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SKG: A Law-Induced Engine from Inference to Induction

Jian An, Chaoshi Wu, Jinhui Wu

arXiv:2610.19505v1cs.LG

TL;DR

Data-driven learning lacks extrapolation guarantees when test distributions differ from training distributions. SKG makes laws the core of inference and uses data for calibration; its results include task-independent Fisher-based kernel bandwidth and strictly positive group-direction OOD gain under a stated condition.

  • Problem

    Data-driven models have no extrapolation guarantee when test distributions differ from training distributions, despite success in several application areas.

    Source: §1.1 · p. 4

  • Method

    SKG makes laws the core of its inference engine and uses data only for calibration, representing the framework with an octuple mapped to four levels of laws.

    Source: §1.2 · p. 4, §1.4 · p. 5

  • Results

    When LM · d > C + εID(K), group-direction OOD gain is strictly positive, and Fisher information uniquely determines a task-independent kernel bandwidth matrix.

    Source: §3.7 · p. 27, §8.5 · p. 65

  • Takeaways & Limitations

    The residual isolates the dissipative force because the symplectic part cancels exactly, while the geometric force is produced by the state-dependent metric.

    Source: §6.8 · p. 52

    The OOD advantage theorem depends on orthogonal representations; regularization for non-orthogonal representations remains an open problem.

    Source: §11.5.3 · p. 92

Abstract

from arXiv · show

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. We organize symplectic geometry, GENERIC, kernel methods, RKHS, group-equivariant networks, and constrained mechanics into a unified, analyzable, implementable, verifiable framework. 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: 1D damped oscillator and 2D rotation-invariant system; 6 unit tests, 10 experiments (paired t-test, p < 0.01).

1 Introduction

The paper contrasts data-driven inference, which learns correlations without an extrapolation guarantee, with law-induced inference, where laws are given and data provide instances. SKG formalizes law-induced inference and induction as geometric evolution under constraints, with data used to calibrate unknown task components.

1.1 Two Paradigms of Inference

  • Law-induced inference starts from a priori laws and uses data as instances, whereas data-driven methods learn training-distribution correlations that may fail when distributions change.The paper identifies the lack of a systematic, provable law-induced inference engine as a gap.Source: §1.1 · p. 4

1.2 Limitations of Existing Methods

  • Existing methods incorporate laws locally, but SKG instead makes laws the inference engine's core and uses data only for calibration.The paper contrasts this framing with PINNs, Hamiltonian and Lagrangian networks, group-equivariant networks, Neural ODEs, and RKHS methods.Source: §1.2 · p. 4

1.3 Core Idea of This Paper

  • SKG divides inference into law, geometric, and inference levels, encoding a priori system laws as geometric structures that determine evolution.Data calibrate unknown framework components, including the task-dependent part of the energy.Source: §1.3 · p. 5

1.5 Closed Loop from Inference to Induction

  • Inference and learning share a geometric structure, with inference evolving under a metric and energy while learning determines the task energy.The paper presents the two flows as a closed loop with a shared geometric skeleton.Source: §1.5 · p. 6, Table 2 · p. 6

1.6 The Pullback Principle

  • Pullback is the common mechanism in both flows; task-independent metric properties carry across the inference and learning levels and support induction decoupling.The paper describes the state-space and parameter-space metrics as two instances of this mechanism.Source: §1.6 · pp. 6–7

1.7 Why the Symplectic–Dissipative Structure

  • SKG uses symplectic–dissipative structure as an instance because it offers a complete theory, exposes law-induced OOD advantages, and draws on mature mathematical tools.The framework is stated to extend beyond these systems to law structures with a known-part-plus-estimated-part decomposition.Source: §1.7 · p. 7

1.8 Comparison with PINN

  • The paper frames SKG and PINN as different paradigms: SKG is law-driven with data for calibration, while PINN is data-driven with law regularization.The comparison is described as paradigmatic rather than a controlled-variable experiment.Source: §1.8 · p. 7

2 Geometric Foundations

The chapter grounds SKG in GENERIC's reversible–irreversible decomposition, defines its law-based system structure, and specifies its geometric dynamics. It resolves a degeneracy conflict with a symplectic-step projection and explains how state-dependent geometry contributes to the residual.

2.1.1 Core Claim of GENERIC Theory

  • GENERIC decomposes dynamics into reversible and irreversible parts, with degeneracy conditions encoding energy conservation and nondecreasing entropy.In SKG's mechanical instance, the Hamiltonian, entropy, symplectic map, and inverse metric instantiate the GENERIC energy, entropy, reversible operator, and dissipative operator.Source: §2.1.1 · p. 9, §2.1.2 · p. 9, §2.1.3 · p. 9

2.1.4 Violation of the Degeneracy Condition

  • The momentum-only entropy choice S = S(p) conflicts with the first GENERIC degeneracy condition under the standard symplectic structure.The paper identifies this as a structural conflict in dissipative mechanics, where momentum-dependent entropy represents dissipation acting on momentum rather than position.Source: §2.1.4 · p. 10

2.1.5 Resolution: Symplectic-Step Projection

  • The symplectic-step projection removes the component along the entropy gradient, restoring the degeneracy condition without changing the residual because the symplectic term cancels there.The projection is orthogonal, symmetric, and idempotent.Source: §2.1.5 · p. 10

2.2.1 Definition

  • The SKG octuple packages the configuration manifold, metric, symplectic structure, energy and entropy, dissipative generator, invariants, and three projections.Its law-related components correspond to conservation, symmetry, axioms, and constitutive relations; data calibrate G, the dissipative-metric generator.Source: §2.2.1 · pp. 10–11, §2.2.2 · p. 11, §2.2.3 · p. 11

2.2.4 Objective–Task Decomposition of Energy

  • The objective energy is known from physical laws while the task energy is learned from task data, so learning reduces to the task-energy component.This decomposition separates the inference flow from the learning flow.Source: §2.2.4 · pp. 11–12

2.3.1 From First to Second Order

  • The second-order SKG flow includes covariant acceleration, a corrected symplectic force, a dissipative force, and Rayleigh damping; the connection term supplies the geometric force.That force vanishes for a constant metric but is nonzero for a state-dependent metric, where it is part of the curved-geometry dynamics.Source: §2.3.1 · p. 12, §2.3.2 · p. 12, §2.3.3 · p. 12

2.4.1 Geometric Force and Inertial Force

  • The geometric force is a geometric acceleration correction rather than an externally applied force, and its relation to curvature is expressed through geodesic deviation.The chapter relates the growth of geodesic deviation to curvature.Source: §2.4.1 · p. 13, §2.4.2 · p. 13

2.4.3 Role of Geometric Force in the Residual Equation

  • In the residual, the symplectic term cancels but the metric-dependent geometric force remains; it vanishes in the constant-metric case used for the one-dimensional observable formula.The chapter summary covers the constant and state-dependent metric cases and the geometric force's residual role.Source: §2.4.3 · p. 13, §2.5 · p. 14

3 Invariant-Induced Metrics and Intrinsic Metric Reconstruction

This section develops invariant-induced state-space metrics and methods for reconstructing intrinsic metrics from multiple embeddings, including rank and equivariance criteria. It also derives the kernel bandwidth matrix’s Fisher-information basis, task-independence, and error behavior, and discusses related engineering implications.

