AI RNG: Practical Systems That Ship
A LaTeX document can look polished while hiding a logical gap. Typesetting is a powerful form of camouflage: clean notation makes shaky reasoning feel stable, and a well-formatted lemma can be wrong in a way that is hard to see. Proofreading for logical gaps is different from proofreading for grammar. You are not asking, “Is this readable?” You are asking, “Is this true, and is every dependency stated?”
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AI can help by extracting structure, checking consistency, and flagging suspicious jumps. But it cannot replace the human obligation to verify. The goal is a workflow that makes gaps visible and then forces them to be closed.
Separate three kinds of proofreading
A strong pass distinguishes these layers:
- LaTeX correctness: compiles, references resolve, macros behave
- Exposition clarity: definitions introduced before use, notation consistent
- Logical validity: every implication justified, every case covered
Treat them as separate passes. Mixing them creates fatigue and missed gaps.
Start by extracting the proof skeleton
Before you reread paragraphs, rewrite each proof as a short outline:
- Goal statement
- Main tool or lemma used
- Key reduction step
- Case split or induction step
- Conclusion and where each hypothesis was used
AI is useful here: give it the LaTeX source of a proof and ask for a bullet skeleton that preserves the logical moves. Then compare the skeleton to the written proof. Skeleton mismatches often reveal missing steps.
Run an assumption and quantifier audit
Logical gaps often hide in quantifiers and domains.
Common failure patterns:
- A statement that should be “for all” is used as “there exists”
- A variable silently changes domain mid-proof
- A bound like “nonzero” is used without being stated
- A parameter limit or boundary case is omitted
A practical audit checklist:
- List all variables and their domains at the start of the proof
- Identify every “choose” step and whether the choice is valid
- Mark each use of an existence statement and where it came from
- Check boundary cases explicitly when a parameter can be 0, 1, or empty
Ask AI to produce a variable table for the proof, then check it manually against your text.
Verify that every lemma is used with its hypotheses satisfied
A classic hidden gap is invoking a lemma without verifying its conditions. This happens in papers when the writer knows the lemma is “usually true” and forgets the edge conditions that make it fail.
Create a dependency table as you read:
| Invoked result | Hypotheses required | Where verified in the proof | Risk if missing |
|---|---|---|---|
| Lemma A | condition list | line or paragraph reference | statement may be false |
| Theorem B | condition list | line or paragraph reference | wrong domain or case |
| Estimate C | parameter bounds | line or paragraph reference | inequality direction breaks |
If you cannot point to where a hypothesis is checked, you have found a gap or a missing statement.
Look for “miracle sentences”
A miracle sentence is a line where multiple things happen at once:
- “It follows immediately that…”
- “Therefore we may assume…”
- “By standard arguments…”
- “After simplification…”
These are not always wrong, but they are where gaps hide. For every miracle sentence, force a local expansion:
- What exact lemma is being used?
- What computation is being skipped?
- What case is being assumed away?
AI is good at expanding these sentences into explicit steps. Then you check whether those steps are valid under your assumptions.
Check notation consistency like a compiler would
Small notation drift causes big logical drift. Watch for:
- Reusing a symbol for a different object later
- Switching between similar norms or inner products without warning
- Writing “≤” where “<” is required for a later step
- Using “O(·)” without specifying which variable tends to what
Ask AI to list every defined symbol and every time it appears. This is mechanical work that AI can do well. Your job is to decide whether the uses are consistent.
Close the loop with a “gap closure paragraph”
When you find a gap, do not patch it with a vague sentence. Close it with a paragraph that contains:
- The missing claim stated explicitly as a lemma or subclaim
- A short proof or a citation with verified hypotheses
- A sentence explaining how it connects to the next step
This makes the fix durable and makes future readers trust the document.
Use AI to propose checks, not to certify truth
A safe way to involve AI:
- Ask it to flag likely gap locations
- Ask it to expand miracle sentences into steps
- Ask it to produce a hypothesis checklist for each invoked lemma
An unsafe way:
- Ask it, “Is this proof correct?” and accept confidence as evidence
The purpose is to make you faster at seeing what needs proof, not to outsource proof.
Keep Exploring AI Systems for Engineering Outcomes
• Turning Scratch Work into LaTeX Notes
https://ai-rng.com/turning-scratch-work-into-latex-notes/
• AI Proof Writing Workflow That Stays Correct
https://ai-rng.com/ai-proof-writing-workflow-that-stays-correct/
• 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/
• Writing Clear Definitions with AI
https://ai-rng.com/writing-clear-definitions-with-ai/
Books by Drew Higgins
