Neuro-symbolic compliance frameworks integrating SMT solvers will reduce coalition-based deviations in decentralized green hydrogen markets by enforcing logical consistency in multi-agent bidding strategies, where the exponent-range precision barrier (validated in FP32-BF16 LMC transitions) constrains the scalability of equilibrium computations in low-precision hardware deployments.

In a decentralized multi-agent green hydrogen market simulation (N ≥ 20 agents, repeated double-auction or Vickrey-style bidding over ≥500 rounds), embedding an SMT-solver-based symbolic compliance layer (e.g., Z3/CVC5 encoding of market rules, capacity constraints, and non-collusion invariants) into each agent's action-selection pipeline will reduce the frequency of detected coalition-based bidding deviations (measured via a collusion-detection statistic, e.g., bid-correlation clustering or mechanism-design regret) by ≥30% relative to a purely neural (RL/LLM-policy) baseline with no symbolic layer, at equivalent or better social welfare. Separately and independently, when agent policy networks and SMT solver floating-point encodings are executed in BF16 vs FP32, the reduced exponent-range/mantissa precision of BF16 induces measurable equilibrium-computation divergence (Nash-gap or bid-consistency error) that grows superlinearly with agent count N, establishing a scalability ceiling below which BF16 deployment is safe. These are two coupled but separable claims that must each be falsifiable independently.

Design: 2×2×5 factorial: {Neuro-symbolic vs Neural-only baseline} × {FP32 vs BF16} × {N ∈ 5,10,20,50,100}, each cell run for 20 random seeds, 500 auction rounds per run.

Independent variables: compliance framework (symbolic-augmented vs none), numerical precision, agent count.

Dependent variables: (a) coalition-deviation rate (detector-flagged incidents per 1000 rounds), (b) social welfare ratio vs first-best, (c) Nash-gap / bid-inconsistency error, (d) wall-clock time per round, (e) SMT solver timeout rate.

Controls: identical agent initialization seeds across conditions; identical market clearing mechanism; ablation with symbolic layer present but disabled (logging-only) to isolate enforcement effect from monitoring effect.

# Pseudocode: Neuro-symbolic bidding agent with SMT compliance layer

class NeuroSymbolicAgent:
    def __init__(self, policy_net, smt_constraints, precision='fp32'):
        self.policy = policy_net  # small transformer/MLP, cast to precision
        self.constraints = smt_constraints  # Z3 formula templates
        self.precision = precision

    def propose_bid(self, market_state, history):
        with autocast(self.precision):
            raw_bid = self.policy(market_state)  # neural proposal
        return raw_bid

    def compliance_filter(self, raw_bid, coalition_window):
        solver = z3.Solver()
        solver.add(self.constraints.capacity(raw_bid))
        solver.add(self.constraints.rationality(raw_bid))
        solver.add(self.constraints.anti_collusion(raw_bid, coalition_window, tau=TAU))
        if solver.check() == z3.sat:
            return raw_bid
        else:
            return self.project_to_feasible(raw_bid, solver)  # repair via optimize

class Market:
    def run_round(self, agents):
        bids = [a.compliance_filter(a.propose_bid(state, hist), window)
                for a in agents]
        clear_price, allocations = self.clear(bids)
        deviation_flag = self.detector.check_collusion(bids, planted_coalitions)
        nash_gap = self.compute_equilibrium_gap(bids, precision_mode)
        log(deviation_flag, nash_gap, clear_price, latency)

# Outer loop: factorial sweep over {framework, precision, N}, 20 seeds each
for framework in [symbolic, neural_only]:
  for precision in [fp32, bf16]:
    for N in [5,10,20,50,100]:
      for seed in range(20):
        run_market_simulation(framework, precision, N, seed, rounds=500)