The problem
Classification, not just calculation
Existence and uniqueness (A3-25-P1-E3) ↗ produced one of the sharpest gaps in the entire benchmark. It begins with
(y″ + yʳ) cos t = 1, r ∈ ℤand later replaces the nonlinear term with
(y″ + |y|ᵅ) cos t = 1, α ∈ ℝ.The task is not merely to find an explicit solution. A complete answer must distinguish local from global existence, local uniqueness from uniqueness of the maximal solution, domain components separated by the poles of sec t, even from odd powers, time-reflection symmetry, Lipschitz from non-Lipschitz regimes, and piecewise solutions joined at zeros.
That theoretical layering—more than the algebra itself—is what separates the models.
Observed scores
No overlap with the rest of the sample
The current matched panel contains 24 responses from 16 model-mode configurations. Four configurations have three repetitions; the remaining twelve have one.
| Model | Runs | Mean | Min–max |
|---|---|---|---|
| GPT-5.6 Sol | 3 | 95.83 | 88.00–100 |
| Gemini 3.5 Flash | 3 | 74.58 | 72.50–77.25 |
| Claude Fable 5 | 3 | 73.25 | 70.75–75.25 |
| Gemini 3.1 Pro Preview | 3 | 66.08 | 62.00–70.50 |
| DeepSeek V4 Pro | 1 | 64.75 | — |
| Grok 4.3 | 1 | 50.25 | — |
| Grok 4.5 | 1 | 45.00 | — |
| DeepSeek V4 Flash | 1 | 42.50 | — |
| DeepSeek V4 Flash no-thinking | 1 | 41.25 | — |
| OpenAI o3 | 1 | 40.50 | — |
| Mistral Small 4 | 1 | 40.00 | — |
| Mistral Medium 3.5 | 1 | 36.50 | — |
| Gemma 4 31B IT | 1 | 36.00 | — |
| Claude Sonnet 4.5 | 1 | 31.00 | — |
| OpenAI GPT-4.1 | 1 | 27.25 | — |
| Claude Haiku 4.5 | 1 | 23.50 | — |
Data & methodSee Atlas A3-25-P1-E3 ↗ for the response-level evidence and download the current-panel exercise-score CSV ↗. The table groups repetitions by configuration.
The unweighted mean across the 16 model-mode configurations is 49.27/100. GPT-5.6 Sol leads the second configuration by 21.25 points.
In the current matched panel, none of the three GPT-5.6 Sol runs overlaps with any of the twenty-one responses produced by the other models.
Scientific control
The exercise also challenged the reference
Before interpreting the gap, one methodological fact must be made explicit. The original official solution contained incomplete or incorrect conclusions in the nonlinear regimes. It did not properly distinguish odd powers r ≥ 3 from even powers, treated the loss of Lipschitz continuity for 0 < α < 1 as sufficient evidence of non-uniqueness, and did not fully formalize global continuation and matching for α = 1.
The original adjudication batch contained 20 responses. The table now shows the 24 responses in the current 16-configuration panel and excludes historical or out-of-panel responses, including Gemini 3.1 Flash-Lite. Every included response uses the corrected reference and frozen rubric.
The initial reference did not support all nonlinear conclusions.
The mathematics was reviewed and corrected in a versioned authority.
Every affected response was judged again under the same corrected rubric.
The scores in this article are exclusively the post-correction results. GPT-5.6 Sol's lead is therefore not an artifact of the older reference or of inconsistent criteria.
A scientific benchmark must not treat its reference as infallible. When a mathematical conflict emerges, the reference must be corrected, versioned and applied again to every affected response.
Rubric decomposition
Where the gap is produced
The exercise has four subproblems: (a) existence, uniqueness and continuation for integer r; (b) time-reflection symmetries; (c) an explicit solution for r = 1; and (d) the extension to |y|ᵅ.
| Subproblem | GPT-5.6 Sol | Other current-panel responses |
|---|---|---|
| (a) Global classification | 85.19 | 43.61 |
| (b) Symmetries | 100 | 84.94 |
| (c) Linear solution | 100 | 70.24 |
| (d) |y|ᵅ regimes | 99.43 | 26.64 |
Data & methodSubproblem scores are rubric-weighted aggregates from the adjudicated responses in Atlas A3-25-P1-E3 ↗; aggregation follows the benchmark method ↗.
The decisive separation occurs in part (d), followed by the global classification in part (a). Gemini 3.5 Flash and Claude Fable 5—the next two configurations—score 100% on both the symmetries and the explicit linear solution. They do not lose because they cannot manipulate the formula. They lose when they must decide which theorem applies and what truly follows from its hypotheses.
