# Saturating Gradient Implementation Summary

## Implementation Components

### 1. Gradient Mode Configuration (`zeroproof/autodiff/grad_mode.py`)
- **`GradientMode`**: Enum with `MASK_REAL` and `SATURATING` options
- **`GradientModeConfig`**: Global configuration for gradient computation
  - Default mode: `MASK_REAL`
  - Configurable saturation bound
  - Context manager support for temporary mode changes
- **Convenience functions**: `use_mask_real()`, `use_saturating(bound)`

### 2. Saturating Operations (`zeroproof/autodiff/saturating_ops.py`)
- **`saturate_value`**: Smoothly saturates values using `x/(|x| + bound)`
- **`saturating_reciprocal`**: Computes `1/x` with saturation near zero
  - Uses `sign(x) / sqrt(x² + bound²)` to avoid explosion
- **Operation-specific gradients**:
  - `saturating_div_grad`: Bounded gradients for division
  - `saturating_log_grad`: Bounded gradient for log (1/x)
  - `saturating_sqrt_grad`: Bounded gradient for sqrt
  - `saturating_pow_grad`: Bounded gradients for negative powers
  - `saturating_rational_grad`: For TR-Rational layers

### 3. Backward Pass Integration (`zeroproof/autodiff/backward.py`)
- Modified to check `GradientModeConfig.get_mode()`
- Routes to appropriate gradient computation based on mode
- Mask-REAL: Zeros gradients for non-REAL forward values
- Saturating: Computes bounded gradients even for non-REAL values

### 4. Saturating Rational Layer (`zeroproof/layers/saturating_rational.py`)
- **`SaturatingTRRational`**: Extended TR-Rational with mode support
  - Configurable gradient mode per layer
  - `forward_with_mode`: Override mode for specific forward passes
  - `compare_gradient_modes`: Utility for ablation studies
- **`create_saturating_rational`**: Factory function

## Key Features

### Gradient Computation
```python
# Mask-REAL (default)
if forward_tag != REAL:
    gradient = 0

# Saturating
if operation involves 1/x:
    gradient = x/(x² + bound²)  # Bounded near x=0
```

### No Epsilon Hacks
- Uses transreal arithmetic throughout
- Smooth transitions without discontinuities
- Principled bounding based on mathematical properties

### Flexibility
- Global mode setting
- Per-layer configuration
- Context manager for temporary changes
- Easy comparison between modes

## Usage Examples

### Basic Mode Switching
```python
# Global change
zp.autodiff.use_saturating(bound=1.0)

# Context manager
with zp.autodiff.gradient_mode(GradientMode.SATURATING, bound=5.0):
    loss.backward()

# Back to default
zp.autodiff.use_mask_real()
```

### Layer with Saturating Gradients
```python
model = SaturatingTRRational(
    d_p=4, d_q=3,
    gradient_mode=GradientMode.SATURATING,
    saturation_bound=10.0
)
```

### Ablation Study
```python
results = model.compare_gradient_modes(x_batch)
# Returns gradients for both modes on same inputs
```

## Benefits

1. **Continuous Gradient Flow**: Maintains gradients through singular regions
2. **Bounded Behavior**: Prevents gradient explosion without hard cutoffs
3. **Research Tool**: Enables study of different gradient strategies
4. **Smooth Learning**: Can learn functions with poles more effectively
5. **No Arbitrary Constants**: Uses principled TR arithmetic

## Testing
- Unit tests in `tests/unit/test_saturating_grad.py`
- Tests for:
  - Saturation functions
  - Mode switching
  - Gradient comparison
  - Integration with training

## Documentation
- Detailed guide: `docs/archive_tr/saturating_grad_guide.md`
- Example: `examples/saturating_grad_demo.py`
- Shows comparison between modes on pole learning

## Compliance with Specification
✅ Replaces singular growth (e.g., 1/Q²) with bounded form  
✅ Uses TR arithmetic (no epsilon hacks)  
✅ Provides continuous transitions near poles  
✅ Optional alternative to Mask-REAL  
✅ Suitable for ablation studies  

## Integration Points
- Works with all TR operations
- Compatible with adaptive loss policy
- Integrates with existing optimizers
- Supports all layer types

The saturating gradient mode provides a valuable alternative for research and applications where maintaining gradient flow through near-singular regions is important, while still preventing the numerical instabilities of unbounded gradients.
