literally as compressed representations of much larger bodies of numerical information (Wolff, 2013, Wolff, 2018).

A second mechanism is matching and unification of names. In the example

$x = 4,\qquad y = 5,\qquad z = x + y,$

computing $z$ requires matching the symbolic names $x$ and $y$ in the third equation with their earlier definitions and unifying them so that the correct values are used. Function invocation is treated similarly: a call such as $\log 1000$ matches the schema $E = mc^2$0, with $E = mc^2$1 unified into the variable position (Wolff, 2013, Wolff, 2018).

A third mechanism concerns numbers themselves. The SP system in its core form has no built-in concept of number and no native arithmetic operators, but unary notation is treated as an especially transparent example of redundancy. Using Peano’s successor notation,

$E = mc^2$2

corresponds to unary forms such as $E = mc^2$3, $E = mc^2$4, $E = mc^2$5. This is practical only for small values because the repeated symbols are highly redundant; higher bases such as binary, octal, or decimal are then interpreted as compression of unary repetition. The same point is illustrated in the later paper by examples such as /////// $E = mc^2$6 7 and ///////////////// $E = mc^2$7 17 (Wolff, 2013, Wolff, 2018).

The literature then organizes these observations into three named compression variants. Chunking-with-codes uses a short code to stand for a larger chunk of structure. Examples include “TFEU” for “Treaty on the Functioning of the European Union,” ordinary names such as “New York” or “Nelson Mandela,” and named mathematical or computational functions. Schema-plus-correction stores a reusable template plus variable fillers; the papers illustrate this with the menu schema Menu1: Appetiser (S) sorbet (M) (P) coffee-and-mints, encoded as Menu1:(3)(5)(1), and with parameterized functions such as SQRT(number), BIN2DEC(number), and COMBIN(count_1, count_2). Run-length coding compresses repetition by indicating a count or boundary, as in “touch toes (×15)” or “start...stop,” and is used to reinterpret multiplication as repeated addition, division as repeated subtraction, exponentiation as repeated multiplication, factorials as repeated multiplication and subtraction, and notations such as

$E = mc^2$8

as shorthands for repeated operations, often combined with schema-plus-correction because a variable changes on each iteration (Wolff, 2013, Wolff, 2018).

The same compression logic is also extended to class-inclusion hierarchies and part-whole hierarchies. Shared attributes stored once at a higher level and inherited downward are treated as compression, just as shared parts embedded in a whole avoid unnecessary repetition. In this sense, mathematics is presented not only as economical description but also as structured transformation: expressions are manipulated by matching, unification, repeated application, and replacement (Wolff, 2018).

3. Logic, computing, and the reversibility problem

A major extension of the compression model is the claim that logic and computing are not merely adjacent to mathematics but instantiate the same underlying mechanism. The papers treat Post’s Canonical System and the transition function of a Universal Turing Machine as systems operating largely through matching and unification of patterns. Computer memory retrieval is described as a match between an address held by the CPU and the corresponding address in memory, with implicit unification of the two. Systems such as Prolog and query-by-example are likewise said to work largely through matching and unification of patterns (Wolff, 2013, Wolff, 2018).

In logic, the same claim is illustrated by truth tables. For XOR, the output is obtained by selecting the row that best matches the inputs, with the “best match” understood as the one yielding the greatest compression. Similar reasoning is applied to NAND, which is noted as sufficient, in principle, to build any general-purpose digital computer. The later paper also interprets deriving a set from a multiset, computing unions and intersections, and Prolog’s unification mechanism as instances of matching and unification (Wolff, 2018).

The compression model also addresses the apparent paradox of “decompression by compression.” If a system is fundamentally compression-based, it may seem unable to reconstruct or generate detailed output from a shorter code. The SP account resolves this by stating that the same mechanism operates in both directions. A phrase such as “Treaty on the Functioning of the European Union” may be encoded as “TFEU,” and “TFEU” together with the stored pattern TFEU Treaty on the Functioning of the European Union can then retrieve the longer phrase. The stated condition is that the code retain some residual redundancy: it must be “at least slightly bigger than the theoretical minimum for the process to work.” The paper compares this bidirectionality to a Prolog program running forwards and backwards, or a car driven in either direction with the same engine (Wolff, 2013).

This directly addresses a common misconception. The compression model does not imply that redundancy is absent or undesirable in all contexts. The SP paper states that computing and cognition as information compression is compatible with the uses of redundancy in such things as backup copies to safeguard data and understanding speech in a noisy environment (Wolff, 2013).

