SKG: A Law-Induced Engine from Inference to Induction

Jian An; Chaoshi Wu; Jinhui Wu

Home/Archive/arXiv

arXiv · 2610.19505

SKG: A Law-Induced Engine from Inference to Induction

Jian An·Chaoshi Wu·Jinhui Wu

View original source ↗Read source document ↗Export metadata ↓

Abstract

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).

Explore related subjects

cs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jian An, Chaoshi Wu, Jinhui Wu. 2026-10-14. SKG: A Law-Induced Engine from Inference to Induction. https://arxiv.org/abs/2610.19505

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Re-Label Method For Data-Centric Machine Learning

In industry deep learning application, our manually labeled data has a certain number of noisy data. To solve this problem and achieve more than 90 score in dev dataset, we present a simple method to find the noisy data and re-label the noisy data by human, given the model predictions as references in human labeling. In this paper, we illustrate our idea for a broad set of deep learning tasks, includes classification, sequence tagging, object detection, sequence generation, click-through rate prediction. The dev dataset evaluation results and human evaluation results verify our idea.

cs.LG↗

arXiv2026

Parallel Layer Normalization for Universal Approximation

Layer Normalization (LN) is usually studied for its role in optimization, but it also acts as a multivariate nonlinearity between affine layers. We investigate how grouping LN units affects the approximation power of neural networks as width grows. In Parallel Layer Normalization networks (PLN-Nets), fixed-dimensional LN units are applied independently to groups between two affine layers. We prove that a shallow LN-Net with one LN unit is non-universal regardless of width, whereas PLN-Nets become universal as the number of parallel groups grows. The mechanism extends to deep networks, Root Mean Square Normalization (RMSNorm) variants, position-wise feed-forward networks, and convolutional networks under suitable grouping structures. Controlled MLP and physics-informed neural network (PINN) experiments support PLN's practical viability, while CNN and ViT experiments show that group count shapes the finite-sample fitting envelope but does not act alone: the grouped axes, replacement site, and label structure lead to distinct fitting and generalization behavior.

cs.LG↗

arXiv2026

Thinking-Free Policy Initialization Makes Distilled Reasoning Models More Effective and Efficient Reasoners

Reinforcement Learning with Verifiable Reward (RLVR) effectively solves complex tasks but demands extremely long context lengths during training, leading to substantial computational costs. While multi-stage training can partially mitigate this, starting with overly short contexts often causes irreversible performance degradation, ultimately failing to reduce overall training compute significantly. In this paper, we introduce Thinking-Free Policy Initialization (TFPI), a simple yet effective adaptation to RLVR that bridges long Chain-of-Thought (CoT) distillation and standard RLVR. TFPI employs a simple ThinkingFree operation, explicitly discarding the thinking content via a direct append, to reduce token usage during inference. Training with ThinkingFree-adapted inputs improves performance and lowers token consumption, even in the original slow-thinking mode. Extensive experiments across various benchmarks have shown that TFPI accelerates RL convergence, achieves a higher performance ceiling, and yields more token-efficient reasoning models without specialized rewards or complex training designs. With TFPI only, we train a 4B model to reach 89.0% accuracy on AIME24 and 65.5% on LiveCodeBench using less than 4K H20 hours.

cs.LG↗