## Chart: Performance Comparison of Goedel-Prover-SFT and Kimina-Prover-Preview-Distill-7B ### Overview The image presents two line charts comparing the performance of different models based on "Pass budget (K)" and "Accuracy (%)". The left chart focuses on "Goedel-Prover-SFT" with and without "Apollo", while the right chart focuses on "Kimina-Prover-Preview-Distill-7B" with and without "Apollo". ### Components/Axes **Left Chart:** * **Title:** Performance of Goedel-Prover-SFT * **X-axis:** Pass budget (K) - log scale * Scale markers: 32, 80, 150, 306, 1.0K, 25.6K * **Y-axis:** Accuracy (%) * Scale markers: 58, 59, 60, 61, 62, 63, 64, 65 * **Legend:** Located in the bottom-right corner. * Blue line: Goedel-Prover-SFT * Orange line: Goedel-Prover-SFT + Apollo **Right Chart:** * **Title:** Performance of Kimina-Prover-Preview-Distill-7B * **X-axis:** Pass budget (K) * Scale markers: 0, 200, 400, 600, 800, 1000 * **Y-axis:** Accuracy (%) * Scale markers: 64, 66, 68, 70, 72, 74 * **Legend:** Located in the top-right corner. * Red line: Kimina-Prover-Preview-Distill-7B * Green line: Kimina-Prover-Preview-Distill-7B + Apollo ### Detailed Analysis **Left Chart (Goedel-Prover-SFT):** * **Goedel-Prover-SFT (Blue):** The line slopes upward, indicating increasing accuracy with a larger pass budget. * (32, 57.7%) * (80, 59.2%) * (150, 60.5%) * (306, 61.3%) * (1.0K, 62.2%) * (25.6K, 64.7%) * **Goedel-Prover-SFT + Apollo (Orange):** The line slopes upward, indicating increasing accuracy with a larger pass budget. * (32, 57.6%) * (80, 60.7%) * (150, 63.5%) * (306, 65.1%) **Right Chart (Kimina-Prover-Preview-Distill-7B):** * **Kimina-Prover-Preview-Distill-7B (Red):** The line slopes upward, indicating increasing accuracy with a larger pass budget. * (0, 63.2%) * (1000, 71.0%) * **Kimina-Prover-Preview-Distill-7B + Apollo (Green):** The line slopes upward, indicating increasing accuracy with a larger pass budget. * (0, 63.1%) * (100, 68.8%) * (200, 74.1%) * (300, 74.8%) ### Key Observations * In the Goedel-Prover-SFT chart, the "Apollo" addition consistently improves accuracy across all pass budget values. * The x-axis on the left chart is logarithmic, while the x-axis on the right chart is linear. * In the Kimina-Prover-Preview-Distill-7B chart, the "Apollo" addition significantly improves accuracy, especially at lower pass budget values. The green line plateaus after 200K. ### Interpretation The charts compare the performance of two different models, "Goedel-Prover-SFT" and "Kimina-Prover-Preview-Distill-7B", with and without the addition of "Apollo". The data suggests that adding "Apollo" generally improves the accuracy of both models. However, the impact of "Apollo" is more pronounced for "Kimina-Prover-Preview-Distill-7B", especially at lower pass budget values. The logarithmic scale on the left chart indicates that the pass budget has a diminishing return on accuracy for "Goedel-Prover-SFT". The Kimina model with Apollo plateaus quickly, suggesting a saturation point.

