Can GPT5 Pro Outperform GPT-4? The $200 vs $20 Math Proof

TL;DR: Explore a real-world test comparing GPT-5 Pro and GPT-4 in solving a complex theorem in random matrix theory, revealing that the more expensive GPT-5 Pro not only failed to outperform its cheaper counterpart but also produced confident yet incorrect answers.

• Why This Test Matters for AI Users
• The Researcher’s Goal: Automate Theorem Proving
• The Mathematical Challenge: A Theorem in Random Matrix Theory
• Notation and Context Provided in the Prompt
• GPT-4 (ChatGPT Plus) Performance: $20/Month Model
• Thinking Time and Error Detection
• Proof Strategy: Problem Decomposition
• Clarity and Pedagogical Value
• GPT5 Pro Performance: $200/Month Model
• Extended Thinking Time and Initial Approach
• Redundant Restatements Waste Time
• Critical Failure: Hallucinated Equation
• False Confidence vs. Actual Utility
• Head-to-Head Comparison: $20 vs $200 Model
• Key Insight: AI Reasoning Is Still Unreliable in Technical Domains
• Why Extended Thinking Time Didn’t Help
• The “AI Slop” Problem: When Models Fake Understanding
• Practical Advice for Researchers Using AI
• Is GPT5 Pro Ever Worth $200/Month?
• Broader Implications for AI Development
• What This Means for “Can GPT5 Pro” Questions
• Future Testing Plans
• Final Takeaways: What You Should Do Next
• Conclusion: The $20 Model Won This Round

In a real-world test pitting the premium $200-per-month GPT5 Pro against the standard $20-per-month GPT-4 (ChatGPT Plus), a mathematics PhD researcher put both AI models to the ultimate challenge: proving a complex theorem in random matrix theory. The goal? To determine whether paying 10 times more delivers 10 times the performance—or if the upgrade is overhyped. The results were surprising, revealing critical insights into AI reasoning, hallucination risks, and the true value of extended thinking time.

This article dives deep into every detail of that experiment, dissecting methodology, outputs, errors, and practical implications for researchers, developers, and AI users. Spoiler: the expensive model didn’t just underdeliver—it hallucinated nonsense while pretending to succeed.

Why This Test Matters for AI Users

As AI models become integral to research, coding, and problem-solving, users face a growing dilemma: is upgrading to premium tiers worth the cost? With GPT5 Pro priced at $200/month—ten times the cost of ChatGPT Plus at $20/month—expectations are high. This real-world test in advanced mathematics offers a rare, granular look at whether higher cost translates to higher capability, especially in domains requiring precision, logic, and deep reasoning.

The Researcher’s Goal: Automate Theorem Proving

The tester, a PhD candidate in mathematics, sought to offload part of their research workload to AI. Specifically, they wanted to know: “Can ChatGPT prove this theorem for me so I can focus on other aspects of my work?” This isn’t about replacing human insight—it’s about augmenting research efficiency by delegating well-defined, technical tasks to AI.

As the researcher noted: “It’s always good if I can allocate some work to ChatGPT and then I can work on something else. Kind of helps in the research process.”

The Mathematical Challenge: A Theorem in Random Matrix Theory

The test centered on a non-trivial problem from random matrix theory, specifically involving the circular unitary ensemble (CUE). The user provided:

• Standard notation used in the field
• Known theorems as background
• A final theorem to be proven

The prompt was comprehensive: “I’m going to give you some known theorems and some notation that we’re going to use across the whole problem. And I’d like you to prove the final theorem that I’m stating to you.”

This setup tested the AI’s ability to integrate context, apply mathematical logic, and produce a valid proof—not just regurgitate information.

Notation and Context Provided in the Prompt

The prompt included detailed mathematical notation, which the researcher acknowledged might seem “confusing if you don’t do maths.” This ensured both models operated with the same foundational knowledge. Crucially, the prompt contained a typo: certain terms involving the variable “z” were accidentally omitted in one section, though they appeared correctly in earlier theorems.

This unintentional error became a key test of the models’ attention to detail and contextual understanding.

GPT-4 (ChatGPT Plus) Performance: $20/Month Model

Thinking Time and Error Detection

The $20/month model (referred to as “GPT Plus”) took 4 minutes to generate its response. Remarkably, it detected the missing “z” terms in the prompt, stating: “You’re missing the scale factor Z.”

This demonstrated strong contextual awareness—recognizing an inconsistency by cross-referencing the provided theorems. As the researcher noted: “That’s quite impressive to be fair.”

Proof Strategy: Problem Decomposition

GPT-4 approached the proof by splitting the main expression into two parts: one involving variable “z” and another involving “w.” Due to symmetry in the problem, solving one effectively solves both. This is a legitimate and efficient mathematical strategy.

The model justified this by referencing differentiation of the kernel (the core component inside a determinant), which determines key numerical coefficients in the final formula.

Clarity and Pedagogical Value

While the proof wasn’t “super clear,” the researcher admitted: “I was pretty impressed… I felt like I got some understanding.” However, it lacked depth for a novice: “It hasn’t really explained it to a dum dum.” Thus, it offered moderate utility as a research aid—not a standalone solution, but a useful starting point.

GPT5 Pro Performance: $200/Month Model

Extended Thinking Time and Initial Approach

GPT5 Pro took 42 minutes to respond—over 10 times longer than GPT-4. It adopted a structured approach: first proving two lemmas (intermediate results) to build toward the main theorem. Like GPT-4, it split the problem into “z” and “w” components, leveraging symmetry.

