1. This result extends your Bayesian optimization and machine learning surrogate models by providing a rigorous mathematical framework to handle path-dependent, risk-sensitive control problems under stochastic uncertainty.

The truncated discovery text asserts: "Existing Bayesian optimization (BO) and ML-surrogate-based control frameworks can be rigorously extended to path-dependent, risk-sensitive control problems under stochastic uncertainty via a formal mathematical framework." Restated as a falsifiable claim: There exists a provably convergent, computationally tractable extension of BO/surrogate-model-based control — formalized as a risk-sensitive (e.g., entropic/CVaR-regularized) stochastic control problem over path-dependent (non-Markovian) state spaces — such that (a) the surrogate-augmented policy converges to within ε of the true risk-sensitive optimal value function under stated regularity conditions, and (b) this extension yields measurably better risk-adjusted performance (e.g., ≥10% reduction in CVaR@95 of cumulative cost, or ≥15% reduction in regret) than a Markovian/risk-neutral BO baseline on a benchmark suite of path-dependent control tasks (e.g., optimal execution, hedging with transaction costs, epidemic/queueing control). Note: the "TARGET HIERARCHY" biomedical content in the supplied context is NOT part of this discovery and is excluded from the falsifiable claim; it is retained below only as an architectural analogy where explicitly marked.

Minimum viable test (MVT): implement the risk-sensitive path-dependent BO framework on 2 canonical benchmarks — (a) optimal execution with price impact and CVaR objective (Almgren-Chriss extension), (b) American-style option hedging under transaction costs with entropic risk. Compare against 2 baselines: risk-neutral BO, and dynamic programming with discretized state (where tractable at small scale). Run 30 random seeds per method per task; report mean±CI on value-function gap, CVaR@95, and wall-clock time. This constitutes the minimum bar before scaling to the full 8+ task benchmark suite.

# Pseudocode: Risk-Sensitive Path-Dependent BO Control

class PathDependentSurrogate:
    def __init__(self, kernel="path-attention"):
        self.model = GPyTorch.ExactGP(kernel=PathAttentionKernel())
        # or: LSTM/Transformer encoder -> GP head

    def posterior(self, path_history, action):
        phi = self.encode_path(path_history)  # sufficient statistic
        return self.model(phi, action)  # mean, variance

def risk_sensitive_acquisition(surrogate, path, candidate_actions, risk_measure="entropic", theta=1.0):
    scores = []
    for a in candidate_actions:
        mu, sigma2 = surrogate.posterior(path, a)
        if risk_measure == "entropic":
            score = mu + (theta/2) * sigma2  # risk-adjusted UCB-type acquisition
        elif risk_measure == "cvar":
            score = cvar_estimate(mu, sigma2, alpha=0.95)
        scores.append(score)
    return candidate_actions[argmin(scores)]

def outer_loop(env, surrogate, horizon, n_bo_iters):
    for t in range(n_bo_iters):
        path = env.reset_and_replay_history()
        for step in range(horizon):
            a_t = risk_sensitive_acquisition(surrogate, path, env.action_space)
            next_state, cost, path = env.step(a_t)  # path updated with new state
        surrogate.update(path, cumulative_risk_adjusted_cost(path))
    return surrogate.best_policy()

# Theoretical component (separate, symbolic/proof track):
# 1. Define V(t, path) = inf_a rho_t[ C(path, a) + V(t+1, path U {s_{t+1}}) ]
# 2. Prove DPP holds under filtration F_t via BSDE representation:
#    dY_t = -f(t, Y_t, Z_t) dt + Z_t dW_t,  Y_T = xi (path-dependent terminal cost)
# 3. Bound |V_surrogate - V_true| <= L_rho * (surrogate posterior contraction rate)