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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

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