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  "note": "network output a_hat enters F(z;a_hat,u)=z^3+a_hat*z-u nonlinearly; breaking network weights changes outputs on every seed (RULE 0.1 self-check)"
 },
 "limitations": "Synthetic track generated exactly from the idea's equation class (z = root(z^3+a(x)z-u), a hidden, x noisy sigma=0.15, z noiseless). The advantage is the physics inductive bias: idea learns only a(x); baseline must learn z(x,u) directly with the same budget and grid. No observation noise on z; single architecture family; CPU full-batch Adam."
}