## Histogram: Steps to KL-based Threshold by Category ### Overview The image presents four histograms, each displaying the distribution of "Steps to KL-based Threshold" for different categories: "high school mathematics", "philosophy", "logical fallacies", and "moral scenarios". Each histogram compares a "Default" setting with a "Cont. CoT" (Continuous Chain-of-Thought) setting. The y-axis represents "Density", and the x-axis represents "Steps to KL-based Threshold". ### Components/Axes * **X-axis:** "Steps to KL-based Threshold", ranging from 0 to 30 in each histogram. * **Y-axis:** "Density", ranging from 0.00 to 0.08. * **Histograms:** Four histograms, one for each category: * High school mathematics * Philosophy * Logical fallacies * Moral scenarios * **Legend:** Located at the top of each histogram. * "Default": Represented by a light gray color with black outlines. * "Cont. CoT": Represented by a distinct color for each category (green, yellow, red, blue). * **Mean (µ) values:** Provided in the legend for both "Default" and "Cont. CoT" settings within each category. ### Detailed Analysis **1. High School Mathematics** * **Default (µ=12.7):** The light gray histogram shows a distribution that peaks around 5-10 steps and then gradually decreases. * **Cont. CoT (µ=11.9):** The green histogram shows a similar distribution to the default, peaking around 5 steps and then decreasing. * **Trend:** Both distributions are right-skewed, with the Cont. CoT slightly shifted to the left compared to the Default. **2. Philosophy** * **Default (µ=14.6):** The light gray histogram shows a distribution that peaks around 10-15 steps and then gradually decreases. * **Cont. CoT (µ=13.5):** The yellow histogram shows a similar distribution to the default, peaking around 10 steps and then decreasing. * **Trend:** Both distributions are right-skewed, with the Cont. CoT slightly shifted to the left compared to the Default. **3. Logical Fallacies** * **Default (µ=15.6):** The light gray histogram shows a distribution that peaks around 10-15 steps and then gradually decreases. * **Cont. CoT (µ=14.4):** The red histogram shows a similar distribution to the default, peaking around 10 steps and then decreasing. * **Trend:** Both distributions are right-skewed, with the Cont. CoT slightly shifted to the left compared to the Default. **4. Moral Scenarios** * **Default (µ=16.2):** The light gray histogram shows a distribution that peaks around 20-25 steps. * **Cont. CoT (µ=16.0):** The blue histogram shows a similar distribution to the default, peaking around 20 steps. * **Trend:** The distributions are less skewed compared to the other categories, with a more pronounced peak. ### Key Observations * For all categories, the "Cont. CoT" setting has a lower mean (µ) value than the "Default" setting. * The distributions for "high school mathematics", "philosophy", and "logical fallacies" are right-skewed, indicating that most cases require fewer steps to reach the KL-based threshold. * The distribution for "moral scenarios" is less skewed and has a more pronounced peak, suggesting a more consistent number of steps required. ### Interpretation The histograms compare the number of steps required to reach a KL-based threshold under "Default" conditions versus using a "Continuous Chain-of-Thought" (Cont. CoT) approach across four different categories. The consistent trend of lower mean values for "Cont. CoT" suggests that this method generally reduces the number of steps needed to reach the threshold, potentially indicating a more efficient or direct path to the solution or conclusion. The varying shapes of the distributions across categories suggest that the nature of the problem influences the number of steps required, with "moral scenarios" showing a more consistent step count compared to the other, more skewed distributions.

