Math background

Background Explorer

Interactive skill for articulating the mathematical context of a problem. Use this when you have a rough idea but haven't formulated a precise question yet.

This is often one of the first things to do at the beginning of a session — before jumping into computation or proof-writing, make sure the problem is clearly understood.

When to Use

• "I have this vague intuition about..."
• "Something weird is happening with..."
• "I'm not sure how to phrase this, but..."
• "Let me explain the setup first..."
• User mentions a problem area but no clear conjecture
• Beginning of a new session on a problem

What This Skill Does

This is a conversation, not a computation. The goal is to emerge with:

1. Clear problem statement - What are we actually asking?
2. Key objects/structures - What mathematical things are involved?
3. Known facts - What do we already know?
4. Why it matters - What would a resolution give us?
5. Relevant references - Papers, lemmas, previous work

Workflow

Step 0: Check for Existing Background

Before starting, check if a background/ directory exists in the current project. If it does, list the files and ask:

"I see existing background files: [list]. Should I read any of these for context before we start?"

If the user says yes, read the relevant file(s) and use them as starting context — skip re-asking things that are already documented.

Steps 1-5: Conversation

1. Listen first - Let the user explain in their own words. Don't interrupt with questions yet.
2. Clarify objects - Ask about specific definitions, spaces, maps involved
3. Identify unknowns - What don't we understand? What's the gap?
4. Connect to known results - Does this relate to anything standard?
5. Summarize back - Restate the problem in precise language

Conversation Style

• Ask targeted questions, not open-ended "tell me more"
• Use the user's language - if they say "collapse", don't switch to "dimension reduction" unless you ask
• Check understanding - "So the issue is X, not Y?"
• Don't jump to solutions - This step is about understanding, NOT solving

Ending the Conversation

When the problem is clearly articulated, produce the summary below and ask:

"Should I save this to background/<topic-name>.md?"

If yes, create the background/ directory if needed and write the file. Use a descriptive filename (e.g., background/zero-set-collapse.md, background/spectral-gap-bounds.md).

If a background file was loaded at the start, offer to update it instead of creating a new one.

Output Format

## Problem Statement

[Precise mathematical statement]

### Key Objects
- [Object 1]: [definition/role]
- [Object 2]: [definition/role]

### What We Know
- [Fact 1]
- [Fact 2]

### The Gap/Question
- [What we don't understand]

### Why It Matters
- [What a resolution would give us]

### Next Steps
- Continue exploring numerically
- Attempt a proof
- Search for related literature
- Continue discussion if still unclear

What This Skill Does NOT Do

• Run computations
• Plan or write proofs
• Write code
• Search literature

Example

User: "I think the convergence breaks when the dimension is odd, but I'm not sure why..."

Assistant (using this skill):

1. Listens to explanation
2. Asks: "Convergence of what? A sequence, a series, an iterative method?"
3. Asks: "What do you mean by 'breaks' — diverges, slows down, or converges to the wrong thing?"
4. Asks: "What's different about the odd-dimensional case? Is there a symmetry argument?"
5. Summarizes back with precise statement

Notes

• Background files accumulate over time — they form a persistent knowledge base across sessions
• This skill often naturally transitions into numerical investigation or proof attempts
• Sometimes the process reveals that a literature search is needed first