3.1.1 Problem; 3.1.4 Why State-Dependence Is Required

  • Direct metric parameterization risks coordinate dependence, while state dependence is required to represent differing local physical behavior.The section also motivates state dependence by variation in data density and curvature of invariant spaces.Source: §3.1.1 · p. 14, §3.1.4 · p. 15

3.3.2 Kernel Hessian Gives the Metric

  • At the invariant reference point, the kernel Hessian yields the state-space metric g(q) = J^T Sigma^-1 J; it is positive definite when J has full column rank and otherwise requires regularization.The kernel Hessian is negative definite at its maximum, so taking its negative gives the positive-semidefinite pullback metric.Source: §3.3.2 · p. 17

3.4.2–3.4.3 Regularization and Equivariance

  • Adding ε_G I makes the metric positive definite, but preserves equivariance only for orthogonal representations; non-orthogonal regularization remains an open problem.The stated orthogonal cases include SO(d), SE(3), and the translation group.Source: §3.4.2 · p. 18, §3.4.3 · p. 18

3.5.1–3.5.4 Multiple-Embedding Reconstruction

  • For multiple embeddings of the same invariant space, normalized Jacobians align coordinates, and consistency requires their pullbacks to yield the same state-space metric.The reconstruction framework applies to coordinate representations, alternate functional forms, and different invariant sets, provided they map to the same M-dimensional space.Source: §3.5.1 · p. 18, §3.5.2 · p. 19, §3.5.3 · p. 19

3.5.6 Vectorization and Solvability

  • Unique reconstruction of the intrinsic metric holds if and only if the stacked linear system has rank M(M + 1)/2.Equation-count sufficiency alone is not enough; the actual condition is the rank of the system matrix.Source: §3.5.6 · pp. 20–21, §3.5.5 · p. 20

3.5.12 Engineering Applications

  • For OOD assessment, intrinsic-space Mahalanobis distance locally approximates geodesic distance, with error O(∥R∆∥·∥δ∥^3) when curvature is small and points are close.For state-dependent metrics, the exact distance is the geodesic distance.Source: §3.5.12 · p. 23
  • If two systems have identical intrinsic metrics, that metric transfers across systems; when J = I, inversion can instead be performed in intrinsic space, and the residual uses the covariant metric.The section also states that using the contravariant metric would give an incorrect residual direction.Source: Transfer learning · p. 23, Inverse-free · pp. 23–24

3.6.1 Origin of the Kernel Bandwidth Matrix

  • The kernel bandwidth matrix Σ is not a free parameter: it is uniquely determined by Fisher information for the invariant distribution, rather than by the task.Under the natural parameterization η = Σ^-1, the section identifies Fisher information as the origin of Σ.Source: §3.6.1 · p. 24

3.6.3 Objectivity Theorem

  • Under task-independent invariant, distribution-family, and unsupervised-estimation assumptions, the Fisher metric g∆ = F(θ̂)^-1 depends on data and invariants, not task labels or losses.The distribution family is assumed to follow the invariant-space structure, and θ̂ is estimated by unsupervised maximum likelihood.Source: §3.6.3 · p. 25

3.6.4 Error Decomposition and Condition Number Amplification

  • Metric reconstruction error decomposes into statistical, distribution-family, invariant-selection, and parameterization errors, while statistical amplification is dominated by κ(Σ*)^2.The section notes that for M = 6 and κ = 100, the stated sample-size example requires N ≥ 6 × 10^10, making condition-number reduction more effective than simply increasing sample size.Source: §3.6.4 · p. 26

3.5.12 Engineering Applications; 3.7 Chapter Summary

  • The forward metric is the Jacobian pullback g(q) = J(q)^T g∆ J(q), with εG I regularization for rank deficiency; regularization preserves equivariance for orthogonal, but not non-orthogonal, representations.The effective dimension obeys rank(g) ≤ min(rank(g∆), rank(J)); under nonlinear group actions the joint rank condition need not hold at all states, and reconstruction remains open.Source: Inverse-free · p. 24, §3.7 · p. 27

3.5.6 Vectorization and Solvability; 3.5.7 Consistency Condition; 3.5.8 Uniqueness and Condition Number

  • Reconstruction requires rank(A) = M(M + 1)/2; consistency holds iff vech(g) ∈ Col(A), and full rank yields a unique g∆, while deficient rank yields infinitely many solutions.When the system is inconsistent, the text recommends a least-squares approximate solution.Source: §3.5.7 · p. 21, §3.5.8 · p. 21, §3.7 · p. 27

3.5.9 Underdetermined Case; 3.5.10 Sufficient Number of Embeddings; 3.5.11 Solution Methods

  • When reconstruction is underdetermined, the section presents minimum-norm, ridge, and sparse solutions, requires at least d(d + 1)/2 embeddings, and recommends positive-definite manifold optimization.The embedding-count bound is necessary but not sufficient: embeddings must be diverse enough to span the symmetric-matrix space.Source: §3.5.9 · p. 22, §3.5.10 · p. 22, §3.5.11 · p. 23

4 State-Dependent Intrinsic Metric

This section develops a function-space framework for reconstructing state-dependent intrinsic metrics, connecting RKHS methods to geometry, identifiability, sample complexity, existence, OOD bounds, and transfer. It also covers applications and numerical safeguards, with stated conditions and limitations.

4 State-Dependent Intrinsic Metric

  • The chapter extends constant-case intrinsic metric reconstruction to a complete theory for state-dependent metrics g_∆(∆).The preceding theory assumes g_∆ is a constant matrix; this chapter relaxes that assumption.Source: §4 · p. 28

4.1.2 Necessity of State Dependence

  • State dependence is necessary because invariant distributions vary by state and a constant metric implies vanishing Christoffel symbols, conflicting with generally curved invariant spaces.The paper concludes that the intrinsic metric must depend on ∆; state dependence is a requirement of the Riemannian geometry described.Source: §4.1.1 · p. 28, §4.1.2 · p. 28

4.3.3 Equivariance Inheritance

  • The state-space metric is formed by pulling back g_∆(∆) through the invariant Jacobian, and equivariance of g_∆ yields equivariance of the pulled-back metric.The pullback includes the regularization term ε_G I; the equivariance result is stated in Proposition 4.5.Source: §4.3.3 · p. 30, §4.3.1 · p. 30

4.4.3 Key Difficulties

  • Function reconstruction is an infinite-dimensional problem constrained pointwise at sampled invariant values, leaving behavior between samples to regularization.Parameterization affects convergence and generalization; Cholesky parameterization automatically guarantees positive definiteness.Source: §4.4.3 · p. 31

