s1148 — D1239/A1558 house verification (disjoint path)
Part A — exact channel algebra (sympy)
  [PASS] A1 H_sgn Hermitian, trace 0, Frobenius norm 1  
  [PASS] A2 unital: W(I2) = I3 for all λ,τ  
   Choi eigenvalues (λ symbolic): {1/2: 2, -sqrt(2)*lambda/4 + 1/2: 2, sqrt(2)*lambda/4 + 1/2: 2}
  [PASS] A3 Choi spectrum at λ=√2 = {0,0,1/2,1/2,1,1}  [0, 0, 1/2, 1/2, 1, 1]
  [PASS] A4 Choi rank at λ=√2 = 4  
  [PASS] A5 Choi PSD at λ=1 (interior)  [1/2 - sqrt(2)/4, 1/2 - sqrt(2)/4, 1/2, 1/2, sqrt(2)/4 + 1/2, sqrt(2)/4 + 1/2]
  [PASS] A6 Choi NOT PSD at λ=3/2 > √2  [1/2 - 3*sqrt(2)/8, 1/2 - 3*sqrt(2)/8, 1/2, 1/2, 1/2 + 3*sqrt(2)/8, 1/2 + 3*sqrt(2)/8]
  [PASS] A7 CP boundary exactly |λ| = √2  [sqrt(2)]
  [PASS] A8 W(ℓQ/√2) = (λ/√2) τ ℓ H_sgn exactly  
  [PASS] A9 the exact D1238 leg τ ℓ H_sgn is reached at λ=√2 and NOT at λ=1  
  [PASS] A10 Gram C_* C_*† = I2 (App-T thm:ucp at r=√5/2: (4r²/5) I2)  
  [PASS] A11 effects diag(C_*†C_*) = (1/4,3/4,1/2,0,1/2), sum I2 pattern  [1/4, 3/4, 1/2, 0, 1/2]
  [PASS] A12 zero effect: Φ(P_abs) = 0 exactly  
  [PASS] A13 composite W∘Φ Choi spectrum at λ=√2 = {0^11, 1/2,1/2, 1,1}, rank 4  [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1/2, 1, 1]
  [PASS] A14 composite unital: W(Φ(I5)) = I3  
  [PASS] A15 Z Q Z = −Q and R Q R† = Q  
  [PASS] A16 rotation covariance: W(R A R†) = W(A)  
  [PASS] A17 reflection covariance: W(Z A Z) = W(A) with H_sgn ↦ −H_sgn (target acts by sgn = the conditional S₃-sign extension)  
  [PASS] A18 gate-6 witness: normalized odd-response norms 1/√2 (λ=1) vs 1 (λ=√2); App-T conditions A10–A12 are λ-independent ⇒ whitening ⇏ unit response  (sqrt(2)/2, 1)
Part B — Albert structure algebra (numpy)
   using product tensor 'jordan_constants'
  [PASS] B0 e=f1+f2+f3 is the unit (L_e = I)  ‖L_e−I‖=0.00e+00
  [PASS] B0' f_i are idempotents, f_i∘f_j=0 (i≠j)  
  [PASS] B1 dim Der(J) = 52  dim=52, sv gap 1.70e+00/1.41e-15
  [PASS] B2 dim str_0(J) = Der ⊕ L_traceless = 78  dim=78
  [PASS] B2' Der ∩ L_traceless = 0 (52+26 independent)  
  [PASS] B3 common fixed-vector nullity on the 27 = 0 (no equivariant point)  smallest sv 1.981e+00
  [PASS] B4 orbit-tangent ranks: f1 → 17 (stab 61), f1+f2 → 26 (stab 52), 1 → 26 (stab 52)  (17, 26, 26)
  [PASS] B5 N(t·1)=t³ on t∈{1/2,1,2,−1} (house cubic from the product)  [0.125, 1.0, 8.0, -1.0]
s1148: 26/26 checks passed; verdict PASS_S1148_ALL
