# ZeroProof Implementation Verification Report

## Verification Against `complete_v2.md` Specification

### ✅ Core Data Model (Phase 0, Section 1)
**Spec**: `TR = (val: float64, tag: {REAL, PINF, NINF, PHI})`

**Implementation**: Correctly implemented in `tr_scalar.py`
- TRScalar with `_value` (float) and `_tag` (TRTag enum)
- Tags: REAL, PINF, NINF, PHI
- Value only meaningful when tag=REAL
- Factory functions: real(), pinf(), ninf(), phi()

### ✅ Arithmetic Semantics (Phase 0, Section 2)
**Spec**: All operations are total with specific tag tables

**Implementation**: Fully implemented in `tr_ops.py`
- tr_add, tr_sub, tr_mul, tr_div follow exact tag tables from spec
- Special cases handled correctly:
  - Division by zero → ±∞
  - 0 × ∞ → PHI
  - ∞ - ∞ → PHI
  - ∞ / ∞ → PHI
- Overflow handling to ±∞
- Signed zero handling in division

### ✅ TR-AD with Mask-REAL Rule (Phase 2)
**Spec**: Zero gradients for non-REAL forward values

**Implementation**: Correctly implemented in `backward.py`
- Lines 94-104: Explicit check for non-REAL tags
- Sets zero gradients for all inputs when forward is non-REAL
- Gradient tape and topological sort implemented
- Complete chain rule for REAL paths

### ✅ TR-Rational Layer (Phase 3)
**Spec**: y = P(x)/Q(x) with identifiability and stability

**Implementation**: Fully implemented in `tr_rational.py`
- Leading-1 constraint in Q (line 105)
- L2 regularization on φ coefficients
- Optional L1 projection for stability region
- lambda_rej parameter for loss policy
- Basis functions (Monomial, Chebyshev)
- Multi-output variant with optional shared Q

### ✅ TR-Norm (Phase 4)
**Spec**: Epsilon-free normalization with zero-variance bypass

**Implementation**: Correctly implemented in `tr_norm.py`
- Epsilon parameter ignored (warning issued)
- Zero variance deterministic bypass (lines 136-147)
- Statistics computed over REAL values only (drop-null)
- Both batch norm and layer norm variants

### ✅ IEEE↔TR Bridge (Phase 5)
**Spec**: Bijective mapping between IEEE-754 and TR

**Implementation**: Complete in `ieee_tr.py`
- Correct mappings:
  - finite ↔ REAL
  - +∞ ↔ PINF
  - -∞ ↔ NINF
  - NaN ↔ PHI
- Round-trip preservation
- Extended with NumPy, PyTorch, JAX bridges

### ✅ Property-Based Testing (Phase 7)
**Spec**: Hypothesis tests for algebraic laws and totality

**Implementation**: Comprehensive test suite
- Totality tests: Operations never raise exceptions
- Commutativity tests for addition and multiplication
- Tag table verification
- Round-trip conversion tests
- Gradient property tests
- Edge case coverage

### ⚠️ Reduction Modes (Phase 0, Section 3)
**Spec**: Every reduction declares strict or drop-null mode

**Implementation**: Partially implemented
- ReductionMode enum defined with STRICT and DROP_NULL
- TR-Norm uses drop-null for statistics
- BUT: No general reduction operations (sum, mean) implemented

### ❌ Wheel Mode (Phase 0, Section 9)
**Spec**: Optional compile-time switch for stricter algebra

**Implementation**: Not implemented
- Only placeholder in `zeroproof/wheel/__init__.py`
- No actual wheel arithmetic (0×∞=⊥, ∞+∞=⊥)
- No mode switching mechanism

### ❌ Saturating-grad Ablation (Phase 2, Section 3)
**Spec**: Optional gradient capping without ε

**Implementation**: Not implemented
- Only Mask-REAL is implemented
- No saturating gradient option mentioned

### ❌ Loss Policy Implementation (Section 9)
**Spec**: Adaptive λ_rej as Lagrange multiplier

**Implementation**: Not implemented
- TR-Rational has lambda_rej parameter
- But no adaptive adjustment mechanism
- No coverage tracking

