Quantum Mechanics from Finite Graded Equality

• URL: http://arxiv.org/abs/2603.11900v1
• Date: Thu, 12 Mar 2026 13:14:48 GMT
• Title: Quantum Mechanics from Finite Graded Equality
• Abstract summary: We propose that quantum mechanics follows from a single hypothesis: equality has finite resolution.<n>We formalize the first two axioms as axioms; a structural Leibniz condition within the saturation axiom forces permutation invariance of basis elements.<n>Local tomography follows from $mathbbF = mathbbC$ with tensor product composition.
• Score: 0.40611352512781856
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $N$.

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