4. Compressed mathematical objects as computational representations

A separate line of work treats compression not primarily as a foundation for notation and reasoning, but as a way to preserve mathematical objects themselves in compressed computational form. The clearest example is the lossless compression of numerical matrices. One paper proposes a model in which a matrix is produced, compressed, and then subjected to mathematical manipulations while still compressed, with decompression deferred “only for human reading.” The aim is to preserve the matrix as a mathematical object—its shape $E = mc^2$9, row/column access, exact integer values, and compatibility with linear algebra operations—while reducing memory footprint. Two schemes are proposed: Supreme Minimum (SM), which uses a uniform bit-width equal to $a^2 + b^2 = c^2$0, and Variable Length Blocks (VLB), which stores each entry using its own minimal bit-length plus a length field. Their efficiencies are written as

$a^2 + b^2 = c^2$1

and

$a^2 + b^2 = c^2$2

with VLB reported as superior in about 99.1% of sampled parameter combinations, while SM is better mainly when bit-lengths are nearly constant or close to the maximum (Paixão et al., 2013).

Compression of combinatorial structure is developed in a different way in work on arithmetic coding for multisets, permutations, combinations, and truncated permutations. There the guiding principle is that one should compress the mathematical object in its own structural form rather than force an arbitrary sequential representation. The relevant probability models are induced by simple assumptions such as i.i.d. draws, forgetting order, or sampling without replacement, and are factorized into univariate conditionals suitable for arithmetic coding. The paper states that the resulting output is within 2 bits of the ideal Shannon code length for every input object, not merely on average (Steinruecken, 2016).

Integer sets are treated similarly through recursive subset-size encoding (rsss). A set $a^2 + b^2 = c^2$3 is represented by recursively encoding subset sizes in a binary decomposition of the universe, and empirical statistics enter through local subtree fractions $a^2 + b^2 = c^2$4. The model is designed to bridge recursive range restriction and learned source statistics while requiring only $a^2 + b^2 = c^2$5 counters rather than exponential storage. In the uniform case, the recursive subset-size model aligns with the entropy of an equivalent independent Bernoulli model; under statistical modeling, it can exploit structure that gap coding or range-narrowing methods may miss (Larsson, 2014).

Compression of abstract simplicial complexes makes the same point in a Boolean-combinatorial setting. Given the facets of a simplicial complex, the Facets-To-Faces algorithm outputs the entire complex in a compressed format using disjoint unions of multivalued rows with wildcards such as $a^2 + b^2 = c^2$6, $a^2 + b^2 = c^2$7, $a^2 + b^2 = c^2$8, and $a^2 + b^2 = c^2$9. The paper states that if a complex has facets $P = k / V$0, then Facets-To-Faces enumerates it as a union of $P = k / V$1 disjoint 012e-rows in time

$P = k / V$2

and that its degree of compression compares favorably to both BooleanConvert and BDDs (Wild, 2018).

Compressed representation also appears in high-dimensional geometry and operator models. For positive semidefinite factorizations and, with normalization constraints, for quantum models, randomized Gaussian Johnson–Lindenstrauss-type projections are used to compress ambient dimension while approximately preserving pairwise inner products $P = k / V$3. The upper bound for general psd factorizations is expressed by the existence of compressed matrices $P = k / V$4 in dimension

$P = k / V$5

such that

$P = k / V$6

while the paper emphasizes that low-rank or low-trace measurements compress well and that normalization constraints create genuine lower bounds for quantum models (Stark et al., 2014).

Large structured matrices supply another operational setting. Toeplitz and Toeplitz-like matrices are described as compressible because low displacement rank becomes, after Fourier transform, a Cauchy-like structure whose off-diagonal blocks are numerically low rank. This yields explicit numerical-rank bounds for submatrices and underwrites hierarchical formats such as HODLR and HSS. In the weakly admissible case $P = k / V$7, the paper gives

$P = k / V$8

and in the strongly admissible case $P = k / V$9,

$PV = k$0

thereby supplying a displacement-based explanation for superfast rank-structured solvers (Beckermann et al., 13 Feb 2025).

Taken together, these models suggest that the phrase “compression model of mathematics” has an operational meaning as well as a foundational one: the compressed object remains mathematically meaningful when its dimensions, entries, access structure, or inferential content are preserved.

