cross-chain-pmm-lps/docs/11-safe-inventory-sizing.md

Safe Inventory Target Sizing (Closed-Form Approximation)

A closed-form approximation for I_T^* (inventory target) per chain so capital allocation can be justified mathematically instead of heuristically. Inputs: expected routed volume, bridge β/γ, refill latency, and target band.

---

1. Objective

Choose I_T^* per chain and per cW token so that:

1. With high probability we do not deplete inventory before bridge/mint can refill.
2. Peg stays inside the target band during normal and stressed flow.
3. Intervention cost (bridge/mint/burn) remains bounded and preferably linear in volume.

---

2. Assumptions

• Epoch: time window over which we measure flow (e.g. 1 hour or “refill cycle”).
• V_epoch: expected net outflow of cW token from the pool during one epoch (routed volume, one-sided; we conservatively use net outflow).
• σ: volatility multiplier for V (e.g. 1.5–2 for “stress”); so worst-case net outflow in one epoch is on the order of V_epoch · σ.
• T_refill: refill latency (time or blocks) for bridge/mint to restore inventory on this chain.
• β: bridge fee (fraction) for refill path; γ: fixed cost per refill (gas + relayer).
• Band: we require that after a shock we do not breach the circuit-break band (e.g. 2% for USD). That effectively caps how far inventory can fall before we must intervene.

---

3. First-order formula

Safe inventory target (units of cW token):

I_T^*  ≥  V_epoch · σ · (1 + T_refill / T_epoch) / (1 - β)  +  γ_buffer

Interpretation:

• V_epoch · σ: worst-case net outflow in one epoch.
• T_refill / T_epoch: number of “epochs” in one refill cycle; we need to cover outflow over that period without going to zero.
• (1 − β): bridge fee reduces effective refill; we size so that after fee we still refill enough.
• γ_buffer: small buffer for fixed costs (e.g. one or a few refill batches); can be set to a multiple of γ or a constant.

Simpler variant (single refill cycle):

If refill is exactly one epoch and we want to absorb one full shock without depleting:

I_T^*  ≥  V_epoch · σ / (1 - β)  +  buffer

buffer = small constant or γ · (typical refill count per epoch).

---

4. Per-chain inputs (from your config)

From config/simulation-params.json and config/token-map.json:

• β = bridgeBeta for the chain (refill path).
• γ = bridgeGammaUnits (nominal; use for γ_buffer if desired).
• V_epoch: not in config; must be estimated or taken from simulation (e.g. expected routed volume per epoch for that chain/token).
• σ: e.g. 1.5–2 for stress.
• T_refill, T_epoch: in blocks or seconds; chain-dependent (e.g. 1 epoch = 1 hour, T_refill = 20 min → T_refill/T_epoch ≈ 1/3).

---

5. Depth consistency (D_0 vs I_T^*)

Your PMM uses D = D_0 · min(1, I_T / I_T^*). For the pool to stay “deep” enough under stress:

• D_0 should be sized so that at I_T = I_T^, the pool can absorb a typical trade size without slipping past the band. A simple rule: D_0 on the order of I_T^ / 2 to I_T^* (so that at target inventory, effective depth is meaningful). So:

D_0  ≈  (0.5 to 1.0) · I_T^*

Then the formula above gives I_T^*; set D_0 accordingly.

---

6. EUR tokens (cWEURC, cWEURT)

• Same formula applies, but use EUR-specific expected volume and, if refill is via a different path, EUR bridge β/γ.
• Use wider band (and possibly higher σ) to reflect FX and dual-source peg risk; that may imply a larger buffer or higher σ in the sizing formula.

---

7. Usage

1. Estimate V_epoch per chain (and per token if needed) from historical or simulated routed volume.
2. Set σ (e.g. 1.5–2) and T_refill / T_epoch from chain/contract data.
3. Read β (and optionally γ) from simulation-params.json.
4. Compute I_T^* from the formula; optionally add γ_buffer.
5. Set D_0 ≈ (0.5–1) · I_T^* for that pool.
6. Put inventoryTargetUnits (and depth) into simulation-params.json or deployment-status.json and run simulations to validate churn and intervention cost.

This gives a justified starting point for capital allocation per chain; tune from there using the three metrics (capture, churn, intervention cost) and stress tests (hub-only, full-quote, bridge shock).

---

8. Example first pass (chains 1, 56, 137, 25)

Assumptions: σ = 1.5 (USD), σ = 2 (EUR); T_refill/T_epoch = 0.33; γ_buffer = 2·γ. Formula: I_T^* ≥ V_epoch·σ·(1 + 0.33)/(1 − β) + γ_buffer; D_0 = 0.75·I_T^*.

Chain	Name	Hub	β	γ	V_epoch (example)	I_T^* (USD)	I_T^* (EUR)	D_0 (USD)
1	Ethereum	USDC	0.001	50	100_000	~200k	~266k	~150k
56	BSC	USDT	0.001	10	80_000	~160k	~213k	~120k
137	Polygon	USDC	0.001	5	60_000	~120k	~160k	~90k
25	Cronos	USDT	0.002	15	30_000	~60k	~80k	~45k

Run node scripts/size-inventory.cjs or node scripts/size-inventory.cjs --v-epoch '{"1":100000,"56":80000,"137":60000,"25":30000}' to regenerate from your simulation-params.json. Use k = 0.15–0.2 on thin chains (e.g. Cronos) if sims show excessive capture; EUR tokens use eurDefaults (higher k, σ, feeBps).