4.6 Parameterization Choices

  • Metric parameterizations include Cholesky functions, matrix-valued kernel expansions, and diagonal-plus-low-rank functions, each with distinct guarantees or tradeoffs.Cholesky guarantees positive definiteness; kernel expansions have RKHS support but need a correction for positive definiteness, while low-rank methods use O(Mr) parameters but assume approximate low rank. Gaussian expansions are suggested for M ≤ 10 and neural networks for larger M.Source: §4.2.3 · p. 30, §4.4.3 · p. 31, §4.5.3 · p. 32

4.7 Identifiability

  • Pointwise metric values are identifiable when sample coverage and the multiple-embedding rank condition hold; the function is unique under the stated RKHS conditions.The pointwise result applies across supp(∆), while the function-space result uses a strictly convex regularized least-squares problem in the RKHS.Source: §4.7.1 · p. 33, §4.7.2 · p. 34

4.7.3 Sample Complexity

  • The sample-complexity bound becomes rapidly infeasible as invariant dimension grows; for M = 20, s = 2, and ε = 0.1, the passage gives ε^(M/s) = 10^-10 and sample size on the order of 10^10.The authors characterize the theorem as a worst-case bound and recommend structural assumptions for high-dimensional cases.Source: §4.7.3 · p. 34

4.8.1 Existence

  • Under the stated conditions, the state-dependent metric reconstruction optimization problem has a unique solution.The conditions are introduced in Theorem 4.12, but their contents are not included in the supplied passages.Source: §4.8.1 · p. 35

4.9.1 Christoffel Symbols; 4.9.2 Invariant-Space Curvature

  • State dependence of g_∆ enters Christoffel symbols through its derivatives, while invariant-space curvature is derivative-determined and vanishes when g_∆ is constant.When g_∆ is constant, its derivative term in the Christoffel symbols is zero; the symbols then depend only on derivatives of J.Source: §4.9.1 · p. 35, §4.9.2 · p. 36

4.9.3 Curvature and Extrapolation Difficulty

  • The proposed curvature–extrapolation link is heuristic: high-curvature regions tend to have larger extrapolation error, but the relationship remains unproved and open.The observation is based on the Jacobi field equation and is not a rigorous result.Source: §4.9.3 · p. 36

4.10.1 RKHS Extrapolation Bound

  • The OOD extrapolation bound controls error using the RKHS norm and kernel, with state-space error scaled by ∥J(q*)∥₂.A pointwise bound can be computed for each OOD point; a more conservative constant bound applies when J is bounded on the domain.Source: §4.10.1 · p. 36

4.10.2 Geodesic OOD Distance

  • For state-dependent metrics, exact OOD distance uses a geodesic rather than a Euclidean approximation; the error is O(∥R_∆∥·∥δ∥³).Here R_∆ is the invariant-space Riemann curvature tensor and δ is the distance between the points.Source: §4.10.2 · p. 37

4.11.1 Transfer Learning

  • If two systems share the same intrinsic metric function, that function can transfer across systems through their respective Jacobians.The application is to learn g_∆(·) on a source task and use it on a target task.Source: §4.11.1 · p. 37

4.11.2 Inverse-Free; 4.11.3 State-Dependent Metric Learning

  • State-dependent metric learning fits g_∆(·) rather than g_q(·), described as coordinate-independent and transferable; when J = I, g_q = Σ⁻¹ + ε_GI.The framework also cites an RKHS extrapolation bound as an advantage.Source: §4.11.3 · p. 37, §4.11.2 · p. 37

4.12.2 Numerical Stability; 4.13 Chapter Summary

  • Cholesky parameterization ensures positive definiteness with a positive diagonal floor and condition-number monitoring; pointwise identifiability holds, but sample complexity is infeasible in high dimensions.The chapter summary reports pointwise identifiability (Theorem 4.8) and function-space uniqueness (Theorem 4.9).Source: §4.12.2 · p. 38

4.8.1 Existence; 4.8.2 Uniqueness

  • Under standard RKHS conditions, the state-dependent metric reconstruction has an existing, unique solution.The chapter summary states existence and uniqueness under standard RKHS conditions (Theorems 4.12 and 4.13).Source: §4.8.2 · p. 35, §4.8.1 · p. 35, §4.13 · p. 38

5 Three-Layer Projections

SKG uses three projections to handle hard state constraints, restore the symplectic-step degeneracy condition, and preserve energy. Together they support valid constrained evolution and tighten the OOD bound when states are projected onto the admissible space.

5 Three-Layer Projections

  • The three projections address distinct constraints: axiom admissibility, symplectic-step degeneracy, and energy conservation.Their order follows the constraint hierarchy: hard axiom constraints, structural degeneracy, then numerical energy conservation.Source: §5 · p. 38, §5.1 · p. 39

5.2.2 Definition of the Axiom Projection

  • The axiom projection maps a violating state to the nearest point in the admissible state space, along its normal direction.It is posed using geodesic distance and can be solved with Lagrange multipliers; independent constraint gradients yield a unique solution.Source: §5.2.2 · p. 40, §5.2.4 · p. 40, §5.2.3 · p. 40

5.2.5 Tangent Map and Pullback Metric

  • The axiom projection’s tangent map sends state-space tangent vectors to admissible-space tangent vectors and pulls back the metric to that space.For a linear admissible subspace, the tangent map is a constant orthogonal projection matrix.Source: §5.2.5 · p. 41

5.3.4 Restoration of the Degeneracy Condition

  • Projecting the symplectic step onto the orthogonal complement of ∇S restores the GENERIC degeneracy condition and prevents entropy production by that step.When ∇S = 0, the projection is defined as the identity because the degeneracy condition is already satisfied.Source: §5.3.2 · p. 41, §5.3.4 · p. 42, §5.3.5 · p. 42

5.3.6 Symplectic-Step Projection and Residual Independence

  • The symplectic-step projection does not affect the residual, which for an exact model depends on the dissipative term g⁻¹∇S.The residual is independent of changes to the projected symplectic step.Source: §5.3.6 · p. 42

5.4.3 Properties of the Conservation Projection

  • The conservation projection adjusts the state along ∇E to return energy to its initial value, preserving energy to first order and acting idempotently to first order.These properties are stated as approximations up to O(∥∆z∥²).Source: §5.4.2 · p. 43, §5.4.3 · p. 43

5.4.4 Conservation Projection and Symplectic Integrators

  • Used with a symplectic integrator, the conservation projection preserves energy to machine precision while preserving the symplectic structure to higher order.The integrator alone can have O(h^p) energy drift, where h is the step size and p is its order.Source: §5.4.4 · p. 43

5.5.2 Tightening the OOD Bound

  • Projecting an OOD point onto the admissible state space reduces extrapolation distance and tightens the OOD bound.The chapter attributes this effect to the axiom projection; the symplectic-step and conservation projections have no direct effect on the OOD bound.Source: §5.5.2 · p. 44, §5.5.3 · p. 44

6 Residual Theory and Identifiability

This section develops residual theory and identifiability, using trajectory data to invert the metric and isolate the metric-gradient force. It specifies observable components, rank and noise effects, structural identifiability, and regularization and potential–metric coupling.

