# Behavioral Stability Analysis

From architecture to **behavioral stability**: what the two-layer + PMM stack enables, how the three metrics interact, what to simulate first, and where the system can break.

---

## 1. What you have systemically

### Layer A — Design graph

- “What could route”: all 11 chains, all potential cW* pools.
- Lets you simulate **future expansion** before deploying.
- Prevents accidental routing exposure when adding pools.

### Layer B — Deployment graph

- “What actually routes”: defined by `config/deployment-status.json`.
- **Kill switch**: remove a pool from deployment-status to disable routing.
- Acts as exposure registry, routing firewall, expansion throttle.

That separation is central-bank-grade control: programmable monetary corridor, not just liquidity.

---

## 2. PMM depth as state variable

Depth:

```
D = D_0 · min(1, I_T / I_T^*)
```

This gives:

- **Self-limiting router exposure**: drain → D drops → marginal rate worsens.
- **Endogenous slippage widening**: no ad-hoc circuit breaker needed for normal stress.
- **Routers drain you → marginal rate worsens → routers leave.**

So the mechanism is **mathematically stabilizing**.

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## 3. Three metrics — economic interpretation

| Metric | Economic meaning | Target / risk |
|--------|------------------|----------------|
| **Router capture ratio** | Are you the primary stable swap venue? | High capture = revenue + inventory risk. Low = safer peg, less fee income. |
| **Inventory churn** | Stress metric: Σ\|ΔI_T\| / I_T^* per epoch | Healthy: 0.3–0.8 normal, <1.5 stress. If churn >1.5–2.0, bot intervenes constantly. |
| **Intervention cost** | “Monetary defense budget”: bridge/mint/burn to keep peg | Linear in volume → OK. Exponential during topology switch → danger. |

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## 4. What to simulate first

### A) Hub-only across all 11 chains

One PMM per cW per chain vs hub.

**Questions:**

- Does router capture stabilize around **10–30%**?
- Does churn remain **<1** under normal volume?
- Is intervention frequency **periodic** rather than constant?

**This is the baseline.**

### B) Full-quote on chains 1, 56, 137

Enable extra quotes (USDT, DAI, etc.) only on Ethereum, BSC, Polygon.

**Watch for:**

- Multi-hop reflexivity
- Increased churn
- Increased router capture
- **Nonlinear** intervention cost

**Rule of thumb:** If churn increases >50% vs hub model → do not deploy full-quote.

### C) Bridge shock scenario

Inject e.g.:

- 5% supply migration 137 → 56 over 24 blocks, or
- 10% whale exit on one chain

**Measure:**

- Time to re-center peg
- Total bot injection required
- Peak deviation

**Interpretation:**

- **Damped oscillator** → good.
- **Resonant feedback loop** → bad.

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## 5. Where it can break (advanced)

### Cross-chain arbitrage loops

If **bridge cost < slippage difference**, routers do: Chain A → bridge → Chain B → swap → bridge back.

**Model must include:** bridge latency + probabilistic settlement delay. Otherwise risk is underestimated.

### k too tight globally

Default k = 0.1, fee = 25 bps. If inventory target is high and k is tight, you can become the cheapest stable venue on thin chains.

**Simulate:** k ∈ {0.05, 0.1, 0.2}. Often **k = 0.15–0.2** is safer cross-chain.

### EUR tokens (cWEURC, cWEURT)

- FX volatility; USD-quoted hubs; dual-source peg risk.
- Need: **slightly higher k**, **slightly wider band**, possibly **higher fee**.
- Do **not** treat EUR and USD tokens symmetrically.

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## 6. System-level framing

The architecture is a **multi-chain programmable monetary corridor**. Each PMM pool is a **localized peg defense membrane**. `deployment-status.json` is the exposure registry, routing firewall, and expansion throttle.

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## 7. Next steps (options)

1. Run a synthetic 30-day stress sim and extract equilibrium metrics.
2. Derive analytical stability conditions for k, fee, D_0.
3. Design adaptive k control law (automatic routing dampener).
4. Model MEV + oracle-lag attack surface.
5. **Derive safe inventory target sizing formula per chain** → see [11-safe-inventory-sizing.md](11-safe-inventory-sizing.md).