Research-Stack/6-Documentation/docs/specs/break_glass_problem_statement.md

# Break-Glass Agent — The Problem Statement

## The Core Tension

We are attempting **transcendental work** — formal verification of novel mathematical
structures (eigensolid compression, spectral gap properties, braid topology) — using
**LLMs that cannot reliably perform division**.

This is not a metaphor. The `cleanMerge_preservesGap` sorry is a concrete example:
the mathematical argument is complete and correct, but the bridge from Q16_16 lists
to boolean patterns requires ~30 lines of list manipulation that no single LLM can
write correctly because:

1. **Infinite vs finite types** — LLMs don't understand that `Q16_16 = Subtype Int`
   is infinite while `Bool` is finite. They keep trying `native_decide` on free
   Q16_16 variables, fail, and loop.

2. **List operations on 8 elements** — `List.zip (List.range 8) [a0,...,a7]` produces
   a term with hundreds of subexpressions. `simp` can't reduce it. LLMs don't know
   this and keep trying `simp [List.zip, List.range, ...]`.

3. **The chicken-and-egg** — To use `native_decide`, you need finite values. To get
   finite values, you canonicalize. To prove canonicalization preserves predicates,
   you need the bridge. The bridge is what you're trying to prove.

4. **No precedent in training data** — This specific pattern (Q16_16 → Bool bridge
   for spectral gap verification) doesn't exist in Mathlib, Lean docs, or any public
   codebase. LLMs can only recombine what they've seen.

## Why Panels Help

The Fusion benchmark showed: **a panel of 3 budget models beats a single frontier
model** on complex reasoning tasks. This is because:

- **Different architectures see different patterns.** DeepSeek's math training
  might see the boolean abstraction immediately. Claude's Lean expertise might
  know the exact `@[simp]` lemma needed. GPT's breadth might recall a similar
  pattern from a different proof system.

- **Diversity > quality.** Three mediocre perspectives from different angles
  beat one excellent perspective from a single angle. This is the same insight
  behind ensemble methods in ML.

- **The judge synthesizes.** The judge model reads all panel responses and
  extracts: consensus, contradictions, unique insights, blind spots. This
  structured analysis is more useful than any single response.

## What Makes This "Transcendental"

The work is transcendental in the mathematical sense: it involves structures
that exist beyond the computational horizon of any single model.

| Property | Single LLM | Panel (8 models) |
|----------|-----------|-------------------|
| Can pattern-match proof tactics | Yes | Yes |
| Can write novel list induction | No | Maybe |
| Can bridge infinite→finite types | No | Maybe |
| Can reason about Q16_16 arithmetic | Barely | Yes (DeepSeek) |
| Can write Lean 4 metaprogramming | No | Maybe (Cohere) |
| Can synthesize diverse approaches | N/A | Yes (judge) |

The "maybe" is the key. A single LLM will always fail on genuinely novel
mathematical reasoning. A panel has a chance — not because any individual
model is smarter, but because the diversity of perspectives can occasionally
triangulate on a solution that no single model would find.

## The Break-Glass Protocol

```
1. AGENT HITS WALL
   ↓
2. "break glass: <problem>"
   ↓
3. DIAGNOSE: What type of problem? (lean_proof, math_novel, etc.)
   ↓
4. ASSEMBLE CONTEXT: Relevant files, build logs, AGENTS.md constraints,
   previous attempts, error messages
   ↓
5. SELECT PANEL: 8 models chosen for problem type
   - lean_proof: Opus + DeepSeek + GPT + Cohere + Gemini + GLM + Kimi + DS-Flash
   - Each model gets the FULL context (files + constraints + history)
   ↓
6. PARALLEL EXECUTION: All 8 models answer simultaneously
   - Each has web search + web fetch enabled
   - Each sees the same problem + context
   - Each produces a structured response
   ↓
7. JUDGE SYNTHESIS: Opus reads all 8 responses
   - Consensus: what all/most agree on
   - Contradictions: where models disagree
   - Unique insights: what only one model raised
   - Blind spots: what no model addressed
   ↓
8. FINAL ANSWER: Structured diagnosis + code + confidence
   - Problem diagnosis
   - Proposed solution with exact file paths
   - Alternative approaches
   - Risks and caveats
   - Cost report
```

## Why This Is Worth the Cost

Each invocation costs $1-3.50. This sounds expensive until you compare:

- **Human time**: A mathematician spending 2 hours on the `cleanMerge_preservesGap`
  bridge costs $200-400.
- **LLM loops**: A single model trying and failing 20 times costs $0.50-1.00
  and produces nothing.
- **Panel**: $2.50 for 8 diverse perspectives + synthesis. If even ONE model
  sees the solution, the judge extracts it.

The break-glass agent is not a replacement for human reasoning. It's a
replacement for LLM flailing. When a single model is stuck in a loop, the
panel breaks the loop by introducing diversity.

## The Sakana Fugu Insight

Sakana Fugu's key finding: **a learned orchestrator model that dynamically
assembles agents from a pool outperforms any fixed pipeline**. The orchestrator
learns non-obvious collaboration patterns — e.g., "for Lean proof problems,
first ask DeepSeek for the mathematical insight, then ask Claude for the
tactic sequence, then ask Cohere for the structured output format."

The break-glass agent doesn't have a learned orchestrator (that would require
training data we don't have). But it does have:

1. **Problem classification** — auto-detect problem type from the prompt
2. **Panel selection** — choose models based on problem type
3. **Judge synthesis** — Opus extracts the best from all responses

This is a fixed-pipeline approximation of what Fugu does dynamically.
As we collect data on which panels work for which problem types, we can
refine the selection.

## Integration with Existing Stack

The break-glass agent plugs into the existing infrastructure:

- **MCP server** → registered in `.mcp.json`
- **Skill** → loaded when "break glass" trigger detected
- **ContextStream** → logs every invocation for cost tracking
- **AGENTS.md** → documents the trigger protocol
- **SilverSight** → the agent respects SilverSight constraints

The agent is the **last resort**. It should be called only when:
1. The local model (Hermes3) has failed
2. A single frontier model has failed
3. Existing skills/tools have failed
4. The problem is genuinely novel (not a rehash of known issues)