6.1.3 Exact Cancellation of the Symplectic Part

  • The symplectic step cancels from the residual, which constrains metric components along ∇S; the geometric force does not cancel and can make the metric enter through its derivatives.When the geometric force is zero, the metric enters only through g^-1∇S; otherwise it enters both through that term and through Γ(˙z, ˙z).Source: §6.1.3 · p. 46, §6.1.4 · p. 46

6.2.2 Observable Formula

  • With one degree of freedom, no damping, S = p^2/2, and zero geometric force, the pp metric component is recoverable from the residual and momentum, but the formula fails at p = 0.The assumptions are strict, and p = 0 corresponds to a turning point.Source: §6.2.2 · p. 47, §6.2.3 · p. 48, §6.2.4 · p. 48

6.3.2 Linear System

  • Stacking residual equations across samples uniquely determines the seven identifiable metric components in the four-dimensional case if and only if the system matrix has full rank 7.The sample states must provide sufficient diversity; concentration in a subspace can make the system rank-deficient.Source: §6.3.2 · p. 49, §6.3.3 · p. 49

6.4.2 Noise Propagation Formula

  • Noise in the one-dimensional observable formula has standard deviation σ/|p|, so it is amplified near turning points; a validation reported maximum relative error of 1.48%.The proposed weight |p|^2/(|p|^2 + σ^2) suppresses low-momentum samples.Source: §6.4.2 · p. 49, §6.4.3 · p. 50

6.5.1 Origin of the Problem

  • When S = S(p), g_qq is structurally unidentifiable because it does not appear in the residual; in four dimensions, the identifiable blocks are g_qp and g_pp, totaling seven components.The four-dimensional metric has ten independent components, of which three belong to the unidentifiable qq block.Source: §6.3.1 · p. 48, §6.5.1 · p. 50

6.6.1 Effect of Potential Error

  • Potential error affects the metric estimate through ε_pp, and the weighted least-squares method accounts for observation quality and potential error while regularizing the estimate.The weighting scheme decreases weights when potential error is large; turning points receive low weights because their observations are noisy.Source: §6.6.1 · p. 51, §6.6.2 · p. 51, §6.7.1 · p. 52

6.7.2 Regularization Term

  • The regularization term R(g) preserves the geometric meaning of g, with regularizations that pull toward a prior metric or preserve Fisher objectivity.The supplied passages identify g_0 as the prior metric and Σ_0 as the Fisher prior.Source: §6.7.2 · p. 52

6.8 Chapter Summary

  • The residual is r = ¨z + Γ( ˙z, ˙z) − P⊥ω♯∇H + α ˙z; for an exact model, r = −g∇S because the symplectic part cancels exactly.The geometric force remains through the state dependence of the metric.Source: §6.8 · p. 52
  • The one-dimensional observable g_obs^2(q_i) = −(¨p_i + q_i)/p_i applies only to one dimension, a single direction, no damping, S = p^2/2, and zero geometric force.These assumptions strictly limit the formula’s applicability.Source: §6.8 · p. 52
  • The four-dimensional system r_i = −A_iw has a unique solution iff rank([A_1; ...; A_N]) = 7; noise propagates as σ_g^2(q_i) = σ/|p_i| and amplifies at turning points.The weighting scheme suppresses this noise amplification.Source: §6.8 · p. 52
  • When S = S(p), g_qq is structurally unidentifiable because it does not appear in the residual; the proposed engineering handling fixes g_qq = I.The number of identifiable components is (3n^2 + n)/2.Source: §6.8 · p. 52

Potential–metric coupling.

  • Potential error pollutes the metric estimate through ε_pp, motivating a downweighting scheme.The section also restates the residual equation as the basis for this potential–metric analysis.Source: Potential–metric coupling · p. 52

7 Kernel Parameterization and OOD Error Decomposition

This section develops RKHS kernel parameterization for intrinsic metrics and analyzes generalization and optimization. It decomposes out-of-distribution error into fitting, approximation, and extrapolation components.

7.1.1 Two Choices for Metric Parameterization; 7.1.3 Specific Forms of Kernel Parameterization

  • The adopted Cholesky kernel expansion parameterizes a lower-triangular matrix function and automatically guarantees positive definiteness.The chapter contrasts kernel and MLP metric parameterizations, both of which can represent arbitrary continuous functions.Source: §7.1.1 · p. 53, §7.1.3 · p. 54

7.1.2 Advantages of Kernel Parameterization

  • Kernel parameterization offers representability, explicit embedded regularizers, tighter same-Lipschitz generalization bounds, and a favorable optimization landscape.The regularizers address commutativity, curvature, and analytic anchors; the generalization and optimization advantages are developed in this chapter.Source: §7.1.2 · p. 53

7.2.1 Definition of the Approximation Rate; 7.2.2 Proof of the Approximation Rate

  • For M = 10 and s = 2, the rate is P^-0.2; achieving 10^-1 accuracy requires P ∼ 10^5 centers, illustrating the curse of dimensionality, since larger M gives slower approximation.Source: §7.2.2 · p. 55

7.3.1 Rademacher Complexity; 7.3.2 Generalization Bound Gap; 7.3.3 Computational Complexity Comparison

  • With equal Lipschitz constraints, kernel metrics have tighter generalization bounds than MLPs because they use O(Mr) rather than O(M^2) parameters; the gap vanishes for low-rank MLPs.The chapter also reports that kernel parameterization outperforms MLPs in backpropagation and regularization computation (Table 13).Source: §7.3.2 · p. 56, §7.3.3 · p. 56, Table 13 · p. 56

7.3.4 Optimization Landscape

  • When the kernel rank r exceeds the true metric rank r*, all saddle points have negative Hessian eigenvalues, and randomly perturbed gradient descent converges to a global optimum almost everywhere.This guarantee is not established at r = r*; benign initialization or second-order methods are then needed.Source: §7.3.4 · p. 57

7.4.1 Sources of OOD Error; 7.4.2 OOD Error Decomposition Theorem

  • OOD error decomposes into hypothesis-space approximation, finite-sample fitting, and distance-dependent extrapolation error.The theorem further separates approximation error within the training distribution from OOD extrapolation error.Source: §7.4.2 · p. 57, §7.4.1 · p. 57

7.4.3 Fitting Error; 7.4.4 Approximation Error

  • Fitting error decays as N^-1/2, while kernel approximation error decays as P^-s/M, with smoothness accelerating and dimension slowing approximation.The fitting term arises from finite samples; the approximation term arises from hypothesis-space restriction.Source: §7.2.2 · p. 55, §7.4.3 · p. 58, §7.4.4 · p. 58