**Language Declaration:** The text in this image is entirely in English. No other languages are present. ## Line Charts: Performance Impact of "Apollo" on Prover Models ### Overview The image consists of two side-by-side line charts comparing the performance (Accuracy) against computational cost (Pass budget) for two different AI models: "Goedel-Prover-SFT" (left) and "Kimina-Prover-Preview-Distill-7B" (right). Both charts demonstrate the baseline model's performance compared to the model augmented with a system or method called "Apollo". ### Component Isolation: Left Chart (Goedel-Prover-SFT) #### Components/Axes * **Positioning:** Left half of the image. * **Title:** "Performance of Goedel-Prover-SFT" (Top center). * **Y-axis:** Labeled "Accuracy (%)". Linear scale with major gridlines at 58, 59, 60, 61, 62, 63, 64, and 65. * **X-axis:** Labeled "Pass budget (K) - log scale". The tick marks are angled and spaced logarithmically. The explicit labels are: 32, 80, 150, 306, 1.0K, 25.6K. * **Legend:** Located in the bottom-right corner of the chart area. * Blue line with circular markers: `Goedel-Prover-SFT` * Orange line with circular markers: `Goedel-Prover-SFT + Apollo` #### Detailed Analysis * **Trend Verification (Blue Line - Baseline):** The blue line slopes upward gradually across a massive span of the x-axis (from 32 to 25.6K). * Point 1: x = 32, y ≈ 57.6% * Point 2: x = 80, y ≈ 59.2% * Point 3: x = 1.0K, y ≈ 62.7% * Point 4: x = 25.6K, y ≈ 64.7% * **Trend Verification (Orange Line - With Apollo):** The orange line slopes upward steeply, achieving higher accuracy at much lower pass budgets, terminating early on the x-axis. * Point 1: x = 32, y ≈ 57.6% (Shares exact starting point with baseline) * Point 2: x = 80, y ≈ 60.7% * Point 3: x = 150, y ≈ 63.5% * Point 4: x = 306, y ≈ 65.1% --- ### Component Isolation: Right Chart (Kimina-Prover-Preview-Distill-7B) #### Components/Axes * **Positioning:** Right half of the image. * **Title:** "Performance of Kimina-Prover-Preview-Distill-7B" (Top center). * **Y-axis:** No explicit text label, but visually shares the "Accuracy (%)" metric from the left chart. Linear scale with major gridlines at 64, 66, 68, 70, 72, and 74. * **X-axis:** Labeled "Pass budget (K)". This is a **linear scale**, unlike the left chart. Major tick marks at 0, 200, 400, 600, 800, 1000. * **Legend:** Located in the top-right corner of the chart area. * Red line with circular markers: `Kimina-Prover-Preview-Distill-7B` * Green line with circular markers: `Kimina-Prover-Preview-Distill-7B + Apollo` #### Detailed Analysis * **Trend Verification (Red Line - Baseline):** The red line slopes upward gradually in a nearly straight line across the linear x-axis. * Point 1: x ≈ 32 (slightly right of 0), y ≈ 63.1% * Point 2: x ≈ 1024 (slightly past 1000), y ≈ 70.8% * **Trend Verification (Green Line - With Apollo):** The green line slopes upward very steeply, then begins to curve (concave down), showing rapid accuracy gains at low pass budgets. * Point 1: x ≈ 32, y ≈ 63.1% (Shares exact starting point with baseline) * Point 2: x ≈ 100, y ≈ 68.8% * Point 3: x ≈ 200, y ≈ 74.1% * Point 4: x ≈ 300, y ≈ 75.0% --- ### Key Observations 1. **Shared Origins:** In both charts, the baseline model and the Apollo-enhanced model start at the exact same accuracy for the lowest pass budget (approx. 32K). 2. **Drastic Efficiency Gains:** The addition of "Apollo" creates a significantly steeper learning/performance curve in both models. 3. **Scale Discrepancy:** The left chart uses a logarithmic scale for the X-axis to show the baseline model requiring up to 25.6K pass budget to reach ~64.7% accuracy. The Apollo version reaches higher accuracy (~65.1%) at a mere 306 pass budget. This is an efficiency gain of nearly two orders of magnitude. 4. **Higher Baseline:** The Kimina model (right) starts at a higher baseline accuracy (~63%) compared to the Goedel model (~57.6%). ### Interpretation The data overwhelmingly demonstrates that the "Apollo" method/system acts as a massive multiplier for computational efficiency (measured here as "Pass budget"). By reading between the lines, "Pass budget" likely refers to the number of attempts, samples, or tokens a theorem-proving model is allowed to generate or evaluate to find a correct solution. Without Apollo, scaling up the pass budget yields diminishing, slow returns (requiring logarithmic scaling on the left chart just to fit the baseline line). With Apollo, the models achieve superior accuracy using a fraction of the computational budget. For example, on the left chart, Apollo achieves in ~300 passes what the baseline cannot achieve in 25,000 passes. On the right chart, Apollo achieves 75% accuracy at 300 passes, while the baseline only reaches ~71% at over 1000 passes. Apollo appears to be a highly effective search heuristic, filtering mechanism, or reasoning enhancement that prevents the models from wasting computational budget on dead ends.

