K < 1: Schrödinger

At subcritical coupling (\(K < 1\)), the order parameter \(r\) is small and a finite fraction of oscillators remain unlocked — they sit in the gaps of the devil’s staircase with no definite winding number. These are the quantum states.

The linearized Kuramoto equation in this regime, with nearest-neighbor diffusive coupling on a spatial lattice, reduces to the Schrödinger equation via the Madelung transform:

\[i\hbar\partial_t\Psi = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi\]

where:

Kuramoto quantity	QM quantity
Unlocked oscillator density \(\rho(x,t)\)	\(\lvert\Psi\rvert^2\)
Accumulated phase perturbation \(S(x,t)\)	Action / phase
Tongue-structure effective potential	\(V(x)\)
Stern-Brocot RG diffusion \(D_{\text{eff}}\)	Quantum potential \(\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}\)

The quantum potential — the term that distinguishes quantum from classical mechanics — arises from the Stern-Brocot renormalization group flow: per-level variance \(\sigma^2(d) \sim \phi^{-4d}\) sums to a convergent constant \(D_{\text{eff}} = D_0 / (1 - \phi^{-4})\).

The full derivation is in Derivation 12: The Two Continuum Limits, Part II.

Three regimes, one equation

Coupling	Regime	Physics
\(K = 1\)	Critical — all oscillators locked	General relativity (Lovelock uniqueness)
\(0 < K < 1\)	Subcritical — partial locking	Quantum mechanics (Madelung / Nelson)
\(K \to 0\)	No coupling	Free particles — no structure

The transition between regimes is controlled by the same self-consistency equation on the Stern-Brocot tree (Derivation 11). There is no quantization postulate.