The Composition Principle Kolmogorov–Arnold Representations, Deep Compositionality, and Structured Dynamics

Complexity isn't fundamentally high-dimensional — it's built by stacking simple pieces.

That's the core claim of this paper, and it's surprisingly far-reaching. Three research traditions arrived at this same idea independently: a 1957 theorem in pure mathematics, a program in machine learning theory, and a framework for what it means to be a persistent agent. The paper argues these aren't three analogies — they're three views of one underlying principle.

The mathematical starting point is the Kolmogorov–Arnold representation theorem, which says any continuous function of many variables can be exactly decomposed into sums and compositions of single-variable functions. No irreducibly multivariate primitives needed. This sounds like a gift for neural networks, but there's a catch the KAN enthusiasm tends to skip: Girosi and Poggio pointed out in 1989 that the theorem's inner functions are necessarily fractal-like and non-smooth, making them useless as a learning prior. Modern KANs escape this by quietly discarding the theorem's exactness and instead learning smooth, spline-parameterized edge functions — they keep the architecture's shape (learnable functions on edges, addition at nodes) while imposing the regularity the theorem never promised. The paper is careful to name this move explicitly.

Poggio's compositionality program attacks the same problem from the other direction. Instead of decomposing a function along its input variables, it decomposes a task along its computational operations. The key insight is that deep networks only beat the curse of dimensionality — the exponential blowup in parameters needed to approximate high-dimensional functions — when the target function is itself hierarchically compositional, meaning it's built from low-arity local pieces. Depth helps not because it's universally more expressive than shallow networks, but because it can match the compositional structure of the world.

The third strand is the most novel. In BCOM's Kolmogorov Theory framework, a persistent agent — one that maintains its own organization over time — must restrict its state transitions to a family of structure-preserving transformations closed under composition. That closure requirement is exactly the group axiom. In the continuous, reversible idealization this family is a Lie group; more realistically it's a pseudogroup (locally defined) or semigroup (irreversible). The punchline is that persistence doesn't merely tolerate group structure — it demands it, by the same compositional logic that governs the other two strands.

The unifying lens is Kolmogorov complexity: a compositional object is one with a short generative program. Depth-efficiency in networks, variable-factoring in the representation theorem, and generator-counting in agent dynamics are all the same compression fact seen from different angles. The paper is honest that this synthesis is a conjecture worth sharpening, not a settled theorem — the discovery problem (how an agent actually finds the right decomposition from data) is left open, and the informal complexity bound in Proposition 1 needs formalization. But the three-way identification itself — KANs decompose along variables, Poggio along operations, KT along the transformations that keep an agent alive — is the paper's genuinely new contribution.