Structural Compression and Hidden Generators: A Conditional Information Invariant for Finite Mathematical Universes

We introduce a finite information-theoretic invariant measuring the structural com- plexity of mathematical objects relative to a chosen family of constraints. Recognizing that mathematical constraints form dense dependency networks, we utilize permutation- averaged conditional self-information to construct an order-independent coherence metric. We prove basic stability and logarithmic representation results for finite con- straint systems. Furthermore, we connect this invariant to algorithmic information theory by defining the Structural Compression Ratio, framing highly structured mathe- matical objects as compression points in a constraint space. The framework formalizes a quantitative principle aligned with the Langlands program: mathematical objects discovered to be massive compression points of apparent independent truths frequently admit simpler generating descriptions in deeper geometric or representation-theoretic categories. 1 Motivation The historical development of mathematics frequently reveals a pattern: complex mathe- matical objects that resist analysis within their apparent domains often yield entirely to the discovery of a new conceptual language that natively absorbs their defining constraints. The purpose of this paper is to construct a decent meta-mathematical invariant that formalizes this structural completeness. We introduce an information-theoretic invariant that measures an object’s structural complexity via conditional probability. By formulating this functional over finite universes and bridging it with algorithmic information theory, we provide a quantitative ranking invariant that shifts the evaluation of mathematical completeness from heuristic observation to formal metric analysis. 1

@article{deceuster2026_90-structural-compre,
  title = {Structural Compression and Hidden Generators: A Conditional Information Invariant for Finite Mathematical Universes},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.21762320},
  url = {https://peterdeceuster.uk/papers/90-structural-compression-and-hidden-generators-a-conditional-information},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/struct (1).pdf}
}