Flux-Calibrated Mode Mixing

1. Integration point: apply this to a low-dimensional latent representation of samples from an energy-based model, score-based diffusion model, or multimodal VAE. First identify two modes by clustering latent samples, then train a scalar separator $s_\psi(x)$; initialize it as a linear classifier and optionally refine it near low-density saddle candidates. Run overdamped Langevin dynamics, or use consecutive reverse-diffusion displacements as the velocity proxy $v_t=(x_{t+1}-x_t)/\Delta t$. 2. Pseudocode: generate a trajectory; compute $s_t=s_\psi(x_t)$; when $s_t<0$ and $s_{t+1}\geq0$, record a candidate crossing; inspect the next $\tau_c$ steps and count it only if all later states satisfy $s>0$; update $\widehat{k}=N/T_A$ over a sliding window; update $T\leftarrow T\exp(\gamma\log(k_\star/(\widehat{k}+\varepsilon)))$ and clip the multiplier. Use $\gamma\in[0.02,0.2]$. 3. Taken from the paper: one-way phase-space flux, basin-normalized rate, saddle-oriented dividing surface, and commitment-time rejection of recrossings. Estimated empirically: $\rho$, $P_A$, $k$, the saddle surface, and the first gap in the crossing-time histogram. 4. Cheap experiment: use a 2D double-well energy, followed by a multimodal toy diffusion model, comparing fixed-temperature Langevin against adaptive flux control. Measure mode occupancy, effective sample size, and crossing rate. The quantitative prediction is that accepted crossing times exhibit a short-time gap after selecting $\tau_c$, and the controller converges to $\widehat{k}=k_\star$ with relative error below 20%. Increasing noise should increase the measured rate over the controlled operating range; failure of a commitment-time plateau or strong nonmonotonicity of $k(T)$ falsifies the proposed controller.