Identity Without Equality: An Informational Reconstruction of Dynamic Systems

In static systems, numbers behave as quantities and the classical equals sign expresses valid identity. But in dynamic and quantum systems, numbers behave as information, not quantities, and the equals sign becomes too rigid to describe evolving states. This work reconstructs the Schrödinger, Dirac, and Klein–Gordon equations as informational refinement processes, where identity is expressed not by equality but by invariance under transformation. Each equation becomes a cycle, attractor, or torsion-stabilised refinement, replacing differential operators with informational ones: torsion, parity, resolution, and informational inertia. This reframing resolves conceptual tensions in quantum theory and reveals a unified structure underlying dynamic systems. Minimal Colab implementations demonstrate convergence to stable informational attractors, providing a reproducible computational foundation. The result is a new mathematical architecture in which dynamic identity — not equality — is the organising principle of physical law. Note on scope: The claim that this framework ‘collapses the many-body explosion’ is used in a conceptual sense: by replacing state-vector enumeration with field-level informational operators, the combinatorial proliferation of basis states is sidestepped in the computational model. A full formal proof of equivalence to standard many-body theory remains an open problem.