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 $N$.

Figures (4)

• Figure 1: The Capacity Deficit. A finite-capacity system with $N$ distinguishable states stores $\log_2 N$ bits (blue). Deterministic outcomes for all measurement contexts require $\Omega(N^2)$ bits of storage (red). The Kochen--Specker bit-count (Proposition \ref{['thm:ks-bits']}) provides the universal bound exceeding $\log_2 N$ for all $N \geq 3$; for prime-power $N$, MUBs sharpen this to $(M{-}1)\log_2 N \sim N\log_2 N$ bits (Lemma \ref{['lem:incompressibility']}(a)). This is a storage deficit: even if non-contextual value assignments existed algebraically, they could not fit in the system's memory. The gap is astronomical for physical systems ($N \sim 10^{23}$): $\sim 10^{46}$ bits required vs. $\sim 76$ available. The deficit forces probabilistic behavior; metric compatibility selects the Born rule $p_k = |c_k|^2$.
• Figure 2: Logical structure: The Relational-to-Statistical Bridge. The two axioms (bold, top row) are the only inputs. $\dagger$Reversible Dynamics (dashed) is derived from Axioms 1--2 (Theorem \ref{['thm:dynamics-derived']}). Three branches follow: (i) cyclic dynamics $\to$ complex numbers; (ii) sheaf holonomy $\to$$U(1)$ gauge and $\mathbb{C} P^{N-1}$; (iii) Capacity Halting $\to$ Probabilistic Response $\to$ Information Isometry $\to$ Born rule (thick coral arrows). The Metric Bridge theorem (Theorem \ref{['thm:metric-bridge']}) makes the coral path explicit: Capacity Halting (Theorem \ref{['thm:capacity-halting']}) shows determinism is impossible; Information Isometry (Theorem \ref{['thm:born-kernel']}) selects the unique form$p_k = |c_k|^2$.
• Figure 3: State space geometry for $N=3$.(a) The probability simplex: vertices are basis states $|0\rangle, |1\rangle, |2\rangle$; a state $|\psi\rangle$ has barycentric coordinates $(|c_0|^2, |c_1|^2, |c_2|^2)$ satisfying $\sum_k |c_k|^2 = 1$. (b) The Born rule from geometry: probability $p_j$ equals the perpendicular height from $|\psi\rangle$ to the face opposite vertex $|j\rangle$. This geometric interpretation underlies the uniqueness proof (Theorem \ref{['thm:born-kernel']}).
• Figure 4: The Born rule as Information Isometry. The map $\psi \mapsto (p_1, \ldots, p_N)$ with $p_k = |c_k|^2$ is the unique probability assignment making quantum distinguishability (Fubini-Study distance $d_{FS}$) equal to statistical distinguishability (Fisher-Rao distance $d_{FR}$). Why must these be isometric? If $g_{FR} \neq c \cdot g_{FS}$, the system would maintain two independent notions of "distance." The $U(N)$-invariance required by Axiom \ref{['ax:relational']} then forces both metrics to be proportional: there is exactly one invariant metric on $\mathbb{C} P^{N-1}$ (Theorem \ref{['thm:fs-unique']}). Equivalently, reversible dynamics preserving $g_{FS}$ would distort $g_{FR}$ for any $\alpha \neq 2$ (Corollary \ref{['cor:dynamics-born']}).

Theorems & Definitions (169)

• Theorem 1: Main Theorem: Characterization of Relational Theories
• Definition 2: Graded Equality
• Proposition 3: Insufficiency of a Single Reference Set
• proof
• Proposition 4: Cyclic Dynamics from Self-Resolution
• proof
• Theorem 5: Dynamical Constitution of Identity
• proof
• Proposition 6: Context-Incompleteness and Stochasticity
• proof