Supernumbers: Invertibility via Structural Identity

Supernumbers introduce a first-class representation of values that carry their full pre-image structure, making invertibility intrinsic rather than imposed externally. Unlike classical numbers, each supernumber encodes the contextual information necessary to reconstruct all inputs that could have produced it under a given transformation. In this framework, shadows are defined as fibers and are allowed to retain their internal structure under composition, yielding nested representations of equivalence classes. This avoids the loss of intermediate structure typically induced by function composition. Supernumbers provide a structural reinterpretation of non-injectivity: information is not destroyed, but collapsed under overly fine identity criteria. By restoring identity at the level of fibers, invertibility becomes universally available modulo structural equivalence.