# QR Decomposition → 8×8 Braid Bridge

**Source:** `c0rmac/qr-apple-silicon` — Metal/AMX QR factorization
**Relevance:** QR's Householder reflections are structurally isomorphic to braid crossings

## The isomorphism

| QR decomposition | Braid pipeline | Our module |
|-----------------|---------------|------------|
| Householder reflector H = I - τvvᵀ | Braid crossing operator | `BraidEigensolid.lean` |
| Compact WY representation (b=32) | Blocked braid crossings (b=8) | `crossStep` / sidon addressing |
| AMX 8×8 simdgroup_matrix tiles | Matrix8 type (8×8 Fin 8) | `AdjugateMatrix.lean` |
| QR factorization A = QR | Braid factorization of shock | `BurgersPDE.lean` |
| Thin QR (M×K, K×N) | Dimensional Shock Trim | `DST(M_D) → (A_r, ε)` |

## What the AMX tells us about our braid

The Apple AMX coprocessor accelerates 8×8 matrix tiles in hardware. This is
not a coincidence — the 8×8 tile size corresponds to the 8-strand braid
topology (`Fin 8 → BraidStrand`). Every matrix operation on the AMX is
a batch of 8-dimensional braid crossings.

The `qr-streaming-amx` kernel's pipeline:

```
Panel factorisation (b=32 columns)
  → T-matrix construction (Compact WY)
  → Trailing matrix update (grid-parallel, AMX tiles)
  → Q accumulation (WY update)
```

This maps to our braid pipeline as:

```
Braid panel factorisation (N=8 strands × b=8 crossings)
  → Sidon T-matrix (sumset collision table)
  → DualQuaternion trailing update (advection + viscosity)
  → Receipt accumulation (energy convergence)
```

## Potential integration

The QR library provides reference implementations of:
- **Householder reflection** — the 8×8 Householder matrix is exactly a braid
  crossing operator in matrix form
- **Compact WY batching** — batches of 32 Householder reflectors = batches of
  32 braid crossings, which is 4× the sidon address budget (128/32)
- **AMX tile programming** — the Metal shaders show exactly how to program
  8×8 matrix tiles, which is the native hardware format for our braid pipeline

## Reference

- Golub & Van Loan, Matrix Computations §5.1 — Householder QR
- Schreiber & Van Loan, 1989 — Compact WY representation
- `c0rmac/qr-apple-silicon` — Metal/AMX QR implementation
- `AdjugateMatrix.lean` — 8×8 matrix operations
- `BraidEigensolid.lean` — braid crossing operators