E7(7) 56/133 SL(8,R) HOUSE BUILD  -- validation battery
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V1  generators=133  independent_rank=133  closure_residual=6.405e-15  -> PASS
V2  Jacobi residual (exact for matrix algebra; 400 triples)=0.000e+00  -> PASS
V3  trace-form signature = (70+, 63-, 00)  diff=7  [Lambda^4 block: 35+/35-]  -> PASS  E7(7)
V4  symplectic max|G^T Om + Om G| = 0.000e+00  -> PASS
V5  quartic-invariant annihilation max|dI4/de| = 5.684e-10 (roundoff-limited)  -> PASS
V5b I4-stabilizer in sp(56,R): dim = 1596 - 1463 = 133  -> PASS  (= E7, 133)
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CERTIFIED: V1-V5b PASS. Ambient E7(7) 56/133 substrate is real and house-verified.
           (V6 P_912 rank=912 is certified by v6() below -- __main__ runs the full V1-V6 battery.)
BUILT S72 (e7_sl8_substrate_house_v2.py): V7 QC+SO(8) regression, V8 closure, V10 70-coset.
STILL STAGED: V9 branchings + kernel E6/27 x-check, V10 54/16 split, Phase-2 L_Theta table
              (pending the E6/F4 Jordan frame -> D751 ASK 2).
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V6  912 projector via e7 Casimir on 56(x)133:
    eigenvalues (minimal poly) = [np.float64(1.5), np.float64(0.5), np.float64(-0.08333)]
    eigenvalue +1.50000 -> dim 56.000
    eigenvalue +0.50000 -> dim 912.000
    eigenvalue -0.08333 -> dim 6480.000
    -> 56(x)133 = 56+912+6480 = 7448; rank(P_912)=912  -> PASS
