## Chart: Accuracy vs. Recursion Depth for Different Models ### Overview The image presents three line graphs comparing the accuracy of different models ("o4-mini", "Goedel-Prover-SFT", and "Kimina-Prover-Preview-Distill-7B") as a function of recursion depth. Each graph plots accuracy (in percentage) on the y-axis against recursion depth on the x-axis. ### Components/Axes * **X-axis (all graphs):** Recursion Depth, with integer values from 0 to 4 (o4-mini) or 0 to 6 (Goedel-Prover-SFT and Kimina-Prover-Preview-Distill-7B). * **Y-axis (all graphs):** Accuracy (%), with a range from approximately 0 to 50 for "o4-mini", 57 to 65 for "Goedel-Prover-SFT", and 67 to 76 for "Kimina-Prover-Preview-Distill-7B". * **Graph Titles:** * Left: "o4-mini" * Middle: "Goedel-Prover-SFT" * Right: "Kimina-Prover-Preview-Distill-7B" * **Data Series:** * "o4-mini": Blue line with circle markers. * "Goedel-Prover-SFT": Red line with downward triangle markers. * "Kimina-Prover-Preview-Distill-7B": Pink line with square markers. ### Detailed Analysis **1. o4-mini (Left Graph):** * **Trend:** The blue line slopes upward, indicating increasing accuracy with recursion depth. * **Data Points:** * Recursion Depth 0: Accuracy approximately 7%. * Recursion Depth 1: Accuracy approximately 38%. * Recursion Depth 2: Accuracy approximately 40%. * Recursion Depth 3: Accuracy approximately 44%. * Recursion Depth 4: Accuracy approximately 46%. **2. Goedel-Prover-SFT (Middle Graph):** * **Trend:** The red line slopes upward, with a steep initial increase, followed by a plateau. * **Data Points:** * Recursion Depth 0: Accuracy approximately 57%. * Recursion Depth 1: Accuracy approximately 61%. * Recursion Depth 2: Accuracy approximately 64%. * Recursion Depth 3: Accuracy approximately 64.5%. * Recursion Depth 4: Accuracy approximately 65%. * Recursion Depth 5: Accuracy approximately 65%. * Recursion Depth 6: Accuracy approximately 65%. **3. Kimina-Prover-Preview-Distill-7B (Right Graph):** * **Trend:** The pink line slopes upward, with a steep initial increase, followed by a plateau. * **Data Points:** * Recursion Depth 0: Accuracy approximately 67%. * Recursion Depth 1: Accuracy approximately 69%. * Recursion Depth 2: Accuracy approximately 72%. * Recursion Depth 3: Accuracy approximately 74%. * Recursion Depth 4: Accuracy approximately 74.5%. * Recursion Depth 5: Accuracy approximately 75%. * Recursion Depth 6: Accuracy approximately 75%. ### Key Observations * All three models show an increase in accuracy with increasing recursion depth, but the rate of increase diminishes as recursion depth increases. * "Kimina-Prover-Preview-Distill-7B" consistently achieves the highest accuracy across all recursion depths. * "o4-mini" has the lowest initial accuracy and the smallest overall range of accuracy values. * "Goedel-Prover-SFT" and "Kimina-Prover-Preview-Distill-7B" exhibit a similar trend, with a sharp initial increase in accuracy followed by a plateau. ### Interpretation The graphs suggest that increasing recursion depth generally improves the accuracy of these models, but there are diminishing returns. The "Kimina-Prover-Preview-Distill-7B" model appears to be the most effective, achieving the highest accuracy across all recursion depths. The "o4-mini" model is the least accurate. The plateauing effect observed in "Goedel-Prover-SFT" and "Kimina-Prover-Preview-Distill-7B" indicates that there is a limit to the benefits of increasing recursion depth for these models. The data implies that the "Kimina-Prover-Preview-Distill-7B" model is the most sophisticated or well-optimized among the three, as it consistently outperforms the others.

