AI RNG: Practical Systems That Ship
Mathematical writing rewards confidence, but it punishes unearned certainty. A proof can look clean and still be wrong because one definition shifted, one quantifier was mishandled, or one case was silently assumed away. AI can help you draft and organize proofs, but only if you use a workflow that keeps correctness in charge.
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This workflow treats proof writing like engineering: track assumptions, isolate dependencies, verify boundary cases, and only then polish the exposition.
Start by pinning the statement to definitions
Before you prove anything, rewrite the theorem in your own words and attach every symbol to a definition. Many proof failures begin with ambiguity.
A practical pre-proof checklist:
- Define every object and space that appears in the statement
- State every hypothesis explicitly, even if it feels obvious
- Identify which conclusions are local, global, or existence claims
- Note which theorems or lemmas you intend to rely on
If you use AI at this stage, ask it to rewrite the statement with explicit definitions and to list the minimum assumptions needed. Then you decide which assumptions are permitted.
Build an assumption ledger
An assumption ledger is a short list of facts you are allowed to use. It keeps the proof honest, especially when AI drafts intermediate steps.
Include:
- Definitions and conventions
- Standing hypotheses from the theorem
- Known lemmas you will invoke
- Constraints on parameters and domains
When AI proposes a step, you check whether it uses only the ledger. If it uses something else, it must either be added explicitly as a new hypothesis or replaced with a valid argument.
Draft a dependency outline before full prose
Many proofs become difficult because the dependency structure is implicit. A simple outline makes the structure visible.
A useful outline includes:
- The main claim
- A short chain of subclaims that imply the main claim
- The lemmas needed for each subclaim
- Where each hypothesis is used
This can be captured as a small table.
| Target claim | Depends on | Uses which hypotheses | Verification check |
|---|---|---|---|
| Main theorem | Lemma A, Lemma B | H1, H2 | Boundary cases, uniqueness |
| Lemma A | Definition D, inequality | H1 | Check extreme parameters |
| Lemma B | Compactness argument | H2 | Confirm topology assumptions |
The point is not bureaucracy. The point is to prevent hidden leaps.
Write the proof in proof obligations
Instead of writing a long narrative immediately, write in obligations: small steps that must each be justified.
A helpful pattern:
- Claim
- Reason
- Where the reason comes from: definition, lemma, or prior step
- What conditions must hold for the reason to apply
AI is useful for producing candidate justifications and alternate routes. Your responsibility is to verify that every justification is valid under the assumption ledger.
Stress-test the proof before polishing
Before you format anything, try to break your own proof. This is where correctness is won.
Stress tests that catch many errors:
- Boundary cases: smallest values, degenerate cases, empty sets
- Symmetry checks: invariance under natural transformations
- Dimensional checks: are quantities comparable
- Counterexample search: does the claim fail if a hypothesis is removed
- Alternate derivation: can you reach the conclusion by a different route
If a proof survives these checks, it becomes much more trustworthy.
Convert the final proof into clear exposition
Once the logic is stable, polish the writing:
- Make quantifiers explicit where confusion is likely
- Keep notation consistent across sections
- State where each hypothesis enters the argument
- Replace long chains of equalities with short explained moves
- Add a short intuition paragraph that matches the proof, not a different idea
AI is excellent at improving readability at this stage. The main guardrail is simple: do not let AI rewrite the math in a way that changes meaning.
A workflow summary you can reuse
- Pin the statement to explicit definitions
- Maintain an assumption ledger
- Outline dependencies before writing full prose
- Write in proof obligations with explicit reasons
- Stress-test with boundary cases and counterexamples
- Only then polish into clear exposition
Used this way, AI becomes a drafting and organization tool that serves correctness, rather than a confidence amplifier that hides mistakes.
Common failure modes and guardrails
Most wrong proofs fail in recognizable ways. Naming them helps you detect them early.
| Failure mode | What it looks like | Guardrail that catches it |
|---|---|---|
| Silent strengthening | You prove a stronger claim than stated without noticing | Compare each step against the original quantifiers |
| Hidden regularity | You assume continuity, differentiability, or finiteness | Ledger check: every regularity must be stated |
| Case omission | A degenerate or boundary case is skipped | Boundary sweep: test smallest and extreme values |
| Illicit interchange | Limits, sums, integrals swapped without conditions | Explicit theorem citation and condition check |
| Notation drift | A symbol changes meaning mid-proof | Notation table and consistent definitions |
| One-way implication | You use an equivalence when only one direction holds | Write directions explicitly, prove both when needed |
How to use AI without losing correctness
AI is most helpful when you ask it for constrained outputs that you can verify.
Good requests include:
- Produce a proof outline with named lemmas and where each hypothesis is used.
- List the proof obligations that must be justified, one per line.
- Suggest alternate arguments for a specific step, with cited conditions.
- Generate boundary-case checks and attempt counterexamples when a hypothesis is removed.
- Rewrite for clarity without changing any symbols or logical structure.
Risky requests include:
- Prove this theorem end-to-end with no structure.
- Fill in the details for all missing steps.
- Make it more elegant by simplifying assumptions.
Elegance is a reward for correctness, not a replacement for it. The safest way to use AI is to keep the proof modular and to verify each module against your assumption ledger.
Keep Exploring AI Systems for Engineering Outcomes
• How to Check a Proof for Hidden Assumptions
https://ai-rng.com/how-to-check-a-proof-for-hidden-assumptions/
• Proof Outlines with AI: Lemmas and Dependencies
https://ai-rng.com/proof-outlines-with-ai-lemmas-and-dependencies/
• AI for Building Counterexamples
https://ai-rng.com/ai-for-building-counterexamples/
• Turning Scratch Work into LaTeX Notes
https://ai-rng.com/turning-scratch-work-into-latex-notes/
• AI Unit Test Generation That Survives Refactors
https://ai-rng.com/ai-unit-test-generation-that-survives-refactors/
