---
name: math-background
description: Interactive conversation to understand the mathematical background and context of a problem. Use when user has a vague idea, intuition, or half-formed thought and needs to articulate it precisely.
license: Apache-2.0
metadata:
  version: "1.0"
  category: research
---

# 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