DeepSeek Math & STEM Reasoning: Proofs, Calculations & Science

Prove the following statement using thinking mode (effort=max).
Show your COMPLETE reasoning, including false starts and corrections.

STATEMENT: [Theorem or proposition to prove]

REASONING REQUIREMENTS:
1. RESTATE the theorem in your own words
2. IDENTIFY the proof strategy (direct, contradiction, induction, etc.)
3. STATE assumptions and definitions explicitly
4. DERIVE each step, justifying it with the previous step, a definition, or a known theorem
5. CHECK each step: "Does this follow logically? What could go wrong?"
6. If you encounter a contradiction or dead end, backtrack and try another approach
7. VERIFY the final proof against edge cases

Your reasoning_content should show the full exploration process.
Your final content should present the polished proof.

Verify the following proof. Use reasoning mode to check each step.

THEOREM: [Statement]
PROOF: [The proof to verify]

In your reasoning:
1. For EACH step, determine:
   - What is being claimed?
   - What justifies it? (previous step, definition, theorem?)
   - Is the justification valid?
2. Flag any step where:
   - The justification is missing or implicit
   - The logical connection is unclear
   - An assumption is unstated
3. If you find an error, explain:
   - What the error is
   - Why it breaks the proof
   - Whether the proof can be salvaged

Final output:
- VALID: The proof is correct (explain why)
- FLAWED: The proof has an error at step [N] (explain the error)
- INCOMPLETE: The proof is missing justification for steps [N, M, ...]

Solve the following problem. Show your work completely.

PROBLEM: [Problem statement with all given values and units]

REASONING:
1. IDENTIFY what's being asked — restate the problem
2. LIST known values with units
3. IDENTIFY the relevant formulas or principles
4. SOLVE step by step, keeping units throughout
5. CHECK dimensional consistency at each step
6. VERIFY the answer: plug it back into the original problem
7. State the answer with appropriate precision and units

If using approximations:
- State the approximation explicitly
- Estimate the error introduced

Analyze this engineering problem using reasoning mode.

SYSTEM: [Description of physical system with parameters]

ANALYSIS STEPS:
1. DRAW the system mentally — identify all forces/components
2. WRITE governing equations (conservation laws, constitutive relations)
3. SIMPLIFY with justified assumptions (neglect friction? small angle?)
4. SOLVE the simplified system analytically or numerically
5. CHECK boundary conditions and limiting cases
6. VERIFY with dimensional analysis
7. STATE the result with engineering context (is this reasonable?)

For each assumption: "I assume [X] because [justification].
This assumption is valid when [condition]. If [condition] is violated,
the analysis must be revised."

Analyze this experimental data using reasoning mode.

DATA: [Dataset with variables, units, uncertainties]

ANALYSIS:
1. IDENTIFY relationships: What variables depend on what?
2. PLOT mental trends: increasing, decreasing, periodic, random?
3. SELECT model: linear, exponential, power law? Justify your choice.
4. FIT parameters and report uncertainties
5. ASSESS goodness of fit: R², residuals analysis
6. IDENTIFY outliers and explain whether to include or exclude
7. DRAW conclusions: What does this data tell us? What CAN'T it tell us?
8. SUGGEST follow-up experiments to resolve ambiguities

For statistical tests:
- State the null hypothesis
- Report the test statistic and p-value
- Interpret in context (not just "p < 0.05 therefore significant")

Synthesize these [N] scientific papers on [topic].
Use reasoning mode to identify patterns and contradictions.

For each paper:
- Key finding (1 sentence)
- Methodology (1 sentence)
- Stated limitations (1 sentence)

SYNTHESIS:
1. CONSENSUS: What do all/most papers agree on?
2. CONTRADICTIONS: Where do findings conflict?
   - What might explain the disagreement? (methodology, sample, era)
3. GAPS: What obvious questions do NONE of these papers address?
4. EVOLUTION: If papers span years, how has understanding changed?
5. STRENGTH OF EVIDENCE: Which findings are most robust? Why?

response = client.chat.completions.create(
    model="deepseek-v4-pro",
    messages=[{"role": "user", "content": "Prove that sqrt(2) is irrational."}],
    reasoning_effort="max",
    extra_body={"thinking": {"type": "enabled"}}
)

reasoning = response.choices[0].message.reasoning_content
proof = response.choices[0].message.content

# Verify reasoning quality
checks = [
    ("Assumes rationality", "assume" in reasoning.lower()),
    ("Derives contradiction", "contradiction" in reasoning.lower()),
    ("Handles fraction reduction", "lowest terms" in reasoning.lower() or "coprime" in reasoning.lower()),
    ("Considers both sides", "even" in reasoning.lower() and "odd" in reasoning.lower()),
]

for check, passed in checks:
    print(f"{'✓' if passed else '✗'} {check}")