Canonical Symbolic Collapse: A Limit-Theoretic Foundation for Expression-Level Simplification over ℤ, ℝ, and ℂ

This paper presents a complete, limit-theoretic framework for symbolic simplification in classical mathematics, defining symbolic collapse as the reduction of any expression to a simpler or canonical form using the standard tools of arithmetic, algebra, calculus, logic, and set theory. It enumerates 500 such collapses, claiming to exhaust the class of valid simplifications under these rules. Crucially, the system operates on bounded domains and relies on limit-based equivalence, meaning expressions that are undefined at isolated points (like x∣x∣\frac{x}{|x|}∣x∣x at x=0x = 0x=0) are treated as valid when their limiting behavior matches the canonical form. This is essential because it allows the system to incorporate real-world mathematical behavior—like derivatives, discontinuities, and removable singularities—without contradiction. The discussion above shows that even seemingly problematic identities (like Collapse 14) are not contradictions, but deliberate and correct applications of limit theory within a tightly defined mathematical structure. This validates the paper’s central claim: that symbolic collapse, when grounded in classical limit logic and domain-awareness, forms a complete and consistent foundation for expression-level simplification across the reals, integers, and complex numbers.