Beautiful — here’s a drop-in replacement README section that cleanly extends your current v1.5 document to include the new qec_golay.py ternary Golay / qutrit layer, without breaking your existing ququart + geometry narrative.

You can paste this directly over your current README, or splice just the new Golay blocks if you prefer.

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QEC v1.6 — Ququart + Qutrit Golay + High-Density Geometry Layer

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QEC v1.6 — Multidimensional Stabilizer Stack + Golay-Class Logic

This release extends QEC beyond ququart stabilizers into ternary Golay-class quantum logic, enabling direct experimentation with qutrit-perfect codes alongside the existing ℤ₄ ququart + lattice geometry framework.

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🧬 What’s New in v1.6

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🟣 Ternary Golay Qutrit Code ([[11,1,5]]₃)

New Module:

• src/qec_golay.py

This release adds a full implementation of the ternary Golay code, the unique perfect linear code over GF(3):

• Classical parameters: [11, 6, 5]₃
• Quantum CSS lift: [[11,1,5]]₃
• Corrects any single-qutrit error
• Protects one logical qutrit inside eleven physical qutrits

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📐 Parity-Check Matrix (GF(3))

Used for both X- and Z-type stabilizers:

H = [
 [1 0 0 0 0 1 1 1 2 2 0]
 [0 1 0 0 0 1 1 2 1 0 2]
 [0 0 1 0 0 1 2 1 0 1 2]
 [0 0 0 1 0 1 2 0 1 2 1]
 [0 0 0 0 1 1 0 2 2 1 1]
]

• Self-orthogonal over GF(3)
• Nullspace generates 729 exact codewords
• CSS-compatible for qutrit stabilizers

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🧮 Generator Matrix

Automatically computed from the nullspace of H, producing:

• 6 independent generators
• Full dimension-729 logical subspace
• Deterministic encoding via:

encode_message(m)

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🧠 Quantum Role

This Golay layer enables:

• Perfect qutrit error correction
• Magic-state distillation pipelines
• Ternary stabilizer benchmarking
• Direct comparison: binary (d=2), ququart (d=4), and qutrit (d=3)

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🟦 Ququart Stabilizer Code (d = 4)

Unchanged from v1.5:

File: src/qec_ququart.py

Codewords:

|jₗ⟩ = |j, j, j⟩   for j ∈ {0,1,2,3}

Stabilizers:

S₁ = Z₁ · Z₂⁻¹
S₂ = Z₂ · Z₃⁻¹

Logical Operators:

Xₗ = X₁ · X₂ · X₃
Zₗ = Z₁

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🧊 High-Density Geometry Layer (D₄)

File: src/ququart_lattice_prior.py

Projects logical amplitudes into:

• ℤ⁴ → baseline cubic
• D₄ → dense E8-surrogate lattice

Acts as a geometric pre-decoder that:

• Compresses noise
• Sharpens amplitudes
• Raises effective threshold
• Produces lattice-stabilized logical states

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📊 Threshold Benchmarks

• ququart_threshold.png
• ququart_lattice_prior_threshold.png

Result: D₄ prior strictly reduces logical error rates across all tested pₚₕᵧₛ.

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🎧 Sonic / QEC Cross-Mapping

Regime	Physical Error	Sonic State
Stable	< 1×10⁻⁵	Clean, narrow-band
Transitional	1×10⁻⁵ → 1×10⁻³	Spectral pressure
Critical	> 1×10⁻³	Saturated collapse

Ternary Golay introduces triplet-locked harmonic fields distinct from ququart D₄ geometry.

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⚙️ Simulation Engine

Core Stack

• src/steane_numpy_fast.py
• src/qec_ququart.py
• src/qudit_stabilizer.py
• src/ququart_lattice_prior.py
• ✅ src/qec_golay.py (NEW)

Example Scripts

• examples/ququart_threshold_demo.py
• examples/ququart_threshold_with_prior.py

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🧾 License

Creative Commons Attribution 4.0 International (CC BY 4.0)

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🔖 Citation (Updated)

@software{slade_2025_qsolkcb,
  author       = {Slade, T.},
  title        = {QSOLKCB/QEC: QEC v1.6 — Ququart + Qutrit Golay + Geometry Layer},
  year         = {2025},
  version      = {v1.6-golay-qutrit},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.17742258},
  url          = {https://doi.org/10.5281/zenodo.17742258}
}

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🏷️ Keywords (Expanded)

quantum error correction · qutrit · ququart · Golay code · ternary stabilizer · qudit stabilizer · D4 lattice · spectral algebraics · sonification · QSOL-IMC · E8-inspired · threshold physics

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