## Line Charts: Accuracy vs. Recursion Depth Across Three Models ### Overview The image consists of three side-by-side line charts arranged horizontally (left, center, right). Each chart illustrates the relationship between "Recursion Depth" and "Accuracy (%)" for a different computational model. While the x-axis scale is similar across the charts, the y-axis scales differ significantly, indicating varying baseline performances among the models. ### Components/Axes * **Layout:** Three distinct subplots in a 1x3 grid. * **X-Axis (All Charts):** * **Label:** "Recursion Depth" (located at the bottom center of each chart). * **Scale:** Linear integer scale. The left chart ranges from 0 to 4. The center and right charts range from 0 to 6. * **Y-Axis (Varies per Chart):** * **Label:** "Accuracy (%)" (located vertically on the left side of each chart). * **Left Chart Scale:** Major gridlines at 10, 20, 30, 40. * **Center Chart Scale:** Major gridlines at 58, 60, 62, 64. * **Right Chart Scale:** Major gridlines at 68, 70, 72, 74. * **Grid:** All charts feature a light gray background grid corresponding to the major axis ticks. * **Legend:** There is no explicit floating legend. Instead, the title of each chart identifies the data series, and each uses a distinct color and marker shape. --- ### Detailed Analysis #### 1. Left Chart: o4-mini * **Position:** Far left. * **Title:** "o4-mini" (Top center). * **Styling:** Thick blue line with large circular markers outlined in black. * **Visual Trend:** The line exhibits a massive, steep upward slope from depth 0 to 1, followed by a moderate, steady upward slope from depth 1 to 4. There is no plateau reached within the visible data. * **Data Points (Approximate):** * Recursion Depth 0: ~7.0% (± 1.0%) * Recursion Depth 1: ~37.0% (± 1.0%) * Recursion Depth 2: ~40.0% (± 1.0%) * Recursion Depth 3: ~44.0% (± 1.0%) * Recursion Depth 4: ~47.0% (± 1.0%) #### 2. Center Chart: Goedel-Prover-SFT * **Position:** Middle. * **Title:** "Goedel-Prover-SFT" (Top center). * **Styling:** Thick red line with large downward-pointing triangular markers outlined in black. * **Visual Trend:** The line slopes steeply upward from depth 0 to 2. The slope then decreases significantly, forming a gradual curve that flattens into a near-horizontal plateau between depths 4, 5, and 6. * **Data Points (Approximate):** * Recursion Depth 0: ~56.5% (± 0.5%) * Recursion Depth 1: ~60.8% (± 0.5%) * Recursion Depth 2: ~63.5% (± 0.5%) * Recursion Depth 3: ~64.3% (± 0.5%) * Recursion Depth 4: ~64.7% (± 0.5%) * Recursion Depth 5: ~65.1% (± 0.5%) * Recursion Depth 6: ~65.2% (± 0.5%) #### 3. Right Chart: Kimina-Prover-Preview-Distill-7B * **Position:** Far right. * **Title:** "Kimina-Prover-Preview-Distill-7B" (Top center). * **Styling:** Thick pink/light-purple line with large square markers outlined in black. * **Visual Trend:** The line shows a steady, linear upward slope from depth 0 to 3. After depth 3, the curve sharply flattens, reaching a horizontal plateau by depth 5 and 6. * **Data Points (Approximate):** * Recursion Depth 0: ~67.2% (± 0.5%) * Recursion Depth 1: ~68.8% (± 0.5%) * Recursion Depth 2: ~72.1% (± 0.5%) * Recursion Depth 3: ~74.2% (± 0.5%) * Recursion Depth 4: ~74.8% (± 0.5%) * Recursion Depth 5: ~75.1% (± 0.5%) * Recursion Depth 6: ~75.1% (± 0.5%) --- ### Key Observations 1. **Universal Improvement:** Across all three models, increasing the "Recursion Depth" from 0 results in an increase in "Accuracy (%)". 2. **Diminishing Returns:** Both the "Goedel-Prover-SFT" and "Kimina-Prover-Preview-Distill-7B" models demonstrate a clear pattern of diminishing returns. The accuracy gains become marginal after a recursion depth of 3 or 4, eventually plateauing. 3. **Varying Baselines and Ceilings:** * `o4-mini` has the lowest overall accuracy (peaking around 47%) but shows the most dramatic relative jump from depth 0 to 1 (+30%). * `Goedel-Prover-SFT` operates in the mid-range (peaking around 65%). * `Kimina-Prover-Preview-Distill-7B` has the highest baseline (starting at ~67%) and the highest ceiling (peaking at ~75%). 