The Mirror Theorem: Central Incidence Saturation in Keith Recurrences

Can a mirror emerge when numbers return? Some numbers reappear in sequences generated from their own digits. Known as Keith numbers since 1987, they are the starting point for Federico Ignacio Zamponi’s Keith mirrors, a class defined by how terms occur in the recurrence equations. Nearly four decades later, the Mirror Theorem classifies this family across all number bases from 3 onward: exactly 35 mirrors with three or more digits, alongside two infinite families of two-digit mirrors. The paper includes mathematical proofs and code to reproduce the finite classification.