SKG: A Law-Induced Engine from Inference to Induction

This paper proposes the SKG (Symplectic-Kernel Gradient) reasoning engine for law-induced inference and induction.

Laws are a priori; data provide instances.

Inference is geometric evolution under law constraints.

Framework Organization

We organize symplectic geometry, GENERIC, kernel methods, RKHS, group-equivariant networks, and constrained mechanics into a unified, analyzable, implementable, verifiable framework.

SKG Octuple

We propose the SKG octuple S = (M, g, omega, H, S, G, Delta, Pi), whose four components correspond to four levels of laws: conservation laws, constitutive relations, symmetry groups, and axiom systems.

Main Results

1. Invariants enter metric construction via the Jacobian: g = J^T Sigma^-1 J + epsilon_G I. For constant cases, we establish inverse reconstruction theory: solvability rank condition (Thm 3.9), consistency criterion (Thm 3.11), and uniqueness theorem (Thm 3.14).
2. The kernel bandwidth matrix is uniquely determined by Fisher information and satisfies task-independence (Thm 3.19).
3. The metric reconstruction error decomposes into four terms (Thm 3.21), with squared condition number kappa(Sigma^*)^2 as the dominant amplification factor (Thm 3.22).
4. For state-dependent cases, we establish function-space reconstruction theory on matrix-valued RKHS (Thms 4.8, 4.9).
5. Three-layer projections handle axiom, symplectic-step, and conservation constraints.
6. The residual equation yields the observable formula, four-dimensional rank condition, and noise propagation, while g_qq is structurally unidentifiable.
7. Kernel parameterization yields the RKHS extrapolation bound (Thm 4.15) and strict OOD gain theorem in group directions (Thm 8.15).
8. The learning-decoupling theorem (Thm 9.1) proves the sole object of induction is the parameters of the task energy.

Validation

Validation: 1D damped oscillator and 2D rotation-invariant system; 6 unit tests, 10 experiments (paired t-test, p < 0.01).