Invariant Density in Algebraic Semantics: Reducing Equational Presentations while Preserving Semantic Invariants

This paper studies redundancy in equational presentations of algebraic semantics, showing how axioms can be removed without altering equational consequences or varieties. The approach is syntactic, bounded, and witness-based, illustrated on groups and semigroup-like structures. Version 2: This version corrects and improves the SymPy witness in the Algebraic Semantics paper (proper inv Function, refined substitution chain illustrating collapse to unit then variable); applies small LaTeX polishing (math spacing, indentation, consistency) across the suite for better clarity and reproducibility. Version 3. This version expands the semantic framework with additional illustrative examples across groups, semigroups, and monoids, refines the executable SymPy witnesses, and clarifies bounded approximations of invariant sets and description measures. Core claims and scope remain unchanged. This work belongs to a suite of papers developing invariant density across multiple representational layers. The present paper investigates invariant density within algebraic semantics, emphasizing preservation of semantic invariants under syntactic reduction. Companion papers address geometric perspectives, general syntactic frameworks, bounded derivability in equational presentations, and formal realizations in proof assistants, all archived on Zenodo. https://doi.org/10.5281/zenodo.18265909 (geometric formulation)https://doi.org/10.5281/zenodo.18301563 (syntactic framework)https://doi.org/10.5281/zenodo.18404261 (bounded derivability, equational presentations)https://doi.org/10.5281/zenodo.18301480 (proof assistants)