Invariant Density in Mathematical Representations: A Syntactic Framework for Presentation Reduction

This work introduces a purely syntactic framework for reducing equational presentations by identifying and eliminating redundant relations while preserving all derivable invariants. It formalizes invariant density as a ratio between derivable relations and syntactic description length, and demonstrates how invariant-preserving transforms induce monotonic density gains under explicit derivability bounds. The framework is constructive, deterministic, and grammar-relative: all reductions are mechanically witnessed, no semantic interpretation is assumed, and no claims of optimality or completeness are made. Version 2. This version expands the operational framework and executable examples, clarifying invariant-preserving presentation reduction as a self-contained syntactic method. Core definitions, criteria, and scope remain unchanged. This paper is part of a coordinated series investigating invariant density as a measure of structural compression in mathematical representations. The present work develops a general syntactic framework independent of semantic interpretation. Related papers in the suite specialize this framework to geometric formulations, bounded derivability in equational presentations, algebraic semantics, and proof assistant environments, each available as separate Zenodo records. https://doi.org/10.5281/zenodo.18265909 (geometric formulation)https://doi.org/10.5281/zenodo.18404261 (bounded derivability, equational presentations)https://doi.org/10.5281/zenodo.18301522 (algebraic semantics)https://doi.org/10.5281/zenodo.18301480 (proof assistants)