# ZeroProof Implementation Summary

## Core Transreal Arithmetic ✓
- `TRScalar`: Transreal scalar type with value and tag (REAL, PINF, NINF, PHI)
- Complete arithmetic operations (add, sub, mul, div) with tag propagation
- Unary operations (abs, sign, neg, log, sqrt, pow_int) with domain awareness
- IEEE-754 bridge for seamless interoperability
- Comprehensive unit and property-based tests

## Autodifferentiation with Mask-REAL ✓
- `TRNode`: Computational graph nodes with gradient tracking
- `TRGradientTape`: Context manager for recording operations
- Backward propagation with topological sorting
- **Mask-REAL rule**: Zero gradients when forward pass produces non-REAL tags
- High-level gradient functions (tr_grad, tr_value_and_grad)
- Gradient checking utilities
- Full operator overloading for natural syntax

## TR-Rational Layers ✓
- `TRRational`: Single-output rational layer y = P(x)/Q(x)
- `TRRationalMulti`: Multi-output with optional shared denominator
- Basis functions:
  - Monomial basis (simple but less stable)
  - Chebyshev basis (numerically stable for high degrees)
  - Fourier basis (interface defined, implementation pending)
- Features:
  - Leading-1 constraint for identifiability
  - L2 regularization on denominator coefficients
  - Optional L1 projection for stability
  - Minimum |Q(x)| tracking

## Training Orchestration ✓
- `Trainer` and `HybridTRTrainer` coordinate epochs, coverage control, and hybrid gradient schedules
- Logging cadence controlled via `log_interval`; robotics examples expose `--log_every`
- Per‑epoch bench metrics recorded in training summaries under `bench_history`:
  - `avg_step_ms`, `data_time_ms`, `optim_time_ms`, and `batches`

## TR-Norm (Epsilon-Free Normalization) ✓
- `TRNorm`: Batch normalization without epsilon
- `TRLayerNorm`: Layer normalization without epsilon
- Key features:
  - Deterministic bypass when variance = 0
  - Statistics computed over REAL values only (drop-null)
  - Limit equivalence to BN(ε) as ε→0⁺
  - Affine parameters (γ, β) with proper initialization
  - Full gradient support through both branches

## Testing Coverage
- Unit tests for all core operations
- Property-based tests using Hypothesis
- Focused tests for Mask-REAL rule
- Tests for rational layers at poles
- Tests for normalization with zero variance
- Edge case coverage (signed zeros, infinity arithmetic, etc.)

## Documentation
- Comprehensive API documentation
- Mathematical specifications
- Implementation guides
- Example scripts demonstrating all features
- Comparison with traditional approaches

## Key Achievements

1. **Total Operations**: No operation ever throws an exception
2. **No Epsilon Hacks**: Clean handling of singularities and zero variance
3. **Stable Gradients**: Mask-REAL prevents gradient explosions
4. **IEEE Compatibility**: Seamless bridge with standard floats
5. **Production Ready**: Comprehensive tests and documentation

## Architecture Highlights

```
zeroproof/
├── core/           # TR arithmetic foundation
│   ├── tr_scalar.py
│   ├── tr_ops.py
│   └── ...
├── autodiff/       # Gradient computation
│   ├── tr_node.py
│   ├── gradient_tape.py
│   ├── backward.py
│   └── ...
├── layers/         # Neural network layers
│   ├── basis.py
│   ├── tr_rational.py
│   ├── tr_norm.py
│   └── ...
├── bridge/         # IEEE interoperability
└── tests/          # Comprehensive test suite
```

## Design Principles Followed

1. **Totality**: Every operation defined for all inputs
2. **Determinism**: No hidden randomness or thresholds
3. **Transparency**: Clear semantics via tag system
4. **Composability**: Layers work together seamlessly
5. **Pythonic**: Natural syntax with operator overloading

## Next Steps (Optional)

- GPU acceleration for tensor operations
- Integration with PyTorch/JAX/TensorFlow
- Additional basis functions (Hermite, Laguerre)
- Specialized loss functions for TR outputs
- Performance optimizations
- Extended mathematical function library
