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. Part of a suite on invariant density across syntactic, formal, and semantic contexts. This paper applies invariant density to algebraic semantics and equational theories, showing how presentations can be reduced without altering semantic invariants (varieties and equational consequences). Related works in the suite: https://doi.org/10.5281/zenodo.18283944 (syntactic framework), https://doi.org/10.5281/zenodo.18294529 (proof assistants), https://doi.org/10.5281/zenodo.18265909 (geometric formulation).