Research proof index

Triple Euclidean Levi-Civita contraction ∑_h (ε4)_{a,h} · (ε4)_{b,h} = 6 · δ[a,b] at the symbol level: contracting three of the four Fin 4 component slots of ε4 with the naive Kronecker pairing leaves one free pair a, b and the factor 3! = 6. The Lorentz form carries an extra det η = -1.

Double Euclidean Levi-Civita contraction ∑_h (ε4)_{r,s,h} · (ε4)_{t,w,h} = 2 · (δ[r,t]·δ[s,w] - δ[r,w]·δ[s,t]) at the symbol level: contracting two of the four Fin 4 component slots of ε4 with the naive Kronecker pairing leaves two free pairs and the factor 2! = 2. The Lorentz form carries an extra det η = -1.

Full Euclidean Levi-Civita contraction ∑_b (ε4)_b · (ε4)_b = 24 at the symbol level: summing the square of every standard-basis component of ε4 over all four Fin 4 index slots, paired naively (no metric), counts the 4! = 24 permutations. The Lorentz contraction ε^{μνρσ} ε_{μνρσ} lowers one factor with η and equals -24 instead.