7.4.5 Extrapolation Error; 7.4.6 Complete OOD Error Decomposition

  • Extrapolation error increases with OOD distance and tends to R when the kernel similarity tends to zero; model complexity, sample size, and kernel choice trade off the three error sources.Increasing sample size reduces fitting error only; smoother kernels reduce approximation error but increase extrapolation error, while rougher kernels have the opposite trade-off.Source: §7.4.5 · p. 58, §7.4.6 · p. 58

7.4.7 Experimental Validation of the OOD Error

  • E9’s RKHS extrapolation bound held, with a 6 × 10−4 average tightness ratio, but was about 1700 times larger than actual error.The looseness is attributed to conservative estimates of R and κ∆; cross-validation and kernel ridge regression are proposed to tighten them.Source: §7.4.7 · p. 59

7.5 Chapter Summary

  • Intrinsic metrics can be represented by matrix-valued or Cholesky kernel expansions, with the Cholesky form guaranteeing positive definiteness.This chapter establishes kernel parameterization theory and OOD error decomposition.Source: §7.5 · p. 59
  • Kernel expansion approximation error is O(P^(-s/M)); larger s accelerates decay, while larger M slows it through the curse of dimensionality.This rate is stated in Theorem 7.2.Source: §7.5 · p. 59
  • Kernel parameterization has a tighter generalization bound than MLPs; when r > r∗, strict saddle conditions imply gradient descent reaches the global optimum almost everywhere.The Rademacher-complexity gap ∆(N, r) is positive when r < M.Source: §7.5 · p. 59
  • OOD error decomposes into fitting, approximation, and extrapolation errors, with trade-offs among the three components.The chapter summary also reports that E9 validates the RKHS extrapolation bound, which holds but is overly loose.Source: §7.5 · p. 59

8 Structural OOD Advantages

The chapter analyzes whether invariant kernel metrics have a structural OOD advantage over MLP metrics along group directions. Under stated assumptions, the kernel metric’s OOD error is bounded by its ID error, while a sufficient condition yields strictly positive gain.

8.1.2 SKG OOD Bound

  • The OOD error bound combines approximation, fitting, and extrapolation errors, and the extrapolation error increases with distance from the training distribution.The extrapolation distance ∆inv appears in two terms of the bound.Source: §8.1.2 · p. 60, §8.1.1 · p. 60, §8.1.3 · p. 60

8.2.2 Meaning

  • The symplectic contribution ratio is a diagnostic without a theoretical guarantee; large values indicate a more significant SKG OOD advantage.It measures the relative weight of conservative and dissipative components.Source: §8.2.2 · p. 61

8.3.3 Three-Step Proof

  • Under orthogonal representations, the kernel metric’s OOD error does not exceed its ID error, while the MLP metric has a Lipschitz lower bound.The theorem’s strict-gain condition is LM · d > C + εID(K).Source: §8.3.3 · pp. 63–64, §8.5 · p. 65
  • A sufficient gain condition is that (LM − L∗) · d covers both models’ ID errors; the lower-bound condition can hold when LM and d are large and εID is small.The bound uses C, the change in the true metric under the group action.Source: §8.3.3 · pp. 63–64, §8.4.3 · p. 65

8.3.4 Specific Forms for Each Group

  • The theorem applies to translation and rotation groups, and to the rotation part of SE(3), but not to the non-orthogonal scale-group representation.Regularization for non-orthogonal representations remains an open problem.Source: §8.3.4 · p. 64

8.4.2 Experimental Validation

  • In experiments, the lower bound was meaningful for zero OOD points, so its substantive role is a sufficient condition rather than a quantitative constraint there.The bound’s form was validated on 1,000 OOD points with zero violations.Source: §8.4.3 · p. 65, §8.4.2 · p. 65

8.4.3 Meaning of the Lower Bound

  • The upper-bound result is the chapter’s substantive contribution: invariant metrics have OOD error no greater than ID error under the stated equivariance setting.The proof transmits metric equivariance to the OOD error bound.Source: §8.3.3 · pp. 62–63, §8.4.3 · p. 65

8.3.1 Implicit Assumptions

  • The strict-gain result depends on linear group action, orthogonal representation, constant intrinsic metric, and an MLP inverse-Lipschitz condition.Without the inverse-Lipschitz or orthogonal-representation conditions, the theorem’s conclusion is not guaranteed.Source: §8.3.1 · p. 61, §8.3.3 · p. 64, §8.5 · p. 65

9 Learning Flow and Induction Decoupling

This section describes how SKG separates geometric inference from induction, distinguishing task-independent metric determination and law-given energy from task-specific supervised learning of task-energy parameters ϕ. It states the conditions and geometric structure under which induction reduces to estimating those parameters.

9.2.1 Metric Determination: Unsupervised Preprocessing

  • The invariant-space metric is uniquely determined by Fisher information, is task-independent, and is estimated once by maximum likelihood rather than gradient descent.Metric determination is unsupervised preprocessing from invariant data and can occur before a task begins.Source: §9.2.1 · p. 67

9.2.2 Energy Determination: Supervised Induction

  • Task-energy determination is supervised: trajectory residuals fit task-specific Stask parameters by gradient descent, making it the framework’s true induction.Different tasks may yield different task energies, learned separately.Source: §9.2.2 · pp. 67–68

9.3.1 Theorem Statement

  • Under a task-independent metric and known law-given objective energy, the sole object of induction is the task-energy parameters ϕ.Theorem 9.1 reduces learning from all parameters to the parameters of Stask.Source: §9.3.1 · p. 68, §9.3.2 · p. 69

9.3.3 Conditions of the Theorem

  • Decoupling holds only under strict conditions; task-trained metrics or an unclear objective/task-energy boundary break it, while second-order flows require modification.The objective/task boundary is set by physical laws; the theorem’s stated flow form is first-order.Source: §9.3.3 · p. 69

9.4.1 Symmetric Structure

  • Inference and learning share a geometric structure, with state space and parameter space, metrics and Fisher information, and energy and loss gradients corresponding.Geometric evolution in inference corresponds to natural-gradient descent in learning.Source: §9.4.1 · p. 69, Table 15 · p. 69

9.4.2 Task Relevance of FS(ϕ)

  • The task-independent metric-space Fisher information differs from task-relevant parameter-space Fisher information, and neither is learned by gradient descent.The parameter-space Fisher depends on the task-data loss; distinguishing “not learned” from “task-independent” is central to the theorem’s scope.Source: §9.4.2 · pp. 69–70

9.5.2 Pullback Structure of the Parameter-Space Metric

  • The learning flow uses a pullback parameter metric built from the state metric and state-to-parameter Jacobian; its loss is residual-fitting error on trajectory data.The described estimation method computes the objective state metric from domain data and obtains the Jacobian by automatic differentiation.Source: §9.5.2 · p. 70, §9.5.1 · p. 70, §9.5.5 · p. 72

9.5.3 Relation Between the Pullback Metric and the Fisher Information

  • The parameter-space pullback metric equals Fisher information when the hidden state is sufficient, loss is negative log-likelihood, and the state metric is Fisher information.Otherwise, the pullback metric is bounded above by Fisher information in Loewner order, and the learning flow uses empirical Fisher information.Source: §9.5.3 · p. 71

