All gradient checks pass. The implementation is complete. Here's a summary:

## Summary

**File:** `./minimax-m2.7/backwards/layer_norm_numpy.py`

### Key Components

1. **Forward Pass** (`layer_norm_forward`):
   - Computes mean, variance, and standard deviation over the feature dimension (D)
   - Normalizes: `x_norm = (x - mean) / sqrt(var + eps)`
   - Scales and shifts: `y = gamma * x_norm + beta`
   - Caches intermediates: `x_centered`, `x_norm`, `mean`, `std`, `gamma`

2. **Backward Pass** (`layer_norm_backward`):
   - **d_gamma**: `sum(dy * x_norm)` over (B, T)
   - **d_beta**: `sum(dy)` over (B, T)
   - **dx**: `(dz - mean(dz) - x_norm * mean(dz * x_norm)) / std`
   
   Where `dz = dy * gamma`. The consolidated formula avoids computing the full Jacobian.

3. **Gradient Check**:
   - Uses central finite differences: `(f(x+h) - f(x-h)) / 2h`
   - Fixed array flattening to use views instead of copies
   - Spot-check for large tensors (>100k elements)

4. **Numerical Stability**:
   - `eps` prevents division by zero: `std = sqrt(var + eps)`
   - Catastrophic cancellation analysis documented
   - Pairwise summation suggestion for fp16

5. **Complexity**:
   - **Time**: O(BTD) for both forward and backward
   - **Space**: O(BTD) for activations (training), O(1) for inference

6. **GPU Fusion Design**:
   - Grid of `(B × T)` blocks, each handling one `(b,t)` position
   - Three phases: mean reduction, variance reduction, normalize+output
   - Warp-level shuffle reductions for efficiency
   - Single kernel replaces 4-5 separate kernels