Python first-principles validation · 2026-09-25

Full 125×125 reconstruction confirms:

• dim ker[(C−6I)(C−30I)] = 47
• P₆ has rank 25 and decomposes into five C₁₂ sectors of rank 5
• 5V₂ ≅ ℂ⁵_L ⊗ V₂
• Z_L X_L = ω X_L Z_L
• X_L⁵ = Z_L⁵ = P₆
• X_L = X₅ ⊗ I_g and Z_L = Z₅ ⊗ I_g
• J_a|_{5V₂} = I_L ⊗ J_a^(2)
• [X_L,J_a] = [Z_L,J_a] = 0
• reduced logical-state error after strong collective SU(2) rotation: 2.69×10⁻¹⁶
• reduced logical-state error after interleaved logical gates and collective rotations: 2.97×10⁻¹⁶

This is the canonical compact statement of the first-principles Python closure; the existing executable validator and machine certificate remain authoritative.

Certificate

MC-E47-NOISELESS-HW-20260925-001 · E0 exact algebra + E1 deterministic numerical reconstruction

Validated structure

• V₂⊗³, dimension 125
• K=(C−6I)(C−30I)
• dim ker K = 47
• E₄₇ ≅ 5V₂ ⊕ 2V₅