Jizhou Guo

Jizhou Guo, Yitao Luo

A convex polygon is called small if its diameter is at most one. Reinhardt proved the universal perimeter bound $\mathrm{perim}(P) \leq U_n := 2n\sin(π/(2n))$, and the bound is attained whenever $n$ has a nontrivial odd divisor. The remaining power-of-two cases have resisted exact solution beyond $n=8$. This paper presents computer-assisted proof candidates for the first three open cases, $n=16,32,64$. In each case, the candidate theorem asserts uniqueness of the maximizing congruence class. The proof architecture is common to all three cases: pass to the difference body $P-P$; encode its reconstruction by a sign code; prove that every global maximizer is saturated, so all difference-body vertices lie on the unit circle; localize every competitive configuration near the regular angle vector; exhaustively screen the sign codes using exact arithmetic; eliminate all nonwinning dihedral orbits; and prove uniqueness inside the winning code by strong convexity and a quantitative KKT argument. The exact certificates cover $2^{15}$ normalized codes for $n=16$, $2^{31}$ normalized codes for $n=32$, and all $2^{64}$ half-codes for $n=64$, leaving respectively $16$, $96$, and $896$ survivors before orbit elimination. The accompanying source package contains the verifiers, recorded outputs, and separate computational cross-checks. These results have not yet received independent human expert review and are therefore deliberately presented as proof candidates rather than literature-established theorems.

Jizhou Guo, Zhaomin Wu, Hanchen Yang et al.

Enhancing Large Language Model (LLM)'s performance with best-of-N sampling is effective and has attracted significant attention. However, it is computationally prohibitive due to massive, data-hungry text-based reward models. By changing the data source from text to hidden states, we introduce SWIFT (Simple Weighted Intrinsic Feedback Technique), a novel, lightweight technique that leverages the rich information embedded in LLM hidden states to address these issues, which operates on token-level and consists of only linear layers. Extensive experiments show that SWIFT outperforms baselines with less than 0.005% of the parameters of baselines, requiring only a few samples for training, demonstrating significant efficiency improvement. SWIFT's robust scalability, applicability to some closed-source models via logits, and ability to be combined with traditional reward models to yield further performance gains underscore its practical value.

Yaozu Wu, Wei-Chieh Huang, Jizhou Guo et al.

Large language models increasingly operate in settings where humans are active collaborators rather than passive task providers. We introduce HAS-Framework, a graph-based framework that represents humans and LLM-powered agents as first-class participants with explicit roles, permissions, communication paths, and action authority. Building on this framework, HAS-Bench evaluates Human-Agent Systems under configurable human participation across agency levels, interaction channels, and persona policies. The benchmark measures both task outcomes and process-level collaboration behavior, including clarification quality, feedback utilization, control calibration, safety, initiative, and interaction cost. Experiments across six domains show that human participation can substantially improve task completion and failure recovery, but the gains depend on when, how, and by whom human input is exercised.