SKG: A Law-Induced Engine from Inference to Induction

The construction of state-space metrics from invariants via Jacobian pullback is geometrically elegant, and the error analysis for intrinsic metric reconstruction is concrete. Theorem 3.22 correctly identifies the squared condition number $\kappa(\Sigma^*)^2$ as the dominant amplifier of statistical error, and the four-term decomposition in Theorem 3.21 follows standard perturbation arguments. The philosophical distinction between objective physical law and task-specific calibration is a valuable corrective to the default everything-from-data mindset.

A central internal tension weakens the framework. Section 2.2.3 claims that data are used only to calibrate $G$, the generator of the dissipative metric, calling it the 'only part to be estimated.' Yet the later learning-decoupling narrative asserts that the metric $g_q$ is completely objective and determined by Fisher information, leaving only $S_{\text{task}}$ to be learned. It is never clarified how $G$ relates to $g_q$, or whether the metric itself is partly learned. Moreover, the symplectic-step projection $P_\perp = I - \nabla S \nabla S^\top / \|\nabla S\|^2$ depends on the full entropy gradient $\nabla S = \nabla S_{\text{obj}} + \nabla S_{\text{task}}$, which includes the unknown task component during learning, creating a circular dependency that the paper does not resolve. The experiments are far too narrow to validate any of the grand OOD or transfer claims.

The comparison with PINNs and related methods (Table 3) is philosophical rather than empirical. The authors rightly note that PINNs treat laws as soft regularization, but their claim that SKG 'mathematically forbids' law violation ignores the fact that any error in the learned $G$ or the estimated Fisher metric produces violations just as surely as a PINN loss term. The RKHS OOD bound (Theorem 4.15) and the strict OOD gain theorem (Theorem 8.15) are interesting, but they are derived under the strong assumption that the invariant space and its geometry are perfectly known—an assumption PINNs do not require. The cited 'experimental validation' on toy physical systems does not include any high-dimensional or real-world OOD test, so the bounds remain unverified outside of hand-designed settings.

Reproduction is effectively impossible from the manuscript alone. No code repository is mentioned, no dataset links are provided, and no implementation details—network architectures for $S_{\text{task}}$, optimizers, learning rates, Fisher estimation procedures, or how the projections are enforced numerically—are disclosed. The paper describes '6 rigorous unit tests (machine precision)' and '10 core experiments' without defining the metrics, baselines, or even the exact parameterization of $S_{\text{task}}$. The absence of hyperparameters and experimental detail blocks independent reproduction entirely.