# 07: P-AMM Redemption Curve

## Source
- **Protocol**: Gyroscope GYD
- **Files**: `PrimaryAMMV1.sol`, `IPAMM.sol`, `Flow.sol`
- **Paper**: `gyrostable/technical-papers/P-AMM/P-AMM technical paper.pdf`

## Rationale for MYCO

Every bonding curve needs a redemption mechanism. The P-AMM provides a sophisticated, bank-run-resistant approach:

1. **Piecewise pricing**: Full redemption at par when fully backed, smooth degradation when underbacked
2. **No cliff**: Unlike simple bonding curves where selling drains the reserve linearly, the parabolic discount protects remaining holders
3. **Floor guarantee**: Even deeply underbacked, there's a minimum redemption rate (θ̄)
4. **Outflow memory**: Tracks recent redemptions, dampening cascade effects
5. **Separating redemption from swap pricing**: The bonding surface determines minting price; the P-AMM independently determines redemption price

For MYCO: this is the "exit" side of the bonding curve. It determines how much a $MYCO holder gets back when selling, ensuring the system degrades gracefully rather than catastrophically.

## Invariant / Pricing Function

Three regions by backing ratio ba = reserve / supply:

**Region I: Fully backed (ba ≥ 1)**
```
rate = 1.0  (redeem at par)
```

**Region II: Parabolic discount (xu < x ≤ xl)**
```
b(x) = ba - x + α(x - xu)² / 2
```
where x = cumulative recent redemptions as fraction of supply.

**Region III: Linear floor (x > xl)**
```
rate = θ̄  (minimum guaranteed)
```

## Parameters

| Parameter | Symbol | Range | Effect |
|-----------|--------|-------|--------|
| Curvature | ᾱ | > 0 | Steepness of discount. Higher = sharper drop |
| No-discount threshold | x̄_U | [0, 1] | Fraction of supply that can redeem at par |
| Floor rate | θ̄ | [0, 1] | Minimum redemption rate |
| Outflow memory | | (0, 1) | Decay rate for flow tracking. Higher = longer memory |

## Properties

- **Monotonically decreasing**: More redemption → worse rate (protects remaining holders)
- **Continuous**: No jumps between regions (smooth transitions)
- **Bounded**: Rate always in [θ̄, 1.0]
- **Path-dependent**: Rate depends on recent history (outflow memory), not just current state

## Implementation

See `src/primitives/redemption_curve.py`