We resolve Open Question 1 from Paper XII (Zenodo:18714643): does a cubic analogue of the Oppermann zero-skew theorem exist? The answer is no. For primes p = 2n−1 with p ≡ 1 (mod 6), the negation involution x ↦ p−x preserves the cubic character (since χ₃(−1) = 1), yielding two structurally distinct half-interval constraints: - Right half: every cubic phase count is even (perfect orbit pairing under the involution).- Left half: exactly one phase count is odd — the phase of the midpoint m = n(n−1) ≡ p−r (mod p), whose involution partner is the puncture r = 4⁻¹. The symmetry breaking between k = 2 and k = 3 is controlled by the parity of (p−1)/k: odd exponent causes phase swapping (zero skew for quadratic); even exponent causes phase preservation (parity constraint only, no zero skew for cubic). Additionally, for k = 4 with p ≡ 5 (mod 8), the involution forces complementary pairing: N(Φ₀) = N(Φ₂) and N(Φ₁) = N(Φ₃), verified 78/78. All results verified for 146 non-degenerate qualifying primes up to n = 1000. Source code: https://github.com/Ruqing1963/cubic-parity-half-intervals This is Paper XIII of the Titan Project.
Ruqing Chen (Fri,) studied this question.