\n ## Histograms: Density Distributions for Different Scenarios ### Overview The image presents four histograms, each representing the density distribution of "Steps to KL-based Threshold" for different scenarios: high school mathematics, philosophy, logical fallacies, and moral scenarios. Each histogram displays two distributions, labeled "Default" and "Cont. CoT" (Chain of Thought), with associated mean (μ) values. ### Components/Axes * **X-axis:** "Steps to KL-based Threshold" - ranging from 0 to 30, with increments of 5. * **Y-axis:** "Density" - ranging from 0.00 to 0.08, with increments of 0.01. * **Histograms:** Four separate histograms, one for each scenario. * **Legend:** Each histogram has a legend in the top-left corner indicating the two distributions: * "Default" (Green for high school mathematics, Yellow for philosophy, Red for logical fallacies, Blue for moral scenarios) * "Cont. CoT" (Light Green for high school mathematics, Light Yellow for philosophy, Light Red for logical fallacies, Light Blue for moral scenarios) * **Mean Values (μ):** Each legend also displays the mean (μ) value for each distribution. ### Detailed Analysis or Content Details **1. High School Mathematics (Green)** * **Default (Green):** The distribution is roughly bell-shaped, peaking around steps 5-10. The density decreases as the number of steps increases. μ = 12.7. * Approximate Density Values: * Steps 5: ~0.07 * Steps 10: ~0.06 * Steps 15: ~0.045 * Steps 20: ~0.03 * Steps 25: ~0.015 * **Cont. CoT (Light Green):** The distribution is similar to the "Default" but shifted slightly to the right, peaking around steps 10-15. μ = 11.9. * Approximate Density Values: * Steps 5: ~0.05 * Steps 10: ~0.065 * Steps 15: ~0.05 * Steps 20: ~0.03 * Steps 25: ~0.01 **2. Philosophy (Yellow)** * **Default (Yellow):** The distribution is unimodal, peaking around steps 10-15. The density decreases as the number of steps increases. μ = 14.6. * Approximate Density Values: * Steps 5: ~0.04 * Steps 10: ~0.06 * Steps 15: ~0.05 * Steps 20: ~0.03 * Steps 25: ~0.01 * **Cont. CoT (Light Yellow):** The distribution is similar to the "Default" but shifted slightly to the right, peaking around steps 15-20. μ = 13.5. * Approximate Density Values: * Steps 5: ~0.03 * Steps 10: ~0.04 * Steps 15: ~0.06 * Steps 20: ~0.04 * Steps 25: ~0.01 **3. Logical Fallacies (Red)** * **Default (Red):** The distribution is bimodal, with peaks around steps 5 and 15-20. μ = 15.6. * Approximate Density Values: * Steps 5: ~0.05 * Steps 10: ~0.03 * Steps 15: ~0.06 * Steps 20: ~0.04 * Steps 25: ~0.01 * **Cont. CoT (Light Red):** The distribution is unimodal, peaking around steps 15-20. μ = 14.4. * Approximate Density Values: * Steps 5: ~0.03 * Steps 10: ~0.02 * Steps 15: ~0.06 * Steps 20: ~0.04 * Steps 25: ~0.01 **4. Moral Scenarios (Blue)** * **Default (Blue):** The distribution is roughly bell-shaped, peaking around steps 15-20. The density decreases as the number of steps increases. μ = 16.2. * Approximate Density Values: * Steps 5: ~0.02 * Steps 10: ~0.03 * Steps 15: ~0.07 * Steps 20: ~0.06 * Steps 25: ~0.03 * **Cont. CoT (Light Blue):** The distribution is similar to the "Default" but shifted slightly to the right, peaking around steps 20-25. μ = 16.0. * Approximate Density Values: * Steps 5: ~0.01 * Steps 10: ~0.02 * Steps 15: ~0.05 * Steps 20: ~0.07 * Steps 25: ~0.04 ### Key Observations * The "Cont. CoT" distributions generally have lower peaks and are shifted to the right compared to the "Default" distributions, indicating that using Chain of Thought tends to require more steps to reach the KL-based threshold. * The "Logical Fallacies" scenario exhibits a bimodal distribution for the "Default" setting, suggesting two distinct patterns in the number of steps required. * The mean values (μ) for "Cont. CoT" are consistently lower than those for "Default" across all scenarios, reinforcing the observation that CoT requires more steps. ### Interpretation The data suggests that the use of Chain of Thought (CoT) reasoning in these scenarios generally leads to a need for more steps to reach a certain level of confidence (as measured by the KL-based threshold). This could be because CoT involves more complex reasoning processes, requiring more iterations or steps to converge. The bimodal distribution observed in the "Logical Fallacies" scenario for the "Default" setting is particularly interesting. This could indicate that there are two fundamentally different ways in which the model approaches logical fallacies – one that requires fewer steps and another that requires more. The CoT approach seems to homogenize this, resulting in a unimodal distribution. The differences in distributions across scenarios highlight the varying complexity of the tasks. Moral scenarios and philosophy, for example, seem to require more steps overall compared to high school mathematics, even without CoT. This aligns with the intuitive understanding that these domains involve more nuanced and abstract reasoning. The KL-based threshold likely represents a point where the model's confidence in its answer reaches a certain level. The "Steps to KL-based Threshold" metric, therefore, provides insight into the computational effort required to achieve a reliable outcome in each scenario.