## Line Charts: Performance of Language Models ### Overview The image presents two line charts comparing the performance (Accuracy in %) of different language models against varying "Pass budget" (in thousands, K) on a logarithmic scale for the first chart. The first chart focuses on "Goedel-SFT" and "Goedel-SFT + Apollo", while the second chart focuses on "Kimina-Prover-Preview-Distill-7B" and "Kimina-Prover-Preview-Distill-7B + Apollo". ### Components/Axes **Chart 1: Performance of Goedel-SFT** * **X-axis:** Pass budget (K) - log scale. Markers at 32, 80, 150, 306, 1.0k, 25.6k. * **Y-axis:** Accuracy (%) - Scale from approximately 58% to 65%. * **Legend:** * Blue Line: Goedel-Prover-SFT * Orange Line: Goedel-Prover-SFT + Apollo **Chart 2: Performance of Kimina-Prover-Preview-Distill-7B** * **X-axis:** Pass budget (K) - Scale from 0 to 1000. Markers at 0, 200, 400, 600, 800, 1000. * **Y-axis:** Accuracy (%) - Scale from approximately 64% to 75%. * **Legend:** * Red Line: Kimina-Prover-Preview-Distill-7B * Green Line: Kimina-Prover-Preview-Distill-7B + Apollo ### Detailed Analysis or Content Details **Chart 1: Goedel-SFT Performance** * **Goedel-Prover-SFT (Blue Line):** The line slopes upward, indicating increasing accuracy with increasing pass budget. * At 32K: Approximately 58.5% accuracy. * At 80K: Approximately 60.2% accuracy. * At 150K: Approximately 61.8% accuracy. * At 306K: Approximately 62.5% accuracy. * At 1.0K: Approximately 62.7% accuracy. * At 25.6K: Approximately 64.5% accuracy. * **Goedel-Prover-SFT + Apollo (Orange Line):** The line initially rises sharply, then plateaus. * At 32K: Approximately 63.5% accuracy. * At 80K: Approximately 64.2% accuracy. * At 150K: Approximately 65.0% accuracy. * At 306K: Approximately 65.2% accuracy. * At 1.0K: Approximately 65.1% accuracy. * At 25.6K: Approximately 65.0% accuracy. **Chart 2: Kimina-Prover-Preview-Distill-7B Performance** * **Kimina-Prover-Preview-Distill-7B (Red Line):** The line slopes upward, but at a decreasing rate. * At 0K: Approximately 64.2% accuracy. * At 200K: Approximately 68.5% accuracy. * At 400K: Approximately 69.5% accuracy. * At 600K: Approximately 69.8% accuracy. * At 800K: Approximately 70.0% accuracy. * At 1000K: Approximately 70.2% accuracy. * **Kimina-Prover-Preview-Distill-7B + Apollo (Green Line):** The line rises sharply initially, then plateaus at a higher accuracy than the base model. * At 0K: Approximately 71.5% accuracy. * At 200K: Approximately 74.2% accuracy. * At 400K: Approximately 74.5% accuracy. * At 600K: Approximately 74.4% accuracy. * At 800K: Approximately 74.3% accuracy. * At 1000K: Approximately 74.2% accuracy. ### Key Observations * For both models, adding "Apollo" consistently improves accuracy, especially at lower pass budgets. * The improvement from "Apollo" diminishes as the pass budget increases, suggesting a point of diminishing returns. * The Goedel-SFT model shows a more consistent improvement with increasing pass budget compared to the Kimina model. * The Kimina model with Apollo starts at a significantly higher accuracy than the Goedel model with Apollo. ### Interpretation The charts demonstrate the impact of "Pass budget" and the "Apollo" enhancement on the accuracy of two different language models. The "Pass budget" likely represents the computational resources allocated to the model during a verification or training process. The "Apollo" component appears to be an additional module or technique that boosts performance, particularly when computational resources are limited. The diminishing returns observed with increasing pass budget and "Apollo" suggest that there's a trade-off between computational cost and accuracy gains. The initial steep increase in accuracy with "Apollo" indicates that it effectively leverages limited resources, while the plateau suggests that further investment in pass budget yields smaller improvements. The difference in the overall accuracy levels between the Goedel and Kimina models suggests inherent differences in their architectures or training data. The Kimina model, even without Apollo, achieves higher accuracy than the Goedel model, indicating a potentially more robust or efficient base model. The charts provide valuable insights for optimizing resource allocation and model selection based on desired accuracy levels and computational constraints.

