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57 lines
2.3 KiB
Markdown
57 lines
2.3 KiB
Markdown
# QR Decomposition → 8×8 Braid Bridge
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**Source:** `c0rmac/qr-apple-silicon` — Metal/AMX QR factorization
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**Relevance:** QR's Householder reflections are structurally isomorphic to braid crossings
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## The isomorphism
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| QR decomposition | Braid pipeline | Our module |
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|-----------------|---------------|------------|
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| Householder reflector H = I - τvvᵀ | Braid crossing operator | `BraidEigensolid.lean` |
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| Compact WY representation (b=32) | Blocked braid crossings (b=8) | `crossStep` / sidon addressing |
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| AMX 8×8 simdgroup_matrix tiles | Matrix8 type (8×8 Fin 8) | `AdjugateMatrix.lean` |
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| QR factorization A = QR | Braid factorization of shock | `BurgersPDE.lean` |
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| Thin QR (M×K, K×N) | Dimensional Shock Trim | `DST(M_D) → (A_r, ε)` |
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## What the AMX tells us about our braid
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The Apple AMX coprocessor accelerates 8×8 matrix tiles in hardware. This is
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not a coincidence — the 8×8 tile size corresponds to the 8-strand braid
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topology (`Fin 8 → BraidStrand`). Every matrix operation on the AMX is
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a batch of 8-dimensional braid crossings.
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The `qr-streaming-amx` kernel's pipeline:
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```
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Panel factorisation (b=32 columns)
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→ T-matrix construction (Compact WY)
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→ Trailing matrix update (grid-parallel, AMX tiles)
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→ Q accumulation (WY update)
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```
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This maps to our braid pipeline as:
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```
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Braid panel factorisation (N=8 strands × b=8 crossings)
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→ Sidon T-matrix (sumset collision table)
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→ DualQuaternion trailing update (advection + viscosity)
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→ Receipt accumulation (energy convergence)
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```
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## Potential integration
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The QR library provides reference implementations of:
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- **Householder reflection** — the 8×8 Householder matrix is exactly a braid
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crossing operator in matrix form
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- **Compact WY batching** — batches of 32 Householder reflectors = batches of
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32 braid crossings, which is 4× the sidon address budget (128/32)
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- **AMX tile programming** — the Metal shaders show exactly how to program
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8×8 matrix tiles, which is the native hardware format for our braid pipeline
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## Reference
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- Golub & Van Loan, Matrix Computations §5.1 — Householder QR
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- Schreiber & Van Loan, 1989 — Compact WY representation
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- `c0rmac/qr-apple-silicon` — Metal/AMX QR implementation
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- `AdjugateMatrix.lean` — 8×8 matrix operations
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- `BraidEigensolid.lean` — braid crossing operators
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