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docs/10-behavioral-stability-analysis.md

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+# 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**.
+
+---
+
+## 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. |
+
+---
+
+## 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.
+
+---
+
+## 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.
+
+---
+
+## 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.
+
+---
+
+## 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).