A Singularity-Aware Symbolic Algebra Framework for Physical Equations

Singular points in theoretical physics — black hole singularities, the Big Bang at t=0, infrared and ultraviolet divergences in quantum field theory — are routinely treated as computational breakdowns rather than mathematical objects. We present a symbolic algebra framework based on Wheel Algebra (Carlström 2004) in which division by zero produces a well-defined absorbing element ⊥ ("bottom") instead of an error. Applied to 46 equations across six physical domains (general relativity, cosmology, quantum field theory, classical mechanics, thermodynamics, and pure mathematics), the framework correctly identifies singular points, classifies them into four structural categories, and — for pole singularities — computes residues via the Cauchy theorem. We demonstrate a formal isomorphism between classical resonance and the QFT on-shell condition, and show that the Kretschmann scalar allows the framework to distinguish coordinate artefacts from genuine physical singularities. The complete implementation is available as verifiable Python code.