Invariant Density and the Geometry of Mathematical Representation

Admissible mathematical representations are those that maximize invariant content per unit of description.Under this filter, canonical forms, minimal proofs, and irreducible structures emerge as forced outcomes rather than aesthetic preferences.The framework is domain-agnostic, avoids ontological commitments, and is illustrated with group theory, linear algebra, category theory, and rewrite systems. Version 3 introduced generative symmetric-axiom compression examples (Examples 5–6) and completed invariant-density calculations.Version 4 adds Appendix A, providing an exact, executable formal reproduction of the eliminative derivations in Examples 5–6, with no changes to the core filter. This work is part of a broader suite developing invariant density across syntactic, formal, and semantic contexts. The present paper introduces a geometric formulation of invariant density, focusing on representational compression and structural organization. Complementary works in the suite address bounded derivability in finite equational presentations, general syntactic frameworks, algebraic semantics, and proof assistant realizations, each archived separately on Zenodo. https://doi.org/10.5281/zenodo.18404261 (bounded derivability, equational presentations)https://doi.org/10.5281/zenodo.18301563 (syntactic framework)https://doi.org/10.5281/zenodo.18301522 (algebraic semantics)https://doi.org/10.5281/zenodo.18301480 (proof assistants)