Quantum Mechanics from Finite Graded Equality

Abstract

We propose that quantum mechanics follows from a single hypothesis: equality has finite resolution. Replacing the binary predicate $x = y$ with a graded distinguishability kernel $K(x,y) \in [0,1]$ forces three structural consequences: finite capacity ($N$ perfectly distinguishable states), relational completeness (all structure reduces to $K$-relations, and no measurement orientation is privileged), and reversible dynamics. We formalize the first two as axioms; a structural Leibniz condition within the saturation axiom forces permutation invariance of basis elements, and the full dynamical structure (cyclic evolution of order $N$, complex coefficients, and continuous unitary time evolution) is then uniquely determined. From these axioms (with regularity conditions derived in Appendix B: complex coefficients $\mathbb{C}$ are the unique field supporting cyclic dynamics and relational isotropy; deterministic hidden variables require $Ω(N^2)$ bits of storage (for prime-power $N$; exceeding $\log_2 N$ for all $N \geq 3$); the Born rule $p_k = |c_k|^2$ is the unique probability assignment preserving statistical distinguishability under reversible dynamics; and local tomography follows from $\mathbb{F} = \mathbb{C}$ with tensor product composition. Standard quantum mechanics is the $N \to \infty$ limit; finite $N$ provides a natural UV cutoff. The single free parameter is capacity $N$.