## Line Charts: Performance Comparison of Two AI Models with and without "Apollo" Enhancement ### Overview The image displays two side-by-side line charts comparing the performance (accuracy) of two different AI models against an increasing "Pass budget." Each chart compares a base model against the same model enhanced with a component called "Apollo." The left chart uses a logarithmic scale for the x-axis, while the right chart uses a linear scale. ### Components/Axes **Common Elements:** * **Y-axis (Both Charts):** Labeled "Accuracy (%)". The scale is linear. * **X-axis (Both Charts):** Labeled "Pass budget (K)". The unit "K" likely denotes thousands. * **Left Chart:** Uses a **log scale**. Major tick marks are at 32, 80, 150, 306, 1.0K, and 21.6K. * **Right Chart:** Uses a **linear scale**. Major tick marks are at 0, 200, 400, 600, 800, and 1000. * **Legends:** Each chart has a legend identifying the two data series by color and model name. **Left Chart Specifics:** * **Title:** "Performance of Goedel-Prover-SFT" * **Legend (Located inside the plot area, bottom-right):** * Blue line with circle markers: "Goedel-Prover-SFT" * Orange line with circle markers: "Goedel-Prover-SFT + Apollo" **Right Chart Specifics:** * **Title:** "Performance of Kimina-Prover-Preview-Distill-7B" * **Legend (Located outside the plot area, top-right):** * Red line with circle markers: "Kimina-Prover-Preview-Distill-7B" * Green line with circle markers: "Kimina-Prover-Preview-Distill-7B + Apollo" ### Detailed Analysis **Left Chart: Goedel-Prover-SFT (Log Scale X-Axis)** * **Trend Verification:** Both lines show a positive, upward trend as the pass budget increases. The orange line ("+ Apollo") has a steeper initial slope than the blue line. * **Data Points (Approximate):** * **Goedel-Prover-SFT (Blue):** * At 32K: ~57.5% * At 80K: ~59.3% * At 1.0K: ~62.7% * At 21.6K: ~64.5% * **Goedel-Prover-SFT + Apollo (Orange):** * At 32K: ~57.5% (same starting point as blue) * At 80K: ~60.7% * At 150K: ~63.5% * At 306K: ~65.0% (final data point for this series) **Right Chart: Kimina-Prover-Preview-Distill-7B (Linear Scale X-Axis)** * **Trend Verification:** Both lines show a positive trend. The green line ("+ Apollo") exhibits a very sharp, near-vertical increase at low pass budgets before plateauing. The red line shows a more gradual, steady increase. * **Data Points (Approximate):** * **Kimina-Prover-Preview-Distill-7B (Red):** * At 0K: ~63.0% * At 1000K: ~70.8% * **Kimina-Prover-Preview-Distill-7B + Apollo (Green):** * At 0K: ~63.0% (same starting point as red) * At ~100K: ~68.8% * At 200K: ~74.0% * At 400K: ~75.0% (final data point for this series) ### Key Observations 1. **Apollo's Impact:** In both models, the "+ Apollo" variant significantly outperforms the base model at every measured pass budget beyond the starting point. 2. **Efficiency Gain:** The Apollo enhancement provides a much larger accuracy boost at **lower pass budgets**. This is especially dramatic in the right chart, where the green line reaches near-peak performance (~74%) at just 200K, while the base red line requires 1000K to reach only ~71%. 3. **Performance Ceiling:** The Kimina model with Apollo (green line) appears to hit a performance plateau around 75% accuracy after 400K, suggesting diminishing returns. The Goedel model with Apollo (orange line) does not show a clear plateau within its plotted range. 4. **Scale Context:** The left chart's log scale compresses the high end of the x-axis, making the performance gains of the Goedel model appear more linear. The right chart's linear scale clearly shows the rapid saturation of the Kimina+Apollo model. ### Interpretation This data demonstrates that the "Apollo" component is a highly effective enhancement for increasing the accuracy of these AI models, particularly in **low-compute or early-budget scenarios**. The primary benefit appears to be a dramatic improvement in **sample efficiency**—achieving high accuracy with a much smaller "pass budget." The difference in curve shapes between the two models suggests that Apollo's integration or effect may be architecture-dependent. The Kimina model (right) benefits from an extremely rapid initial gain, making it suitable for applications where the computational budget is severely constrained. The Goedel model (left) shows a more sustained, gradual improvement, which might be preferable in scenarios where scaling the budget further is possible and continued gains are valuable. The charts effectively argue that adding Apollo is not just a minor improvement but a fundamental shift in the performance-vs-budget curve, allowing these models to reach usable accuracy levels far sooner. The choice between the base and Apollo-enhanced versions would depend on the specific operational constraints (available compute/budget) and required accuracy thresholds.