Redundant Restatements Waste Time

A significant portion of GPT5 Pro’s output merely repeated information already given in the prompt. The researcher criticized this: “This whole section… was all in the actual prompt itself. So to me that’s not actually that useful. It’s just regurgitating exactly what I’ve already asked it.”

This redundancy consumed valuable response space without adding insight—especially problematic given the extended processing time.

Critical Failure: Hallucinated Equation

The most damning flaw appeared in Step 4 of GPT5 Pro’s proof. It presented an equation described as “complete mumbo jumbo” and “AI slop hallucination to the max.”

Specifically, the model claimed:

“Then the inner sum in (9) is exactly [nonsense expression involving s^1 = 1].”

Since “s” is defined as a whole number, “s^1 = 1” is mathematically incorrect unless s=1—a condition not stated. Worse, the equation contained multiple notational errors rendering it meaningless. The model invented a false step to pretend it had completed the proof.

False Confidence vs. Actual Utility

Unlike GPT-4, which offered a plausible (if incomplete) path, GPT5 Pro pretended to succeed while failing fundamentally. The researcher emphasized: “It pretends that it can prove it… [but] fundamentally doesn’t manage to prove the identity.”

At best, its response was “only useful in the same sense as a search engine”—providing general ideas, not trustworthy reasoning.

Head-to-Head Comparison: $20 vs $200 Model

The table below summarizes key differences observed in the test:

Key Insight: AI Reasoning Is Still Unreliable in Technical Domains

Both models showed that AI cannot yet be trusted as an autonomous research assistant in advanced mathematics. As the researcher put it: “It’s not consistently good. A lot of the time it’s really not that good. And then you should occasionally get a glimpse and you’re like, ‘Wow, that’s really really useful.’”

The critical takeaway: always verify AI outputs in technical contexts. Hallucinations can be subtle and embedded within otherwise reasonable text—like GPT5 Pro’s single false equation amid a structured proof.

Why Extended Thinking Time Didn’t Help

GPT5 Pro’s 42-minute runtime suggested deeper analysis, but the output revealed diminishing returns on compute investment. Instead of higher-quality reasoning, the model produced:

• Redundant restatements
• Over-engineered structure (lemmas) without substance
• A critical hallucination masked as progress

This raises concerns about AI scalability: if 10x more inference compute yields similar (or worse) results, the path to truly reliable AI reasoning may require more than just increased processing time.

The “AI Slop” Problem: When Models Fake Understanding

The term “AI slop” perfectly describes GPT5 Pro’s failure mode: generating fluent, structured text that looks correct but contains fatal errors. In academic or research settings, this is especially dangerous because:

• It mimics human proof structure
• It uses correct terminology
• It hides errors in technical notation

Only a domain expert would spot the flaw—making such outputs worse than useless for non-experts who might accept them at face value.

Practical Advice for Researchers Using AI

Based on this experiment, here’s how to use AI responsibly in technical work:

1. Never treat AI as a final authority—always verify critical steps.
2. Use AI for idea generation, not proof validation. As the researcher said, it’s “better than a standard search engine” for sparking directions.
3. Beware of verbose outputs—length ≠ quality. Redundancy often masks lack of insight.
4. Test for hallucination** by introducing subtle errors (like the missing “z”) to see if the model catches them.
5. Don’t assume premium = better**—validate performance on your specific use case.

Is GPT5 Pro Ever Worth $200/Month?

For this mathematical proof task, no. The researcher concluded: “I expected like the model to be even one and a half times better. I don’t think it’s showing that much improvement.”

However, the researcher plans to continue testing with more problems: “I will keep sending these to my friend… to give a better idea on an actual long-term sustained use.” This suggests GPT5 Pro might excel in other domains (e.g., code generation, long-form analysis), but in precision-dependent math, it failed to justify its cost.

Broader Implications for AI Development

This test highlights a core challenge in AI: scaling compute doesn’t automatically scale correctness. The fact that both models used similar (flawed) strategies suggests underlying architectural limitations. Until models can:

• Avoid regurgitating prompts
• Admit uncertainty instead of hallucinating
• Provide truly novel, verifiable insights

…their role in research will remain advisory, not authoritative.

What This Means for “Can GPT5 Pro” Questions

When users ask, “Can GPT5 Pro handle advanced math?”, the answer is nuanced: It can attempt it, but you cannot trust its conclusions without expert review. The model’s confidence is not correlated with accuracy—a dangerous combination in technical fields.

For now, GPT-4 remains the better value** for most users, including researchers. Save the $180/month unless you have evidence GPT5 Pro excels in your specific workflow.

Future Testing Plans

The researcher intends to conduct more side-by-side tests: “So long as [my friend] is willing to process the queries… I can give a better idea on long-term sustained use.” This ongoing evaluation is crucial—single-task tests can’t capture full capability, but they reveal critical failure modes.

Final Takeaways: What You Should Do Next

Conclusion: The $20 Model Won This Round

In a direct comparison of mathematical theorem proving, the $20/month GPT-4 outperformed the $200/month GPT5 Pro in accuracy, efficiency, and trustworthiness. While neither model delivered a complete, verifiable proof, GPT-4 avoided catastrophic hallucinations and provided actionable insights—making it the clear winner for research support.

As AI continues to evolve, users must stay skeptical, verify outputs, and remember: more expensive doesn’t mean more correct. For now, save your money—and your research integrity—by sticking with proven, cost-effective tools until premium models demonstrate consistent, measurable superiority.

Can GPT5 Pro Outperform GPT-4? The $200 vs $20 Math Proof

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