arXiv subjects

Ruifeng Cao

Publications and source records attributed to Ruifeng Cao.

3 recordsLinked to original sources

arXiv2026

Ordered Ruzsa-Szemeredi Numbers at Matching Size Two

Bondy and Szwarcfiter defined $\mathrm{ex}^*(n,F)$ as the largest number of edges in an $n$-vertex graph whose edge set partitions into induced copies of $F$; for $F=2K_2$ the deficiency $\binom{n}{2}-\mathrm{ex}^*(n,2K_2)$ is $Θ(n^{3/2})$. We study the ordered relaxation at fixed matching size, in which each part need only be induced in the union of itself with the parts that follow it; write $\mathrm{ORS}_n(r)$ for the largest number of parts, so that $r\,\mathrm{ORS}_n(r)$ is the ordered analogue of $\mathrm{ex}^*(n,rK_2)$. Our main tool is a characterisation valid for every $r$: an ordered decomposition into induced $r$-matchings is a sequence of steps that start from $K_n$ and repeatedly delete a perfect matching from $2r$ vertices currently spanning a clique. Reading a decomposition backwards turns a condition about the ordering into a reachability question that an exhaustive search can settle. For $r=2$ we determine $\mathrm{ORS}_n(2)$ exactly at orders five through nineteen, where it takes the values $1,3,5,8,11,14,19,23,28,34,40,47,54,62,70$, and we confine $\mathrm{ORS}_{20}(2)$ to $\{78,79\}$. The counting bound $\lfloor n(n-4)/4\rfloor$ is attained at orders five through nine and at eleven, and missed by exactly one part at every other order below twenty, so order eleven is an isolated exception, not a parity effect. Across this range the ordered deficiency equals $\frac32n+O(1)$, and along powers of two a dyadic construction keeps it below $O(n\log n)$; whether it is linear for all $n$ is our main open question. The structural results are formalised in Lean 4, and the searches are certified by fail-closed sweeps and an independent checker.

math.CO↗

arXiv2026

Discrete energy as an exact label-free training objective for finite-element surrogates

Supervised training of finite-element (FE) surrogate models requires reference solutions, and each reference solution is obtained by solving the system that the surrogate is intended to replace. The assembled discrete potential energy provides a training signal that requires no reference solution. This note records, with proofs, the identities that make this signal exact for linear elastostatics: the difference between the energy of a prediction and the energy of the reference solution equals one half of the squared stiffness-norm error, and the gradient of the energy equals the stiffness-weighted error. Label-free discrete-energy minimisation and supervised regression in the stiffness norm therefore have the same unique minimiser and identical gradients at every point. Around this central result, the note states a conditioning lemma that bounds the displacement error by the energy gap, a modewise contraction identity that explains why the Euclidean displacement error is an unsuitable primary metric, the Chebyshev bound that governs conjugate-gradient post-processing of surrogate predictions, and a conditional latent-separation proposition for joint-embedding predictive architecture (JEPA) pretraining on a shared stiffness operator, with an explicit numerical counterexample that delimits its scope. Every claim with numeric content is implemented as an executable falsification check; the checks were executed twice, on synthetic test problems and on a probe set of 16 instances from the validation split of a pre-registered experimental run, and every inequality holds, with the measured tightness reported. A closing section explains why the construction does not extend to elastodynamics through direct minimisation of the action functional, and which time-discrete formulation restores exactness.

cs.CE↗

arXiv2025

Beyond Accuracy: A Geometric Stability Analysis of Large Language Models in Chess Evaluation

The evaluation of Large Language Models (LLMs) in complex reasoning domains typically relies on performance alignment with ground-truth oracles. In the domain of chess, this standard manifests as accuracy benchmarks against strong engines like Stockfish. However, high scalar accuracy does not necessarily imply robust conceptual understanding. This paper argues that standard accuracy metrics fail to distinguish between genuine geometric reasoning and the superficial memorization of canonical board states. To address this gap, we propose a Geometric Stability Framework, a novel evaluation methodology that rigorously tests model consistency under invariant transformations-including board rotation, mirror symmetry, color inversion, and format conversion. We applied this framework to a comparative analysis of six state-of-the-art LLMs including GPT-5.1, Claude Sonnet 4.5, and Kimi K2 Turbo, utilizing a dataset of approximately 3,000 positions. Our results reveal a significant Accuracy-Stability Paradox. While models such as GPT-5.1 achieve near-optimal accuracy on standard positions, they exhibit catastrophic degradation under geometric perturbation, specifically in rotation tasks where error rates surge by over 600%. This disparity suggests a reliance on pattern matching over abstract spatial logic. Conversely, Claude Sonnet 4.5 and Kimi K2 Turbo demonstrate superior dual robustness, maintaining high consistency across all transformation axes. Furthermore, we analyze the trade-off between helpfulness and safety, identifying Gemini 2.5 Flash as the leader in illegal state rejection (96.0%). We conclude that geometric stability provides an orthogonal and essential metric for AI evaluation, offering a necessary proxy for disentangling reasoning capabilities from data contamination and overfitting in large-scale models.

cs.AI↗