Start with one axiom: equality has finite resolution. Two states are not simply equal or unequal — they are distinguishable to a degree, measured by a kernel K(x,y) that ranges from 0 (identical) to 1 (perfectly distinct). This is not quantum mechanics. It is a statement about the limits of comparison.
Zilly (arXiv:2603.11900) derives quantum mechanics from this single premise.
The derivation proceeds through three structural requirements that follow from finite resolution. First, finite state capacity: a system described by graded equality can accommodate at most N distinguishable states, where N is set by the resolution of the kernel. Second, relational completeness: every operationally distinguishable property must be representable within the formalism. Third, reversible dynamics: the distinguishability structure must be preserved under time evolution, because a dynamics that created or destroyed distinguishability would violate the resolution constraint.
From these three requirements, enforced by permutation invariance over states, the formalism assembles itself. Complex numbers emerge as the unique field that supports cyclic dynamics — real numbers lack the phase structure needed for reversible evolution, and quaternions overconstrain the system. The Born rule emerges as the unique probability assignment that preserves statistical distinguishability under the reversible dynamics. Standard quantum mechanics appears as the N → ∞ limit, where the finite-capacity cutoff recedes and the Hilbert space becomes infinite-dimensional.
The structural point is what the axiom does not contain. It says nothing about waves, particles, superposition, measurement, observers, collapse, or probability. It says only that comparison is imperfect. Everything else — the entire apparatus of quantum theory — is forced by consistency. Complex amplitudes are not a mysterious feature of nature that we accept because the formalism works. They are the only coefficients compatible with reversible dynamics in a system where equality is graded. The Born rule is not an additional postulate bolted onto the Schrödinger equation. It is the unique way to extract probabilities without violating the resolution structure.
The framework also quantifies the cost of hidden variables. A deterministic completion of the finite-capacity system requires O(N²) classical bits — exceeding log₂N for all N ≥ 3. The overhead is not arbitrary. It is forced by the requirement that the hidden-variable model reproduce the distinguishability kernel exactly. The excess grows polynomially, confirming that quantum mechanics is informationally more compact than any classical simulation of it.
One axiom. One formalism. The weirdness was always the constraint, not the decoration.