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:
The method converges.
Better:
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:
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:
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:
Connectivity is essential for global mixing.
Proof Convenience
Used to simplify analysis but may not be fundamentally required.
Example:
Bounded degree simplifies concentration bounds
but may not be strictly necessary.
Likely Overkill
Stronger than needed.
Example:
Global Lipschitz continuity may only be needed locally.
6. Test Assumption Removal
Ask:
What breaks if this assumption is removed?
Be specific.
Bad:
The proof becomes harder.
Better:
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
# 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.