4. **Incomplete Curve:** The `o4-mini` chart stops at depth 4 and has not yet shown the distinct flattening/plateauing behavior seen in the other two models at depths 5 and 6. ### Interpretation These charts likely represent the performance of automated theorem provers or reasoning models (suggested by "Prover" in the titles). "Recursion Depth" likely refers to the number of iterative steps, search depth, or chain-of-thought expansions the model is permitted to execute before returning an answer. The data demonstrates a fundamental principle in search-based or iterative reasoning AI: allowing a model to "think longer" or search deeper reliably improves its accuracy. However, this benefit is not infinite. The plateauing effect in the Goedel and Kimina models suggests an upper bound to the utility of recursion; beyond a depth of 4 or 5, the models exhaust their capacity to find better solutions, or the search space becomes too noisy to yield further accuracy gains. Furthermore, the charts highlight that while recursion helps, base model capability is paramount. Even at its maximum tested recursion depth (4), the `o4-mini` model (47%) cannot match the zero-recursion baseline accuracy of the `Goedel` (~56.5%) or `Kimina` (~67.2%) models. The `Kimina-Prover-Preview-Distill-7B` is objectively the most performant model shown in this dataset.

\n ## Line Charts: Accuracy vs. Recursion Depth for Different Models ### Overview This image presents three separate line charts, each depicting the relationship between "Accuracy (%)" and "Recursion Depth" for three different models: "o4-mini", "Goedel-Prover-SFT", and "Kimina-Prover-Preview-Distill-7B". Each chart shows how accuracy changes as the recursion depth increases. ### Components/Axes Each chart shares the same axes: * **X-axis:** "Recursion Depth", ranging from 0 to 4 for the first chart, 0 to 6 for the second and third charts. * **Y-axis:** "Accuracy (%)", ranging from approximately 8% to 75%. * Each chart has a title indicating the model being evaluated. * Each chart has a line representing the accuracy of the model at different recursion depths. ### Detailed Analysis or Content Details **Chart 1: o4-mini** * **Line Color:** Blue * **Trend:** The line slopes upward, indicating increasing accuracy with increasing recursion depth, but the rate of increase slows down at higher recursion depths. * **Data Points (approximate):** * Recursion Depth 0: ~9% * Recursion Depth 1: ~38% * Recursion Depth 2: ~41% * Recursion Depth 3: ~44% **Chart 2: Goedel-Prover-SFT** * **Line Color:** Red * **Trend:** The line shows a rapid increase in accuracy initially, then plateaus as recursion depth increases. * **Data Points (approximate):** * Recursion Depth 0: ~58% * Recursion Depth 1: ~60% * Recursion Depth 2: ~62% * Recursion Depth 3: ~63% * Recursion Depth 4: ~64% * Recursion Depth 5: ~64% * Recursion Depth 6: ~64% **Chart 3: Kimina-Prover-Preview-Distill-7B** * **Line Color:** Purple * **Trend:** The line slopes upward, showing increasing accuracy with increasing recursion depth. The rate of increase is more consistent than in the "o4-mini" chart. * **Data Points (approximate):** * Recursion Depth 0: ~68% * Recursion Depth 1: ~70% * Recursion Depth 2: ~72% * Recursion Depth 3: ~73% * Recursion Depth 4: ~74% * Recursion Depth 5: ~74% * Recursion Depth 6: ~75% ### Key Observations * The "Goedel-Prover-SFT" model achieves the highest accuracy overall, plateauing around 64%. * The "Kimina-Prover-Preview-Distill-7B" model shows a consistent increase in accuracy with recursion depth, reaching approximately 75%. * The "o4-mini" model starts with the lowest accuracy and exhibits a diminishing return in accuracy gains as recursion depth increases. * All models demonstrate that increasing recursion depth generally improves accuracy, although the extent of improvement varies significantly. ### Interpretation The data suggests that recursion depth is a crucial parameter for improving the accuracy of these models. However, the optimal recursion depth varies depending on the model architecture. The "Goedel-Prover-SFT" model appears to reach its performance limit relatively quickly, while the "Kimina-Prover-Preview-Distill-7B" model continues to benefit from increased recursion depth throughout the observed range. The "o4-mini" model shows the least improvement with increasing recursion depth, indicating that its performance may be limited by other factors. The differences in performance between the models could be attributed to variations in model size, training data, or architectural design. The plateauing effect observed in the "Goedel-Prover-SFT" and "Kimina-Prover-Preview-Distill-7B" models suggests that there is a point of diminishing returns where further increasing recursion depth does not yield significant accuracy gains.