9.5.5 Estimation of the Pullback Metric and the Fisher Information

  • When the hidden state is a sufficient statistic, the pullback metric applies; otherwise, the empirical Fisher is task-relevant and must be re-estimated for each task.The method choice depends on task structure: structured tasks can use the pullback metric, while unstructured tasks use the empirical-Fisher fallback.Source: §9.5.5 · p. 72

9.5.6 Convergence

  • With a positive-definite, Lipschitz metric, learning converges at the ordinary gradient-descent rate with a smaller constant; in the sufficient-statistic case, it has the classical natural-gradient rate.For non-sufficient tasks, the rate depends on the approximation quality of the empirical Fisher.Source: §9.5.6 · p. 72

9.7.2 Degree of Learning Simplification

  • SKG reduces learning cost from O(d^2 + P) to O(Pϕ) by determining the metric unsupervised and learning only task-energy parameters.The simplification is significant when d is large.Source: §9.7.2 · p. 73

9.7.3 Advantage of Transferability

  • Because gq is shared across tasks in a domain, SKG transfer does not require domain adaptation, unlike standard ML.Standard ML domain adaptation requires target-task data and lacks theoretical guarantees.Source: §9.7.3 · p. 73, §9.6.3 · p. 73

9.8.1 Analogy with Bayesian Inference; 9.8.2 Analogy with Statistical Physics

  • The framework interprets the metric as a Bayesian prior and inertial mass, while energy corresponds to likelihood and Hamiltonian, respectively.The paper describes the learning flow as a geometric realization of Bayesian inference and as statistical-physics dynamics.Source: §9.8.1 · p. 74, §9.8.2 · p. 74

Learning-decoupling theorem.

  • Under three conditions—unsupervised metric determination, energy decomposition, and a determined inference flow—the sole object of induction is the task-energy parameters ϕ.The metric gq is determined objectively from Fisher information, while task data determine Stask.Source: Learning-decoupling theorem · p. 75, §9.10 · p. 75

9.6.3 Separation of the Two Stages

  • Training has two separated stages: offline, task-independent computation determines gq and Sobj; online, task-relevant training learns the parameters ϕ of Stask.Stage 1 is completed once, while Stage 2 is learned per task with few parameters.Source: §9.6.3 · p. 73, Learning-decoupling theorem · p. 75

10 Experimental Validation

Experiments on a damped oscillator and a rotation-invariant system assess SKG’s theoretical conclusions, including reconstruction, OOD performance, bounds, equivariance, and learning-flow decoupling. The validation distinguishes numerical implementation checks from empirical tests and notes bounds and theoretical results that remain limited or unvalidated.

10.2 Unit Tests (Rigorous Conclusions)

  • Unit tests found the 1D observable exact at machine precision, confirmed unique solvability of 7 identifiable components, and verified that the residual is independent of gqq.The T2 rank condition was tested using four momentum points; three different gqq choices produced identical residuals in T3.Source: §10.2.1 · p. 77, §10.2.2 · p. 77, §10.2.3 · p. 77
  • The T5 noise-propagation maximum relative error was 1.48%; T6’s numerical kernel-Hessian error was 5.90 × 10^-9.T5’s noise propagation agreed with theoretical predictions; T6 confirmed that kernel second-order cross-derivatives give the metric.Source: §10.2.5 · p. 78, §10.2.6 · p. 78, Table 26

10.3 Core Experiments

  • SKG’s OOD error was 21% of the black-box error in E1 and 24% of the MLP error in E2; both comparisons were statistically significant.E1 reported t = 8.43, p = 0.0003; E2 reported t = 4.71, p = 0.0061, with parameter counts aligned within 3.4%.Source: §10.3.1 · p. 79, §10.3.2 · p. 79
  • All 5 seeds satisfied the RKHS extrapolation bound in E3, with tightness ratio 0.49; joint estimation’s OOD error in E4 was 21% of independent estimation’s.The E3 bound was about 2 times the actual error; E4 used neural networks for the potential in both approaches.Source: §10.3.3 · p. 80, §10.3.4 · p. 80
  • In group-direction shifts, K’s OOD/ID error ratio was 1.05–1.15, while the M/K OOD-error ratio rose from 101 at 1000 steps to 1589 at 10000 steps.The upper-bound prediction for K was strongly validated, but M’s lower bound was trivial: it held for all seeds, yet was negative for all OOD points.Source: §10.3.5 · p. 81
  • Equivariance errors were < 10^-6 at all tested angles and seeds; state-dependent reconstruction’s OOD error was 26% of constant reconstruction’s and 64% of the unstructured MLP’s.The learned g∆(r) had < 1% interpolation error and < 20% extrapolation error, supporting the explanation for E7’s OOD advantage.Source: §10.3.6 · p. 82, §10.3.7 · p. 82, §10.3.8 · p. 82
  • All 3 seeds in E9 had OOD errors within the state-dependent extrapolation bound, but its average tightness ratio was 6×10^-4.The bound was about 1700 times larger than the actual error, so it held but was overly loose.Source: §10.3.9 · p. 83
  • In E10, SKG’s learning-flow OOD MSE was significantly lower than the standard MLP’s (paired t = −6.581, p = 0.0028), validating learning of task-energy parameters alone when the metric is fixed.The SKG flow learned one task-energy parameter, whereas the MLP learned the full acceleration function with 1185 parameters.Source: §10.3.10 · pp. 83–84

10.3.10 E10: Learning-Flow Decoupling Validation (Theorem 9.1)

  • E10 reports significantly lower OOD MSE for the 1-parameter SKG flow than the 1185-parameter MLP (p = 0.0028), with 29.4% lower OOD error.The reported parameter reduction is a factor of 1185. The experiment supports the directional content of Theorem 9.1, but not its quantitative prediction of absolute values.Source: §10.3.10 · p. 84

10.5.1 Rigorous Conclusions (T1–T6)

  • Six unit tests validate rigorous conclusions at machine precision (< 10^-9), while 10 core experiments provide empirical validation with multi-seed paired t-tests (p < 0.01).The empirical tests use data generated from physical systems and ground truth obtained by independent measurement.Source: §10.5.2 · p. 85, §10.5.1 · p. 85, §10.6 · p. 86

10.5.3 Overly Loose Bound (E9)

  • The E9 OOD extrapolation bound holds for all seeds, but its tightness ratio is 6 × 10^-4—about 1700 times the actual error.The passage says practical use requires improving estimates of R and κ∆.Source: §10.5.3 · p. 85

10.5.4 Triviality of the Lower Bound (E5)

  • E5’s lower bound holds for all seeds with zero violations, but no lower bounds are meaningful, so it provides no quantitative constraint.The passage identifies the upper-bound part of Theorem 8.15 as the substantive contribution.Source: §10.5.4 · p. 85