5. Bounds, obstructions, and non-universality

The compression model is not uniformly optimistic. In analog compression of stationary stochastic processes supported on a shift-invariant set $PV = k$1, the relevant limits are governed by metric mean dimension and mean box dimension, not by entropy rate alone. The main lower bound states that for Borel compression with $PV = k$2-Hölder decompression,

$PV = k$3

is a universal lower bound on worst-case almost-lossless compression rate over all stationary processes supported on $PV = k$4. Upper bounds are then given in terms of mean box dimension and, in a separate result, by a universal rate at most $PV = k$5 via Peano-curve-type surjective Hölder maps. A crucial negative example shows that replacing $PV = k$6-Hölder by merely $PV = k$7-Hölder can destroy the lower bound, so the uniform Hölder constant $PV = k$8 is essential (Gutman et al., 2018).

Direct computation on compressed data is likewise highly scheme-dependent. For simple compressions such as Run-Length-Encoding (RLE), inner product can be done in $PV = k$9 time on vectors of compressed size $x = 4,\qquad y = 5,\qquad z = x + y,$0. For grammar-compressions containing LZ77, LZ78, LZW, Byte-Pair Encoding, and dictionary methods, the paper proves that essentially $x = 4,\qquad y = 5,\qquad z = x + y,$1 or even larger runtimes are necessary in the worst case, under the 3SUM, Strong 3SUM, and Strong $x = 4,\qquad y = 5,\qquad z = x + y,$2SUM conjectures. It also shows that highly compressible input matrices can have products whose output requires $x = 4,\qquad y = 5,\qquad z = x + y,$3 size in any grammar-compression. This rules out any general expectation that compressed linear algebra can always be executed “as if” the inputs were genuinely small (Abboud et al., 2020).

Quantum-model compression introduces a different type of obstruction. If, for a fixed measurement $x = 4,\qquad y = 5,\qquad z = x + y,$4, one defines

$x = 4,\qquad y = 5,\qquad z = x + y,$5

then any exact $x = 4,\qquad y = 5,\qquad z = x + y,$6-dimensional model must satisfy

$x = 4,\qquad y = 5,\qquad z = x + y,$7

and in the approximate setting, if $x = 4,\qquad y = 5,\qquad z = x + y,$8, the bound becomes

$x = 4,\qquad y = 5,\qquad z = x + y,$9

The point is that generic psd factorizations may compress well under random projection, but the normalization constraints $z$0 and $z$1 make quantum models more rigid (Stark et al., 2014).

These results jointly undermine a strong version of the compression thesis. Compression does not, by itself, guarantee efficient direct computation, dimension reduction, or universality of encoding. What it can provide depends on the representation, the regularity class, and the structural invariants preserved.

6. Empirical models of human mathematics

The most explicit empirical test of a compression-based model of mathematics uses MathLib, a large Lean 4 library treated as a proxy for HM. Each library element is assigned a depth, a wrapped length, and an unwrapped length. Depth is the length of the longest path to primitives in the dependency DAG. Wrapped length is the token count of the source-level definition or theorem statement. Unwrapped length is the total number of primitive symbols after recursively expanding all references. The main findings are that unwrapped length grows exponentially with both depth and wrapped length, while wrapped length stays approximately constant across all depths; the paper reports that these observations are consistent with the free abelian monoid model $z$2 and inconsistent with the free non-abelian monoid model $z$3 (Aksenov et al., 20 Mar 2026).

The associated monoid model is mathematically explicit. In the free abelian monoid $z$4, a logarithmically sparse macro set can achieve exponential expansion of expressivity; in one theorem,

$z$5

In the free non-abelian monoid $z$6, by contrast, a polynomially-dense macro set yields only linear expansion, and superlinear expansion requires near-maximal density. This is the formal basis for the paper’s claim that HM behaves like a compressible, low-growth region inside the vastly larger space FM (Aksenov et al., 20 Mar 2026).

The same work also proposes compression-based measures of mathematical interest. Reductive compression

$z$7

measures how much an element compresses when its named form is used, while deductive compression

$z$8

measures how much proof content is packed into a short statement. A PageRank-style score on the dependency graph then combines graph centrality with these compression scores to identify “load-bearing” elements and to direct automated reasoning toward the compressible regions where HM lives (Aksenov et al., 20 Mar 2026).

This suggests a distinctive interpretation of mathematical practice. Definitions, lemmas, and theorems are not only truth-bearing statements; they are also reusable macros that manage exponential primitive content with approximately stable surface length. Under that interpretation, the effectiveness of mathematics is linked to its ability to compress regularity while preserving the capacity for reconstruction, inference, and reuse.