## Line Charts: Performance Comparison of Prover Models with/without Apollo ### Overview The image contains two side-by-side line charts comparing the accuracy performance of two prover models (Goedel-Prover-SFT and Kimina-Prover-Preview-Distill-7B) with and without Apollo enhancements across different pass budgets. Both charts use accuracy (%) on the y-axis and pass budget (K) on the x-axis, with distinct scaling and data patterns. ### Components/Axes **Left Chart (Goedel-Prover-SFT):** - **Title:** "Performance of Goedel-Prover-SFT" - **X-axis:** "Pass budget (K) - log scale" (markers: 32, 80, 150, 306, 1.0K, 25.6K) - **Y-axis:** "Accuracy (%)" (range: 57.5% to 65%) - **Legend:** - Blue line: "Goedel-Prover-SFT" - Orange line: "Goedel-Prover-SFT + Apollo" - **Data Points:** - Blue line: (32K, 58%), (80K, 59.2%), (150K, 60.5%), (306K, 61.8%), (1.0K, 62.7%), (25.6K, 64.5%) - Orange line: (32K, 57.5%), (80K, 60.8%), (150K, 63.4%), (306K, 65%), (1.0K, 63.2%), (25.6K, 64.5%) **Right Chart (Kimina-Prover-Preview-Distill-7B):** - **Title:** "Performance of Kimina-Prover-Preview-Distill-7B" - **X-axis:** "Pass budget (K)" (markers: 0, 200, 400, 600, 800, 1000) - **Y-axis:** "Accuracy (%)" (range: 63.5% to 74.5%) - **Legend:** - Red line: "Kimina-Prover-Preview-Distill-7B" - Green line: "Kimina-Prover-Preview-Distill-7B + Apollo" - **Data Points:** - Red line: (0K, 63.5%), (200K, 66.2%), (400K, 67.8%), (600K, 69.1%), (800K, 70.3%), (1000K, 70.5%) - Green line: (0K, 63.5%), (200K, 74%), (400K, 72.5%), (600K, 68.7%), (800K, 66.5%), (1000K, 70.5%) ### Detailed Analysis **Left Chart Trends:** 1. **Goedel-Prover-SFT (Blue):** Shows a consistent upward trend, increasing from 58% at 32K to 64.5% at 25.6K. The log scale emphasizes performance gains at lower budgets. 2. **Goedel-Prover-SFT + Apollo (Orange):** Starts slightly below the base model (57.5% vs 58%) but surpasses it at 150K (63.4% vs 60.5%). Peaks at 306K (65%) before converging with the base model at 25.6K (64.5%). **Right Chart Trends:** 1. **Kimina-Prover-Preview-Distill-7B (Red):** Steady linear increase from 63.5% at 0K to 70.5% at 1000K, with no plateaus. 2. **Kimina-Prover-Preview-Distill-7B + Apollo (Green):** Sharp initial spike to 74% at 200K (10.5% improvement), followed by a decline to 66.5% at 800K, then recovery to 70.5% at 1000K. ### Key Observations 1. **Apollo's Impact on Goedel-Prover-SFT:** - Provides a 1.3% accuracy boost at 150K but requires 306K pass budget to reach peak performance. - Performance converges with the base model at higher budgets (25.6K), suggesting diminishing returns. 2. **Apollo's Impact on Kimina-Prover:** - Delivers a dramatic 10.5% accuracy boost at 200K but causes overfitting-like behavior (performance drop to 66.5% at 800K). - Final accuracy matches the base model at 1000K, indicating unstable gains. 3. **Model Behavior:** - Goedel-Prover-SFT shows stable scaling with budget. - Kimina-Prover-Preview-Distill-7B exhibits linear scaling without saturation. - Apollo's benefits are context-dependent: helpful for Goedel at mid-budgets but destabilizing for Kimina at higher budgets. ### Interpretation The data suggests that Apollo's effectiveness varies significantly between models: - For **Goedel-Prover-SFT**, Apollo acts as a performance enhancer at mid-range budgets but becomes redundant at higher budgets. - For **Kimina-Prover-Preview-Distill-7B**, Apollo introduces instability, causing performance degradation after initial gains. This could indicate overfitting to specific data patterns or architectural incompatibilities. The contrasting results highlight the importance of model-specific optimization when integrating enhancements like Apollo. While Apollo improves efficiency for Goedel-Prover-SFT, it may require careful tuning or architectural modifications to stabilize performance for Kimina-Prover-Preview-Distill-7B.