10.5.5 Unvalidated Core Theorems

  • Intrinsic metric reconstruction, state-dependent intrinsic-metric sample complexity and convergence, three-layer projections, and symplectic contribution ratio ρH lack experimental validation.Convergence of the learning flow and time-varying Fisher are also unvalidated; the passage distinguishes this from the theorems’ mathematical correctness.Source: §10.5.5 · pp. 85–86

10.6 Chapter Summary

  • The chapter reports 16 passed experiments: 6 unit tests validating rigorous conclusions at machine precision and 10 core experiments validating assumption-dependent and empirical conclusions with multi-seed paired t-test p < 0.01.E5’s upper-bound part is strongly validated while its lower-bound part is trivial; E9’s bound holds but is overly loose, and E10 validates the learning-decoupling theorem (parameter count 1 vs. 1153).Source: §10.6 · p. 86

11 Discussion and Conclusion

SKG centers inference on law-constrained geometric evolution and induction on determining task energy, for systems with symmetry, constraints, and conservation laws. The discussion consolidates the framework’s contributions, guarantees, assumptions, and scope while identifying unresolved theoretical and implementation questions.

11.1.1 Holistic Nature of the Framework

  • SKG makes laws the inference engine and data calibration, formalizing an octuple that applies when symmetry, constraints, and known/estimated law decomposition are present.The symplectic–dissipative system is an instance, not the framework’s boundary.Source: §11.1.1 · p. 86, §11.1.2 · p. 87, §11.1.3 · p. 87

11.2.2 Metric Construction (Chapter 3)

  • Metric construction results establish the kernel-Hessian metric, rank-based solvability, embedding consistency, uniqueness, Fisher objectivity, and a four-term reconstruction-error decomposition.The conclusions include the rank condition rank(A) = M(M + 1)/2 and task-independent g_∆ = F(ˆθ)−1.Source: §11.2.2 · p. 88

11.2.3 State-Dependent Intrinsic Metric (Chapter 4)

  • For state-dependent metrics, the chapter establishes pointwise and RKHS function-space identifiability, sample-complexity and solution guarantees, and transfer across systems with the same metric function.Three-layer projections handle constraints; the conservation projection reduces extrapolation distance.Source: §11.2.3 · p. 88, §11.2.4 · p. 88

11.2.5 Residual Theory (Chapter 6)

  • Residual theory cancels the symplectic step and gives observability, a 4D rank condition, and noise propagation, while g_qq is structurally unidentifiable; kernel parameterization bounds generalization and decomposes OOD error.The residual does not contain g_qq when S = S(p).Source: §11.2.5 · pp. 88–89, §11.2.6 · p. 89

11.2.7 Structural OOD Advantages (Chapter 8)

  • OOD gain in group directions is strictly positive under its stated condition, and learning decoupling leaves only task-energy parameters to induce under three conditions.E1–E10 passed with multi-seed paired t-test p < 0.01; E5 reported K’s OOD/ID ratio as 1.05–1.15 and M’s as 114–1675.Source: §11.2.7 · p. 89, §11.2.8 · p. 89, §11.2.9 · p. 89

11.3 Assumption-Dependent Conclusions

  • These guarantees are conditional: metric reconstruction assumes a constant metric, group-direction OOD gain assumes orthogonal representations, and learning decoupling requires a clear objective/task-energy boundary.The chapter also identifies high-dimensional sample complexity as infeasible without structural assumptions.Source: §11.3.1 · p. 90, §11.3.2 · p. 90, §11.3.3 · p. 90

11.4 Heuristic Observations

  • Heuristic diagnostics link curvature to extrapolation difficulty, symplectic contribution ratio to OOD advantage, and fitting, approximation, and extrapolation errors to trade-offs, without theoretical guarantees.Curvature’s precise relationship to extrapolation error remains open.Source: §11.4.1 · p. 90, §11.4.2 · p. 91, §11.4.3 · p. 91

11.5 Generalization to General Law Structures

  • SKG generalizes to law structures with known and estimated parts by replacing conservative, dissipative, symmetry, or constraint components, but orthogonality, energy decomposition, and dimensionality limit generalization.The generalization paths retain the skeleton of invariant and metric construction, pullback, inference or control, and inverse transformation.Source: §11.5 · p. 91, §11.5.2 · p. 91, §11.5.3 · p. 92

11.6.1 Symmetry Groups and Representations

  • Open symmetry questions include trade-offs among three regularization repairs, reconstruction under nonlinear group actions, and combining embeddings with invariant spaces of different dimensions.No unified optimal repair is established; nonlinear actions may not satisfy the joint rank condition at every state.Source: §11.6.1 · p. 92

11.6.2 Metric Construction and Reconstruction

  • Metric reconstruction remains open on optimal parameterization, convergence-rate validation, time-varying Fisher estimation, non-sufficient statistics, and high-dimensional kernels.The stated convergence rate N^-s/(2s+M) is difficult to validate stably in low-dimensional systems; sparse kernels, random features, and block kernels are possible high-dimensional paths.Source: §11.6.2 · p. 92

11.6.3 Learning Flow

  • Learning-flow research remains needed on global convergence, convergence rates, time-varying Fisher estimation, RKHS-parameterized task energy, and multi-task transfer bounds.The multi-task setting shares metric g_q while each task has its own S_task.Source: §11.6.3 · p. 93

11.6.4 Geometry and OOD

  • Geometry and OOD questions include rigorous curvature-to-extrapolation bounds, guarantees for non-symmetric shifts, and updating metrics online without losing OOD bounds.The invariant method is stated to offer no advantage for noise, occlusion, or adversarial shifts.Source: §11.6.4 · p. 93

11.7.1 Core Contributions

  • SKG formalizes law-induced reasoning and combines Jacobian-based metric construction, Fisher-determined kernel bandwidth, constraint projections, residual theory, and extrapolation and OOD guarantees.The learning-decoupling theorem identifies task energy as the sole object of induction.Source: §11.7.1 · p. 93
  • Pullback of metrics links invariant, state, and parameter spaces; SKG therefore treats inference and learning as instances of one geometric mechanism, with induction determining task energy.The final statement describes laws as a priori and data as providing instances.Source: §11.7.1 · p. 93, §11.7.4 · p. 94

11.7.2 Boundary of the Framework

  • SKG applies to inference tasks with law structure, symmetry, constraints, and conservation laws, but not general-purpose data fitting without those structures.The paper explicitly excludes pure data fitting with no symmetry or constraints.Source: §11.7.2 · p. 94

11.7.3 Future Directions

  • Future directions include stronger theory, a PyTorch/JAX library and optimizer comparisons, real-system validation, intermediate-dimensional extensions, and Transformer integration.Proposed applications include robots, molecular dynamics, and language models; comparisons include PINN and HNN/LNN.Source: §11.7.3 · p. 94

A Construction of the Vectorization Matrix A(k)

The appendix constructs the vectorization matrix used to test whether a state-space metric is consistent with an intrinsic metric. It also outlines the consistency-check procedure and supplements the framework with connection and curvature theory for state-dependent metrics.