\n ## Histograms: Distribution of Steps to KL-based Threshold Across Four Domains ### Overview The image displays four horizontally arranged histograms, each comparing the distribution of "Steps to KL-based Threshold" for two different methods ("Default" and "Cont. CoT") across four distinct domains: high school mathematics, philosophy, logical fallacies, and moral scenarios. The charts share a common x-axis label and y-axis label, but each has its own title and color scheme. ### Components/Axes * **Overall X-Axis Label (Bottom Center):** "Steps to KL-based Threshold" * **Overall Y-Axis Label (Left Center):** "Density" * **X-Axis Scale (All Charts):** Linear scale from 0 to 30, with major tick marks at intervals of 5 (0, 5, 10, 15, 20, 25, 30). * **Y-Axis Scale (All Charts):** Linear scale from 0.00 to 0.08, with major tick marks at intervals of 0.01. * **Chart Titles (Top of each subplot, from left to right):** 1. "high school mathematics" 2. "philosophy" 3. "logical fallacies" 4. "moral scenarios" * **Legends (Positioned in the top-right corner of each subplot):** * **Chart 1 (high school mathematics):** * `Default (μ=12.7)` - Light green fill with diagonal black stripes (\\). * `Cont. CoT (μ=11.9)` - Solid medium green fill. * **Chart 2 (philosophy):** * `Default (μ=14.6)` - Light yellow/beige fill with diagonal black stripes (\\). * `Cont. CoT (μ=13.5)` - Solid golden yellow fill. * **Chart 3 (logical fallacies):** * `Default (μ=15.6)` - Light red/pink fill with diagonal black stripes (\\). * `Cont. CoT (μ=14.4)` - Solid salmon/coral red fill. * **Chart 4 (moral scenarios):** * `Default (μ=16.2)` - Light blue fill with diagonal black stripes (\\). * `Cont. CoT (μ=16.0)` - Solid medium blue fill. ### Detailed Analysis **Chart 1: high school mathematics** * **Trend Verification:** Both distributions are right-skewed. The "Cont. CoT" distribution (solid green) is shifted noticeably to the left (toward fewer steps) compared to the "Default" distribution (striped green). * **Data Points (Approximate):** * The "Cont. CoT" distribution peaks sharply between 5-10 steps, with its highest density bar (~0.08) around 7-8 steps. * The "Default" distribution has a broader peak between 10-15 steps, with its highest density bar (~0.075) around 12-13 steps. * Both distributions taper off, approaching near-zero density by 30 steps. * **Reported Means (μ):** Default = 12.7, Cont. CoT = 11.9. **Chart 2: philosophy** * **Trend Verification:** Both distributions are right-skewed. The "Cont. CoT" distribution (solid yellow) shows a very pronounced, sharp peak at lower step counts compared to the more spread-out "Default" distribution (striped yellow). * **Data Points (Approximate):** * The "Cont. CoT" distribution has its dominant peak between 5-10 steps, with the highest density bar (~0.065) around 6-7 steps. * The "Default" distribution is more dispersed, with a less defined peak region between 10-20 steps. Its highest density bar (~0.08) is around 18-19 steps. * Both distributions approach near-zero density by 30 steps. * **Reported Means (μ):** Default = 14.6, Cont. CoT = 13.5. **Chart 3: logical fallacies** * **Trend Verification:** Both distributions are right-skewed and have similar shapes, but the "Cont. CoT" distribution (solid red) is shifted slightly to the left of the "Default" distribution (striped red). * **Data Points (Approximate):** * The "Cont. CoT" distribution has a primary peak between 15-20 steps, with its highest density bar (~0.08) around 18-19 steps. It also shows a smaller, secondary peak around 5-7 steps. * The "Default" distribution's peak is slightly to the right, between 15-20 steps, with its highest density bar (~0.08) around 19-20 steps. * Both distributions taper off, approaching near-zero density by 30 steps. * **Reported Means (μ):** Default = 15.6, Cont. CoT = 14.4. **Chart 4: moral scenarios** * **Trend Verification:** Both distributions are right-skewed and are very similar in shape and position, with significant overlap. The "Cont. CoT" distribution (solid blue) is only marginally shifted left compared to the "Default" distribution (striped blue). * **Data Points (Approximate):** * Both distributions have their primary peak between 15-20 steps. The highest density bars for both are around 18-20 steps, reaching near 0.08. * Both distributions show a smaller, secondary peak or shoulder around 5-10 steps. * Both distributions approach near-zero density by 30 steps. * **Reported Means (μ):** Default = 16.2, Cont. CoT = 16.0. ### Key Observations 1. **Consistent Direction of Effect:** In all four domains, the "Cont. CoT" method results in a distribution shifted toward fewer steps (lower mean μ) compared to the "Default" method. 