## Line Charts: Model Accuracy vs. Recursion Depth ### Overview The image displays three separate line charts arranged horizontally, each plotting the accuracy of a different AI model against increasing recursion depth. The charts share the same axis labels but have different scales and data trends. All charts show a positive correlation between recursion depth and accuracy. ### Components/Axes * **Common Elements:** * **X-axis Label (All Charts):** "Recursion Depth" * **Y-axis Label (All Charts):** "Accuracy (%)" * **Chart Titles (Top-Center of each plot):** Left: "o4-mini", Middle: "Goedel-Prover-SFT", Right: "Kimina-Prover-Preview-Distill-7B" * **Data Series:** Each chart contains a single data series represented by a colored line with distinct markers. * **Left Chart (o4-mini):** * **Line Color:** Blue * **Marker Style:** Filled circles * **X-axis Range:** 0 to 4 (integer steps) * **Y-axis Range:** Approximately 5% to 45% * **Middle Chart (Goedel-Prover-SFT):** * **Line Color:** Red * **Marker Style:** Filled downward-pointing triangles * **X-axis Range:** 0 to 6 (integer steps) * **Y-axis Range:** Approximately 56% to 65% * **Right Chart (Kimina-Prover-Preview-Distill-7B):** * **Line Color:** Magenta/Pink * **Marker Style:** Filled squares * **X-axis Range:** 0 to 6 (integer steps) * **Y-axis Range:** Approximately 67% to 75% ### Detailed Analysis **1. Left Chart: o4-mini** * **Trend:** The line shows a steep, concave-down increase. Accuracy rises sharply from depth 0 to 1, then the rate of improvement slows but remains positive through depth 4. * **Data Points (Approximate):** * Depth 0: ~8% * Depth 1: ~37% * Depth 2: ~40% * Depth 3: ~43% * Depth 4: ~45% **2. Middle Chart: Goedel-Prover-SFT** * **Trend:** The line shows a steady, concave-down increase. The slope is steepest from depth 0 to 2 and gradually flattens, suggesting diminishing returns at higher recursion depths. * **Data Points (Approximate):** * Depth 0: ~56% * Depth 1: ~61% * Depth 2: ~63.5% * Depth 3: ~64.5% * Depth 4: ~65% * Depth 5: ~65.2% * Depth 6: ~65.3% **3. Right Chart: Kimina-Prover-Preview-Distill-7B** * **Trend:** The line shows a consistent, nearly linear increase with a very slight flattening at the highest depths. It demonstrates the most sustained improvement across the measured range. * **Data Points (Approximate):** * Depth 0: ~67% * Depth 1: ~69% * Depth 2: ~72% * Depth 3: ~74% * Depth 4: ~74.5% * Depth 5: ~74.8% * Depth 6: ~75% ### Key Observations 1. **Performance Hierarchy:** There is a clear hierarchy in baseline (depth 0) and peak accuracy. Kimina-Prover-Preview-Distill-7B > Goedel-Prover-SFT > o4-mini. 2. **Improvement Magnitude:** The model with the lowest starting accuracy (o4-mini) shows the largest relative gain (~37 percentage points from depth 0 to 4). The highest-performing model (Kimina-Prover-Preview-Distill-7B) shows the smallest absolute gain (~8 points from depth 0 to 6). 3. **Diminishing Returns:** All three models exhibit diminishing returns, where each additional unit of recursion depth yields a smaller increase in accuracy. This effect is most pronounced in the Goedel-Prover-SFT chart. 4. **Recursion Depth Tested:** The o4-mini model was only evaluated up to depth 4, while the other two were evaluated up to depth 6. ### Interpretation The data demonstrates that increasing recursion depth is an effective strategy for improving the accuracy of these specific AI models on the evaluated task. The relationship is non-linear, following a pattern of rapid initial gains that taper off. The significant differences in absolute performance suggest these are fundamentally different models or were trained/fine-tuned with different methodologies (as hinted by their names: "SFT" likely for Supervised Fine-Tuning, "Distill" for knowledge distillation). The "o4-mini" model appears to be a smaller or less capable base model that benefits dramatically from added computation (recursion), while the "Kimina-Prover-Preview-Distill-7B" starts from a much higher performance plateau. From a Peircean perspective, the charts are **icons** representing the quantitative relationship between two variables. They are also **indices** pointing to a causal hypothesis: that recursive self-correction or refinement (the implied process behind "Recursion Depth") improves output quality. The consistent trend across three different models strengthens the argument that this is a generalizable phenomenon for this class of model and task, rather than an artifact of a single architecture. The diminishing returns are a critical practical insight, indicating there is an optimal recursion depth beyond which computational cost may outweigh marginal accuracy benefits.

