## 6 Results and Discussions
### Performance Comparison on TSP Problem
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Metric Category** & **FunSearch** & **HillClimb** & **EoH** & **EoH-M** & **EoH-MR** \\ \hline Core Performance Metrics & & & & & \\ \hline Best Score (Peak) & \(-\)6.8240 & \(-\)6.7326 & \(-\)6.6840 & \(-\)6.4494 & \(-\)**6.3967** \\ Std Dev of Scores (Overall) & 5.9192 & 2.2430 & 6.1619 & 5.1762 & _6.6777_ \\ \hline Stability and Robustness Metrics & & & & & \\ \hline Success Rate & **1.0000** & 0.9815 & 0.9600 & **1.0000** & **1.0000** \\ Stability Score & 2.0820 & 1.3808 & 2.2753 & 2.2936 & **2.3804** \\ \hline Efficiency Metrics & & & & & \\ \hline Total LLM Calls & 27 & 23 & **19** & 68 & 112 \\ Total Tokens Used & 25,105 & 19.742 & **10,802** & 61,345 & 107,440 \\ Total Evaluator Calls & 29 & 24 & **25** & 46 & 73 \\ Total Computation Time (s) & **820.21** & 1396.25 & 347.44 & 1492.92 & 2743.37 \\ \hline Solution Diversity Metrics & & & & & \\ \hline Total Unique Scores & **40** & 33 & 19 & 39 & 65 \\ Score Range & 21.5470 & 12.4814 & 20.7595 & 28.7071 & **28.7514** \\ Solution Diversity Score & / & / & 0.7917 & 0.8478 & **0.8904** \\ \hline Baseline Comparison Metrics & & & & & \\ \hline Improvement over Advanced Baseline (\%) & / & / & 0.0000 & 5.4891 & **6.2479** \\ Improvement over Simple Baseline (\%) & / & / & 16.9257 & 21.4857 & **22.1340** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comprehensive comparison results of the TSP optimization task. FunSearch and HillClimb, while efficient (820.21 and 1396.25 s), are ultimately limited by stagnation in exploration. **Diversity and Generalization. ** EoH-MR achieves the highest solution diversity with 65 unique scores, a score range of 28.7514, and a diversity score of 0.8904--surpassing EoH-M (0.8478) and EoH (0.7917). FunSearch and HillClimb follow behind with 40 and 33 unique scores, and diversity scores of 0.8478 and 0.7917, respectively. The narrow range of HillClimb (12.4814) indicates lower exploration strength. Figure 4 visually supports this, showing EoH-MR achieving one of the highest score ranges among all methods. **Strategic Insights from High-Performing Heuristics. ** Evolution trace analysis reveals consistent features among top-scoring heuristics:  
* Hybrid physics-ACO scoring combining proximity, direction, and pheromone strength improves scores by up to 40% over distance-based baselines. * Dynamic distance-direction ratios shift from global guidance to local refinement as the tour unfolds. * Entropy-based diversity preservation penalizes overused edges, preventing early convergence. * Short-term recency penalties reduce pheromone weight for recently visited nodes, improving exploration in late generations. * Operator sequencing matters: initial breakthroughs come from merge/mutate operations (e1, m1), followed by local refinement for final gains. * Open refinement directions include deeper local search along entropy-weighted sub-paths, adaptive score shaping during generation, and tighter operator scheduling using success trace statistics. Together, these mechanisms form a cohesive strategy of hybrid constructive bias, adaptive diversity control, and operator-level tuning. For the TSP task, EoH-MR achieves a best score of \(-6.3967\) with a total runtime of 2743 s. Empirical trends show that a solution within 5% of this value (approximately \(-6.71\)) is typically reached within 1100-1200 s, accounting for less than 45% of the total runtime. This early-stage solution already surpasses the final best scores of traditional baselines such as HillClimb (\(-6.7326\)) and FunSearch (\(-6.8240\)), and is comparable to or better than manually designed heuristics such as nearest-neighbor, insertion, or greedy rules, which typically yield scores between \(-6.5\) and \(-6.9\) for problems of size \(n=50\). While such handcrafted strategies may require several hours to days of iterative design and tuning, EoH-MR produces superior solutions fully automatically within approximately 20 min.