---
name: theory-assumption-extractor
description: Extract explicit and hidden assumptions from a proof, model, or empirical argument. Use for theorem reading, paper review, proposal critique, debugging proofs, conclusions that appear stronger than the stated setup, or determining which premises are logically necessary, proof conveniences, and where the argument breaks.
---

# Assumption Extractor

Make hidden premises visible before trusting the conclusion.

A strong result often depends on assumptions that are:
- implicit;
- buried in notation;
- hidden inside a theorem citation;
- or introduced indirectly through the proof technique.

The goal is not merely to list assumptions.  
The goal is to determine:
- which assumptions are structurally necessary;
- which are artifacts of the proof;
- and what fails if they are removed.

---

# Procedure

## 1. State the Target Claim

Write the conclusion precisely.

Bad:
```text
The method converges.
```

Better:
```text
The estimator converges almost surely to the true parameter
under increasing sample size.
```

The claim should specify:
- variables;
- regime;
- convergence notion;
- probabilistic qualifier;
- comparison target.

---

## 2. Extract Explicit Assumptions

List assumptions stated directly in the theorem, model, or experiment.

Example:
```text
Explicit assumptions:
- graph is connected;
- loss function is convex;
- samples are IID;
- Markov chain is reversible.
```

Separate:
- mathematical assumptions;
- statistical assumptions;
- computational assumptions;
- data-generation assumptions.

---

## 3. Search for Hidden Assumptions

Inspect every proof step, approximation, and empirical claim.

Common hidden assumptions include:

### Regularity Assumptions
- smoothness;
- Lipschitz continuity;
- differentiability;
- bounded gradients.

### Structural Assumptions
- convexity;
- sparsity;
- low rank;
- separability;
- stationarity.

### Probabilistic Assumptions
- independence;
- exchangeability;
- concentration;
- sub-Gaussian tails;
- ergodicity.

### Identifiability Assumptions
- uniqueness of optimum;
- invertibility;
- sufficient excitation;
- non-degenerate covariance.

### Asymptotic Assumptions
- large-sample regime;
- vanishing noise;
- fixed dimension;
- infinite-time horizon.

### Empirical Assumptions
- train/test distributions match;
- benchmark labels are reliable;
- preprocessing does not leak information.

---

## 4. Identify Where Each Assumption Enters

Do not merely list assumptions.  
State where the argument actually uses them.

Example:
```text
Convexity is required when applying Jensen's inequality
during the convergence proof.

Independence is used only in the concentration step.
```

This separates:
- logically necessary assumptions;
- proof-specific assumptions.

---

## 5. Classify Assumptions

For each assumption, label it as:

### Essential
Removing it invalidates the claim itself.

Example:
```text
Connectivity is essential for global mixing.
```

### Proof Convenience
Used to simplify analysis but may not be fundamentally required.

Example:
```text
Bounded degree simplifies concentration bounds
but may not be strictly necessary.
```

### Likely Overkill
Stronger than needed.

Example:
```text
Global Lipschitz continuity may only be needed locally.
```

---

## 6. Test Assumption Removal

Ask:
```text
What breaks if this assumption is removed?
```

Be specific.

Bad:
```text
The proof becomes harder.
```

Better:
```text
Without reversibility, the spectral decomposition step fails
because orthogonal eigenbasis arguments no longer apply.
```

---

## 7. Suggest Weaker Alternatives

Where possible, replace strong assumptions with weaker ones.

Examples:
| Strong Assumption | Possible Relaxation |
|---|---|
| IID samples | Mixing process |
| Global convexity | Local convexity |
| Bounded support | Finite variance |
| Exact sparsity | Approximate sparsity |
| Deterministic graph | Random graph ensemble |

Do not claim the weaker version works unless the proof pathway is plausible.

---

# Rules

- Distinguish logical necessity from proof convenience.
- Hidden assumptions inside cited theorems still count.
- If removing an assumption breaks the argument, state exactly where.
- Avoid treating asymptotic assumptions as automatically realistic.
- Separate:
  - mathematically sufficient;
  - empirically plausible;
  - computationally feasible.
- Do not confuse “common in literature” with “necessary.”

---

# Output Format

```text
# Target Claim

...

# Explicit Assumptions

- ...

# Hidden Assumptions

| Assumption | Where It Enters | Essential or Convenient | What Breaks Without It |
|---|---|---|---|
| ... | ... | ... | ... |

# Likely Over-Strong Assumptions

- ...

# Possible Weaker Alternatives

| Original Assumption | Possible Relaxation |
|---|---|
| ... | ... |

# Main Structural Dependency

State the single assumption the argument depends on most heavily.

# Risk Assessment

Which assumptions are:
- mathematically fragile;
- empirically unrealistic;
- difficult to verify in practice.
```