# Wheel Mode Implementation Summary


## Implementation Components

### 1. Arithmetic Mode Configuration (`zeroproof/core/wheel_mode.py`)
- **`ArithmeticMode`**: Enum with `TRANSREAL` and `WHEEL` options
- **`WheelModeConfig`**: Global configuration for arithmetic mode
  - Default mode: `TRANSREAL`
  - Methods: `set_mode()`, `get_mode()`, `is_transreal()`, `is_wheel()`
- **Context managers**: `wheel_mode()`, `arithmetic_mode()`
- **Convenience functions**: `use_transreal()`, `use_wheel()`

### 2. Extended TR Scalar (`zeroproof/core/tr_scalar.py`)
- Added `BOTTOM` tag to `TRTag` enum
- Added `bottom()` factory function
- Added `is_bottom()` type checker
- Updated string representations for BOTTOM (⊥)

### 3. Modified Arithmetic Operations (`zeroproof/core/tr_ops.py`)
- Updated all operations to handle BOTTOM propagation
- Wheel-specific rules:
  - `0 × ∞ = ⊥` (instead of PHI)
  - `∞ + ∞ = ⊥` (instead of ∞)
  - `∞ - ∞ = ⊥` (instead of PHI)
  - `∞ / ∞ = ⊥` (instead of PHI)
- All operations propagate BOTTOM: `⊥ ⊕ x = ⊥`

### 4. IEEE Bridge Updates (`zeroproof/bridge/ieee_tr.py`)
- BOTTOM converts to NaN in IEEE representation
- Maintains round-trip properties within mode constraints

## Key Features

### Wheel Mode Semantics
```python
# Transreal mode (default)
zp.tr_mul(real(0), pinf())  # → PHI
zp.tr_add(pinf(), pinf())   # → PINF

# Wheel mode
with wheel_mode():
    zp.tr_mul(real(0), pinf())  # → BOTTOM
    zp.tr_add(pinf(), pinf())   # → BOTTOM
```

### Bottom Propagation
```python
with wheel_mode():
    b = bottom()
    b + real(5)    # → BOTTOM
    sqrt(b)        # → BOTTOM
    b / real(2)    # → BOTTOM
```

### Mode Isolation
- No mixing of modes within operations
- Clear mode boundaries with context managers
- Global mode setting for entire computations

## Design Rationale

### Transreal Mode (Default)
- **Philosophy**: Keep computations flowing
- **Use case**: General numerical computation, ML training
- **Example**: `∞ + ∞ = ∞` (limit-like behavior)

### Wheel Mode
- **Philosophy**: Strict algebraic control
- **Use case**: Formal verification, algebraic analysis
- **Example**: `∞ + ∞ = ⊥` (prevents invalid simplifications)

## Usage Examples

### Basic Mode Switching
```python
# Global switch
zp.use_wheel()
result = zp.tr_mul(real(0), pinf())  # BOTTOM

# Context manager
with zp.wheel_mode():
    result = zp.tr_add(pinf(), ninf())  # BOTTOM
```

### Algebraic Verification
```python
def check_identity(expr1, expr2, test_vals):
    """Verify algebraic equivalence."""
    with wheel_mode():
        for x in test_vals:
            r1, r2 = expr1(x), expr2(x)
            if is_bottom(r1) or is_bottom(r2):
                return False  # Algebraic issue
    return True
```

## Testing
- Comprehensive unit tests in `tests/unit/test_wheel_mode.py`
- Tests for:
  - Mode switching and context managers
  - Wheel-specific operations
  - Bottom propagation
  - Comparison with transreal mode

## Documentation
- Detailed guide: `docs/wheel_mode_guide.md`
- Example: `examples/wheel_mode_demo.py`
- Shows practical differences between modes

## Benefits

1. **Algebraic Safety**: Catches operations that violate algebraic laws
2. **Formal Verification**: Suitable for proof systems
3. **Error Detection**: Bottom propagation identifies issues
4. **Mode Flexibility**: Choose strictness level as needed
5. **Clean Semantics**: Well-defined wheel algebra

## Compliance with Specification
✅ Implements wheel laws (0×∞=⊥, ∞+∞=⊥)  
✅ No mixing of modes within operations  
✅ Module-level switch for uniform application  
✅ Bottom propagation through all operations  
✅ Maintains separation from transreal mode  

## Integration Points
- Works with all TR operations
- Compatible with autodiff (gradients of BOTTOM are BOTTOM)
- Integrates with IEEE bridge
- Available through main API

The wheel mode implementation provides a valuable option for users who need stricter algebraic control, while maintaining the default transreal mode for general computation. This completes the full implementation of the ZeroProof specification!
