Turn intuition into a checkable proof scaffold before attempting full formalization.
The objective is not to fake completeness.
The objective is to expose structure, assumptions, and unresolved gaps as early as possible.
Procedure
1. State the Target Claim Precisely
Write the claim in mathematically checkable form. Include:
- quantified variables
- domain restrictions
- asymptotic regime if relevant
- probabilistic qualifiers
- exact conclusion Avoid:
- “behaves like”
- “roughly”
- “should converge”
- “is stable”
- “works well”
Bad:
The walk mixes quickly. Better: For any connected non-bipartite graph (G), the lazy random walk satisfies [ |P^t(v,\cdot)-\pi|_{\mathrm{TV}} \le \varepsilon ] for all [ t \ge C \gamma^{-1}\log(n/\varepsilon), ] where (\gamma) is the spectral gap.
2. Define Objects, Notation, and Assumptions
Explicitly define:
- spaces
- operators
- matrices
- measures
- norms
- graph families
- regularity assumptions
- independence assumptions
- asymptotic assumptions Every symbol used later must already be defined. Separate:
- standing assumptions
- temporary assumptions
- conjectural assumptions Example:
Assumption A1.
G is connected, undirected, and non-bipartite.
Assumption A2.
The transition matrix P is reversible with stationary distribution π.
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3. Decompose the Argument
Split the proof into:
* lemmas
* propositions
* reduction steps
* intermediate estimates
Prefer:
Main theorem
├── Lemma 1: spectral estimate
├── Lemma 2: concentration bound
└── Lemma 3: coupling argument
over a single monolithic proof sketch.
Each sub-result should have:
* explicit input assumptions
* explicit output statement
* clear dependency chain
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4. For Each Stage, Specify the Proof Skeleton
For every lemma or stage, include the following fields.
Goal
State exactly what must be shown.
Candidate Technique
Examples:
* spectral decomposition
* coupling
* martingale inequality
* compactness argument
* variational characterization
* induction
* contradiction
* perturbation analysis
* concentration inequality
* optimal transport duality
Dependencies
List:
* previous lemmas
* external theorems
* hidden assumptions
* required estimates
Likely Weak Points
Identify:
* unjustified limit interchange
* missing regularity
* non-uniform constants
* dependence assumptions
* dimension blow-up
* unproved concentration
* unjustified approximation
Do not hide uncertainty.
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5. Mark Gaps Explicitly
Any unresolved step must be labeled as a gap.
Use labels such as:
GAP 1.
Need a uniform lower bound on conductance.
GAP 2.
The coupling argument only works for reversible chains.
GAP 3.
The concentration step currently assumes sub-Gaussian tails.
Never present an unproved step as established fact.
Separate:
* proved step
* heuristic argument
* conjectural extension
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6. Compare Competing Proof Strategies
If multiple approaches exist, compare them briefly.
Example:
Strategy A: Spectral method
- cleaner constants
- requires reversibility
Strategy B: Coupling method
- more probabilistic
- may extend to directed graphs
- weaker quantitative bounds
Do not expand every branch fully.
Focus on the most plausible route.
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7. End with the Minimal Next Action
Conclude with the smallest concrete step needed to advance the proof.
Good:
Next step:
Prove that the random-walk refinement operator is contractive under the chosen diffusion metric.
Bad:
Next step:
Finish the proof.
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Output Format
Use the following structure.
# Target Claim
...
# Assumptions and Definitions
...
# Proof Structure
## Lemma 1
### Goal
...
### Technique
...
### Dependencies
...
### Weak Points
...
## Lemma 2
...
# Gaps
...
# Alternative Strategies
...
# Minimal Next Step
...
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Rules
* Prefer explicit assumptions over implicit context.
* Prefer incomplete honesty over fake rigor.
* Distinguish theorem, heuristic, and conjecture.
* Avoid narrative filler.
* Avoid undefined notation.
* Avoid skipping dependency chains.
* Do not claim a result is “obvious” without stating why.
* Do not compress multiple logical steps into one sentence.
* Keep the scaffold minimal but structurally rigorous.