## Line Graphs: Model Accuracy vs. Recursion Depth ### Overview The image contains three line graphs comparing the accuracy of three AI models (o4-mini, Goedel-Prover-SFT, Kimina-Prover-Preview-Distill-7B) across varying recursion depths. Each graph uses a distinct color-coded line with markers to represent performance trends. ### Components/Axes - **X-axis**: "Recursion Depth" (integer values from 0 to 4 or 6, depending on the model). - **Y-axis**: "Accuracy (%)" (ranging from ~58% to 75%). - **Legends**: - **Blue line with circles**: o4-mini. - **Red line with triangles**: Goedel-Prover-SFT. - **Purple line with squares**: Kimina-Prover-Preview-Distill-7B. - **Gridlines**: Present for all graphs to aid in value estimation. ### Detailed Analysis #### o4-mini (Blue Line) - **Trend**: Steep initial increase followed by plateau. - **Data Points**: - Depth 0: ~7% accuracy. - Depth 1: ~35% accuracy. - Depth 2: ~40% accuracy. - Depth 3: ~43% accuracy. - Depth 4: ~45% accuracy. #### Goedel-Prover-SFT (Red Line) - **Trend**: Gradual, near-linear improvement with slight deceleration. - **Data Points**: - Depth 0: ~58% accuracy. - Depth 1: ~60% accuracy. - Depth 2: ~63% accuracy. - Depth 3: ~64% accuracy. - Depth 4: ~65% accuracy. - Depth 5: ~65% accuracy. - Depth 6: ~65% accuracy. #### Kimina-Prover-Preview-Distill-7B (Purple Line) - **Trend**: Steady increase with plateau after depth 3. - **Data Points**: - Depth 0: ~68% accuracy. - Depth 1: ~70% accuracy. - Depth 2: ~72% accuracy. - Depth 3: ~74% accuracy. - Depth 4: ~75% accuracy. - Depth 5: ~75% accuracy. - Depth 6: ~75% accuracy. ### Key Observations 1. **o4-mini** shows the most dramatic early improvement but plateaus sharply after depth 1. 2. **Goedel-Prover-SFT** demonstrates consistent, incremental gains but reaches a ceiling at ~65% accuracy. 3. **Kimina-Prover-Preview-Distill-7B** achieves the highest peak accuracy (~75%) but requires deeper recursion to plateau. ### Interpretation The data suggests that recursion depth significantly impacts model performance, with all models improving as depth increases. However, the rate and ceiling of improvement vary: - **o4-mini** prioritizes rapid early gains but offers diminishing returns. - **Goedel-Prover-SFT** balances steady progress with moderate efficiency. - **Kimina-Prover-Preview-Distill-7B** achieves the highest accuracy but requires deeper recursion, indicating potential trade-offs between computational cost and performance. The differences in trends may reflect architectural choices (e.g., model complexity, training data) or optimization strategies for recursion handling. For applications requiring high accuracy, Kimina’s model is optimal despite higher computational demands, while o4-mini suits scenarios with limited recursion depth.