(Bridges: Neuro-Symbolic ODE Discovery × Universal Persistent Brownian Motions × Validated LMC barrier scaling)

The stated title bundles three distinct technical threads without specifying a mechanism connecting them: (1) neuro-symbolic ODE discovery (learning symbolic differential equations from data via neural-guided search), (2) "universal persistent Brownian motions" (a claimed universality class for persistent/correlated random walks, e.g. fractional Brownian motion with memory kernels), and (3) "validated LMC barrier scaling" (Langevin Monte Carlo mixing-time scaling laws across energy barrier heights). As written, no falsifiable claim exists — there is no stated functional relationship, direction of causality, or quantitative prediction linking the three. A minimal falsifiable reconstruction is: "A neuro-symbolic ODE discovery system, when applied to trajectories generated by persistent (memory-kernel) Brownian motion processes, can recover closed-form symbolic drift/diffusion terms whose predicted LMC mixing-time barrier-scaling exponent matches the empirically validated scaling law (e.g., mixing time ∝ exp(c·ΔU) for barrier height ΔU) to within a specified error tolerance (e.g., ±15% on the scaling exponent c) across a benchmark suite of ≥5 synthetic potential landscapes." This restatement must be confirmed or revised with the original authors before further work proceeds.

Minimum viable test (MVT): a controlled synthetic benchmark, not real-world data, to first establish whether the three-way linkage is even coherent before any empirical claim is pursued.

1. Generate synthetic persistent Brownian motion trajectories (fractional Brownian motion, Hurst H ∈ {0.3, 0.5, 0.7, 0.9}) in double-well and multi-well potentials with known analytic barrier heights ΔU ∈ {2, 4, 6, 8, 10} kT.
2. Run an existing neuro-symbolic ODE/SDE discovery tool (e.g., PySR, SINDy, or a neural-guided symbolic regression baseline) on trajectory data to recover drift/diffusion functional forms.
3. Independently compute empirical LMC mixing times via standard MCMC diagnostics (effective sample size, Gelman-Rubin R̂, spectral gap estimation) on Langevin samplers targeting the same potentials.
4. Fit the empirical mixing-time-vs-barrier scaling law and compare its exponent against the scaling implied by the neuro-symbolic-recovered drift term (via Kramers' escape rate formula).
5. Quantify agreement; run ablations removing persistence (H=0.5, standard BM) as a negative control.

# Phase 0: Formalize hypothesis (manual, 1 week)
hypothesis = clarify_with_authors(discovery_id)

# Phase 1: Synthetic data generation
for H in [0.3, 0.5, 0.7, 0.9]:
    for dU in [2, 4, 6, 8, 10]:
        potential = make_multiwell_potential(barrier_height=dU)
        traj = generate_fBM_trajectory(H=H, potential=potential,
                                        n_steps=1e5, n_reps=500, seed=range(3))
        save(traj, key=(H, dU))

# Phase 2: Symbolic discovery
for method in [PySR, PySINDy, NeuralGuidedBaseline]:
    for (H, dU), traj in dataset.items():
        model = method.fit(traj)               # recover drift f(x), diffusion g(x)
        implied_exponent = kramers_exponent(model)  # analytic/numeric derivation
        results.log(method, H, dU, implied_exponent)

# Phase 3: Empirical LMC mixing time
for (H, dU), potential in potentials.items():
    sampler = LangevinMonteCarlo(potential)
    chains = sampler.run(n_chains=4, n_steps=1e6, seed=range(3))
    empirical_exponent = fit_mixing_time_scaling(chains, barrier=dU)
    results.log('empirical', H, dU, empirical_exponent)

# Phase 4: Compare
agreement = compare_exponents(results['method'], results['empirical'])
report(agreement, ci=0.95, controls=['H=0.5 negative control'])