Kinematics of Single-Variable Systems and the Emergence of Quadratic Probability and Quantum Formalism

Quantum mechanics exhibits inherent probabilistic behavior, violations of classical probability, and a formalism seemingly disconnected from spacetime. We show that these features naturally emerge in single-variable systems, which can encode only a single independent unit of information at a time and therefore behave probabilistically. Imposing reversibility and conservation of total probability on the generalized probability formalism leads uniquely to a quadratic probability theory, with linear unitary transformations and probabilities given by squared moduli. From this, the Hilbert-space structure, Born’s rule, and unitary evolution follow directly. This first-principles derivation of quantum theory is consistent with the experimental fact that observing quantum effects requires coherence.