Theoretical obstacles
Four distinctions that determine the solution
The equation is not defined on the whole real line
At every point where cos t = 0, the original equation would require 0 = 1. No solution can cross t = π/2 + kπ. On each domain component Iₖ, the equation becomes y″ = sec t − yʳ. A statement of “global” existence must specify whether it means all of ℝ—which is impossible—or the whole component Iₖ.
Many responses recognize local existence but do not distinguish uniqueness of the maximal solution, existence across the full component and impossibility of crossing the forcing poles.
Even and odd powers have different dynamics
For odd r ≥ 3, the term −yʳ is restorative: it pushes downward for large positive y and upward for large negative y. This prevents interior blow-up and permits continuation across the whole component. For even r, the term remains negative on both sides and may amplify escape toward negative values, so global existence is not guaranteed.
The 88/100 GPT-5.6 Sol run makes this exact mistake: it states generically that superlinear powers may blow up and fails to separate odd from even. The two other runs give the complete classification and score 99.5 and 100. Even the strongest model is not immune to the shortcut “superlinear implies possible blow-up.”
Non-Lipschitz does not automatically mean non-unique
For 0 < α < 1, |y|ᵅ is not locally Lipschitz at y = 0. Picard–Lindelöf therefore cannot be applied directly at that point. It does not follow that the solution is necessarily non-unique.
the theorem's hypotheses faildoes not implythe theorem's conclusion is falseMany answers turn the loss of Lipschitz continuity into an automatic diagnosis of non-uniqueness. The adjudicated reference instead requires analysis of this specific equation and concludes that uniqueness persists even for data with y(t₀) = 0, together with continuation across the full component. GPT-5.6 Sol is the only configuration to handle this step correctly in every repetition.
The α = 1 solution is global but piecewise
When α = 1, the equation contains |y|. Where y > 0, it becomes y″ + y = sec t; where y < 0, it becomes y″ − y = sec t. The solution must use two different linear dynamics and be matched whenever it crosses zero.
Because |y| is globally Lipschitz, the matching is unique. Both y and y′ remain continuous at crossings, and the solution extends through the entire component. Many responses solve only the positive branch or mention a change of form without constructing and justifying the negative branch. The two best GPT-5.6 Sol runs include the matching conditions, uniqueness and maximal domain.
What the item measures
Symbolic calculation is not the decisive skill
The explicit formula in part (c) is not immediate:
y(t) = (a + log|cos t|) cos t + (b + t) sin t.Yet the strongest non-Sol models also derive it correctly. Separation instead comes from six forms of theoretical control:
- Distinguishing a sufficient condition from a necessary one.
- Not confusing the failure of a theorem with the failure of its conclusion.
- Recognizing the qualitative structure of the vector field.
- Classifying growth according to exponent parity.
- Handling piecewise solutions without losing uniqueness or domain.
- Declaring precisely what “global” means.
The item therefore discriminates between symbolic calculation and rigorous control of the solution's mathematical structure.
Interpretation
A strong item result is not a standalone leaderboard
One exercise cannot establish the general superiority of a model. Some configurations have only a single response, and GPT-5.6 Sol itself ranges from 88 to 100.
Recent benchmark research distinguishes the discriminating power of an item from the stability of an overall ranking. An exercise can be highly informative without representing the entire domain. Benchmark² ↗ explicitly proposes measures of item discriminability and consistency; Quantifying Ranking Uncertainty in LLM Benchmarks ↗ emphasizes representing uncertainty instead of reporting point estimates alone.
Correct reading. This is an item-level case study, not an independent model ranking.
On this specific problem, GPT-5.6 Sol is the only configuration that consistently combines explicit calculation, qualitative analysis, global classification and rigorous treatment of non-Lipschitz regimes.
Conclusion
The hardest exercise may be the one that asks when a formula is valid
Almost every model solves at least part of the equation. The observed gap comes from knowing precisely where the solution exists, which hypotheses support uniqueness and what kind of continuation is legitimate.
The separation is not merely finding a formula. It is controlling the formula's entire mathematical structure.
References
Sources and further reading
- Bortot, F. (2026). Existence and uniqueness, A3-25-P1-E3 ↗. UniAIBench Atlas record.
- Bortot, F. (2026). Beyond Correctness, §6.2 ↗. Benchmark paper.
- Qian, Q. et al. (2026). Benchmark²: Systematic Evaluation of LLM Benchmarks ↗. arXiv.
- Neuhof, B., & Benjamini, Y. (2026). Quantifying Ranking Uncertainty in LLM Benchmarks ↗. arXiv.