### ❌ Float64 Default (Phase 0, Section 8)
**Spec**: Default numeric precision: float64

**Implementation**: Uses default Python float
- No explicit float64 enforcement
- Precision control added but not enforcing float64 by default

## Additional Features (Not in Spec)

### ✅ Optimization Tools
- Graph optimization (constant folding, CSE, fusion)
- Profiling and benchmarking
- Caching and memoization
- Parallel processing

### ✅ Extended Framework Bridges
- NumPy arrays (TRArray)
- PyTorch tensors (TRTensor)
- JAX arrays (TRJaxArray)
- Precision control (Float16/32/64, BFloat16)

### ✅ Extended Unary Operations
**Spec mentions**: log, sqrt, pow_int
**Also implemented**: No additional unary ops

## Summary

The implementation correctly follows the core specification with some gaps:

**Fully Compliant**:
- Core TR arithmetic with exact tag tables
- Mask-REAL autodiff rule
- TR-Rational and TR-Norm layers
- IEEE bridge with round-trip guarantees
- Property-based testing

**Missing/Incomplete**:
1. General reduction operations with mode selection
2. Wheel mode implementation
3. Saturating-grad option
4. Adaptive loss policy
5. Explicit float64 enforcement

**Beyond Spec**:
- Comprehensive optimization tools
- Extended framework integration
- Advanced profiling and benchmarking

The implementation is production-ready for the core transreal arithmetic system with autodiff, but lacks some optional features mentioned in the specification.

## Evaluation & Benchmarks (Robotics IK)

- Bucketed near‑pole analysis
  - Evaluate per‑bucket MSE by |det(J)| bins (B0–B4) and report bucket counts
  - Ensures coverage of near‑pole regions and apples‑to‑apples comparisons
- 2D pole metrics
  - PLE (Pole Localization Error) vs analytic θ2∈{0,π}
  - Sign consistency across θ2‑crossing paths
  - Slope error from log‖Δθ‖ vs log|sin θ2| near poles
  - Residual consistency via forward kinematics
- Quick vs full parity profiles
  - Quick: stratified test subset by |det(J)|≈|sin θ2|; DLS aligned to the same subset
  - Full: full dataset; DLS with full iterations
- Bench transparency
  - Per‑epoch timings recorded in training summaries: avg_step_ms, data_time_ms, optim_time_ms, batches (bench_history)

## Automated Verification (Defaults)

For end‑to‑end parity checks on the RR 2R experiments, use the verifier:

```bash
# Aggregate across seeds (90th percentile) with recommended thresholds
python3 scripts/verify_results.py \
  --path results/robotics/paper_suite \
  --method "ZeroProofML-Full" \
  --max-ple 0.30 \
  --max-b0 0.010 \
  --max-b1 0.010 \
  --percentile 90 \
  --require-nonempty-b03

# Strict per-seed bounds
python3 scripts/verify_results.py \
  --glob 'results/robotics/paper_suite/seed_*/comprehensive_comparison.json' \
  --method "ZeroProofML-Full" \
  --max-ple 0.30 \
  --max-b0 0.010 \
  --max-b1 0.010 \
  --no-percentile \
  --require-nonempty-b03
```

Notes:
- These defaults are calibrated for the CPU‑friendly configs in this repo; adapt for different datasets/hyperparameters.
- `--require-nonempty-b03` promotes the near‑pole bucket coverage guardrail (B0–B3) to a hard failure.

## Updates Since Initial Report

- Saturating gradients and Hybrid schedules are implemented (`grad_mode.py`, `hybrid_gradient.py`), with schedule‑driven δ and near‑pole exploration.
- Adaptive loss policy with coverage tracking is implemented (`training/adaptive_loss.py`, `training/coverage.py`).
- Float64 guidance and precision control are documented (`docs/float64_enforcement.md`); default Python floats apply unless otherwise configured.
- Reduction modes are used in TR‑Norm statistics; general reductions remain an area for future work.