2. **Magnitude of Effect Varies:** The reduction in mean steps is most pronounced in "high school mathematics" (Δμ = -0.8) and "philosophy" (Δμ = -1.1). The effect is smaller in "logical fallacies" (Δμ = -1.2) and minimal in "moral scenarios" (Δμ = -0.2). 3. **Distribution Shape:** All distributions are right-skewed, indicating that while most instances require a moderate number of steps, a long tail of instances requires many more steps. 4. **Domain Difficulty:** The overall position of the distributions suggests an ordering of domain difficulty (in terms of steps to threshold), from easiest to hardest: high school mathematics (lowest mean steps) < philosophy < logical fallacies < moral scenarios (highest mean steps). ### Interpretation The data demonstrates that the "Cont. CoT" (likely "Continuous Chain-of-Thought") method consistently reduces the number of steps required to reach a KL-divergence based threshold compared to a "Default" method across diverse reasoning domains. This suggests "Cont. CoT" is a more efficient reasoning or generation process. The **Peircean investigative** reading reveals a clear pattern: the intervention ("Cont. CoT") has a measurable, positive effect (reduction in steps), but its efficacy is **domain-dependent**. The effect is strong in domains with more structured, objective answers (mathematics, philosophy) and weakens in domains involving nuanced judgment or open-ended reasoning (moral scenarios). This implies the mechanism of "Cont. CoT" may be particularly well-suited for optimizing processes in structured problem-solving contexts. The near-identical distributions in "moral scenarios" suggest that for this type of task, the added process does not significantly alter the computational path length, indicating a potential ceiling effect or a fundamental difference in how such problems are solved.

## Histograms: KL-based Threshold Steps by Category ### Overview The image displays four side-by-side histograms comparing two data series ("Default" and "Cont. CoT") across four categories: high school mathematics, philosophy, logical fallacies, and moral scenarios. Each histogram shows the distribution of steps required to reach a KL-based threshold, with density on the y-axis and steps on the x-axis. The histograms use distinct colors for each category and differentiate data series via pattern (solid vs. striped). ### Components/Axes - **X-axis**: "Steps to KL-based Threshold" (range: 0–30, integer increments). - **Y-axis**: "Density" (range: 0–0.08, increments of 0.01). - **Legends**: Positioned at the top of each histogram, with: - **Default**: Light-colored (white/gray) bars with diagonal stripes. - **Cont. CoT**: Solid-colored bars (green, orange, red, blue for respective categories). - **Categories**: 1. High school mathematics (green) 2. Philosophy (orange) 3. Logical fallacies (red) 4. Moral scenarios (blue) ### Detailed Analysis #### High School Mathematics (Green) - **Default (μ=12.7)**: Peaks at ~12–13 steps, with a broad distribution tapering toward 0 and 30. - **Cont. CoT (μ=11.9)**: Slightly narrower peak at ~11–12 steps, overlapping with Default but shifted left. #### Philosophy (Orange) - **Default (μ=14.6)**: Bimodal distribution with peaks at ~10 and ~20 steps. - **Cont. CoT (μ=13.5)**: Single peak at ~13–14 steps, narrower than Default. #### Logical Fallacies (Red) - **Default (μ=15.6)**: Broad peak centered at ~15–16 steps, with a long tail to the right. - **Cont. CoT (μ=14.4)**: Tighter peak at ~14–15 steps, reduced tail length. #### Moral Scenarios (Blue) - **Default (μ=16.2)**: Sharp peak at ~16–17 steps, with a steep decline on both sides. - **Cont. CoT (μ=16.0)**: Nearly identical peak to Default but slightly narrower. ### Key Observations 1. **Cont. CoT Consistently Lower μ**: Across all categories, Cont. CoT has lower mean steps (μ) than Default, suggesting improved efficiency. 2. **Narrower Distributions for Cont. CoT**: Cont. CoT histograms are generally tighter, indicating less variability in step counts. 3. **Bimodal Philosophy**: Philosophy’s Default series shows two distinct peaks, unlike other categories. 4. **Moral Scenarios Symmetry**: Both Default and Cont. CoT for moral scenarios exhibit highly symmetric distributions. ### Interpretation The data suggests that the "Cont. CoT" method reduces the average steps required to reach the KL-based threshold across all categories, with the most pronounced effect in philosophy (Δμ = 1.1). The narrower distributions for Cont. CoT imply more consistent performance, while the bimodal pattern in philosophy’s Default series hints at potential subgroup differences (e.g., easy vs. hard problems). The symmetry in moral scenarios indicates a clear threshold effect, whereas logical fallacies show a longer tail for Default, possibly reflecting complex edge cases. These trends align with the hypothesis that Cont. CoT optimizes step efficiency in reasoning tasks.