A.1 Origin of the Problem; A.4 Construction of the Elements of A(k)

  • A(k) maps the upper-triangular components of the intrinsic metric to those of the state-space metric, with entries determined by the invariant Jacobian J(k).The construction supplies the matrix used in the vectorized pullback equation.Source: §A.1 · p. 94, §A.4 · p. 95

A.5 Verification of the Construction

  • Proposition A.2 verifies that the constructed A(k) reproduces the componentwise vectorized pullback equation.The verification compares the pullback components with the coefficients in the definition of A(k).Source: §A.5 · p. 95

A.6 Vectorization Convention

  • Under the standard vech convention, off-diagonal coefficients are JaiJbj + JajJbi, while diagonal coefficients are JaiJbi.Off-diagonal intrinsic-metric entries contribute two terms to the pullback; diagonal entries contribute one.Source: §A.6 · p. 96

B.1 Algorithm Description

  • The consistency check stacks the matrices A(k) and observations vech(g), solves by least squares, and judges consistency from the residual norm.The algorithm uses the Moore–Penrose pseudoinverse for the least-squares solution.Source: §B.1 · p. 97

B.2 Handling Inconsistency

  • Large residuals can reflect inaccurate Jacobians, noisy metric estimates, or too few embeddings, with residual patterns helping distinguish these sources.The appendix suggests improving precision or the model, using more samples or weighted least squares, or adding embeddings, respectively.Source: §B.2 · p. 97

B.3 Numerical Stability

  • A large condition number κ(A) makes pseudoinverse computation unstable, while SVD truncation avoids amplification from small singular values.The stated truncation threshold is εthresh = 10^-10.Source: §B.3 · pp. 97–98

C.1 Why Connection and Curvature Are Needed

  • Connection and curvature are unnecessary for the first-order dissipative flow but enter second-order dynamics and state-dependent metrics.The connection contributes the geometric force Γ( ˙z, ˙z) to the second-order flow and residual equation.Source: §C.1 · p. 98

C.2.3 Christoffel Symbols in SKG

  • In SKG, Christoffel symbols depend on derivatives of the embedding Jacobian and kernel bandwidth, making the geometric force explicit.The Jacobian derivative is a second derivative of the embedding; bandwidth derivatives encode kernel-bandwidth state dependence.Source: §C.2.3 · p. 99

C.5 Computing Connection and Curvature from the q-Space Metric

This section gives steps for computing connection and curvature from the q-space metric, though the supplied passages only identify parts of that procedure.

C.5.1 Computation Steps

  • The computation includes a step to compute derivatives of the q-space metric.The passage labels this as Step 2.Source: §C.5.1 · p. 101
  • A Cholesky decomposition or linear solve is used in one computation step.The supplied passage does not identify the quantity being solved for.Source: §C.5.1 · p. 101
  • The procedure proceeds to substitute into the Christoffel-symbol definition and compute the Riemann curvature tensor.The passage identifies these as subsequent computation steps.Source: §C.5.1 · p. 101

C.5.2 Computational Complexity

The appendix notes that curvature computation becomes expensive with dimension and relates curvature to out-of-distribution extrapolation error. It gives a heuristic bound involving maximum curvature norm and OOD distance.

C.5.2 Computational Complexity

  • The curvature tensor has O(d^4) components, making curvature computation expensive in high dimensions.The appendix gives O(d^4) = 10^4 for d = 10 and 6 × 10^6 for d = 50.Source: §C.5.2 · p. 101, Table 47 · p. 101

C.6.1 Observation

  • Extrapolation error tends to be larger in regions of high curvature, an observation motivated by the Jacobi field equation.The growth rate of the Jacobi field solution is controlled by the curvature tensor norm.Source: §C.6.1 · p. 101

C.6.2 Heuristic Bound

  • The heuristic bound is expressed using maximum curvature norm, OOD distance, and a constant.The passage identifies these quantities but does not provide the bound's displayed expression.Source: §C.6.2 · p. 101

C.6.3 Degree of Rigor

The curvature-related observation is heuristic, not a theorem or assumption. Its scope is limited by local high-curvature behavior, challenges in bounding curvature, and the experiment's validation target.

C.6.3 Degree of Rigor

  • The curvature-related observation is heuristic, not a theorem or an assumption.The section explicitly characterizes it as an observation and gives reasons for that status.Source: §C.6.3 · p. 102
  • Jacobi-field exponential growth holds only locally in high-curvature regions, while global extrapolation error also depends on geodesic length, conjugate-point distribution, and initial conditions.Source: §C.6.3 · p. 102
  • A lower bound on the curvature tensor is difficult to obtain on general manifolds, and experiment E9 validates the RKHS extrapolation bound rather than the curvature–OOD relation.Source: §C.6.3 · p. 102

C.6.4 Engineering Meaning

Curvature may be used as an OOD diagnostic, but it is not a theoretical guarantee. The SKG OOD guarantee instead comes from the RKHS extrapolation bound and is independent of the appendix's heuristic observation.

C.6.4 Engineering Meaning

  • If computable, the curvature norm ∥R∥ can serve as an OOD-monitoring diagnostic, but not as a theoretical guarantee.Its precise relationship to extrapolation error remains an open problem.Source: §C.6.4 · p. 102
  • The SKG OOD guarantee comes from the RKHS extrapolation bound (Theorem 4.15), not the appendix's heuristic observation.Source: §C.6.4 · p. 102

C.7.1 Constant Metric

The appendix contrasts a constant metric, where geometric effects vanish, with a state-dependent kernel bandwidth, where they are nonzero.

C.7.1 Constant Metric

  • With J = I and constant Σ, Γ = 0 and R = 0, so the geometric force vanishes.Source: §C.7.1 · p. 102

C.7.2 Linear Embedding, State-Dependent Kernel Bandwidth

  • With constant J and state-dependent Σ(q), Γ and R are nonzero, and the geometric force is nonzero.Source: §C.7.2 · p. 102

C.8 Relation to Existing Literature

The appendix presents integration into law-induced inference as SKG's distinctive contribution to OOD analysis, while treating curvature–extrapolation links as open and heuristic rather than guaranteed.

C.8 Relation to Existing Literature

  • SKG's stated distinctive contribution is integrating existing tools into a law-induced inference framework as diagnostic quantities for OOD analysis.Source: §C.8 · p. 102

C.9 Open Problems

  • Open problems are establishing the precise relationship between ∥R∥ and extrapolation error and efficiently computing curvature norms in high-dimensional systems.Suggested paths for computation include sparsification, low-rank approximation, and random projection.Source: §C.9 · p. 102

C.10 Summary

  • The appendix defines Christoffel symbols from metric derivatives and Riemann curvature from connection derivatives, both computed through higher-order kernel derivatives and the pullback formula.It characterizes curvature's relationship to extrapolation error as heuristic; curvature may aid OOD monitoring but is not a theoretical guarantee.Source: §C.10 · p. 103
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