# 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](../03_einstein/12_continuum_limits.md),
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.