Equation 677 Database

Size-35 twisted fiber-bundle magma satisfying Eq 677 (and Eq 255). Carrier Z/5 × F_7 with operation (x, s) ◇ (y, t) = ( x + 4·(y − x) mod 5 , α_d·s + β_d·t in F_7 ), where d = y − x mod 5. The base coefficient c = 4 is a primitive 10th root of unity mod 5 (Φ₁₀(4) = 4⁴ − 4³ + 4² − 4 + 1 ≡ 0 mod 5), which is exactly what Eq 677 forces on the base operation x + c·(y − x). The fiber field is F_7 = Z/7Z (prime field). The fiber coefficient pair (α_d, β_d) depends on the base difference d: d = 0: (α, β) = (4, 1) d = 1: (α, β) = (2, 1) d = 2: (α, β) = (4, 3) d = 3: (α, β) = (4, 6) d = 4: (α, β) = (2, 5) Eq 677 holds iff the pairs satisfy, for every d (writing e₁ = −d, e₂ = cd, e₃ = d(1 − c + c²), e₄ = −d/c, all mod 5): β_e₄·α_e₃ + β_e₄·β_e₃·α_e₂·β_e₁ = 1 and α_e₄ + β_e₄·β_e₃·(α_e₂·α_e₁ + β_e₂) = 0, with the d = 0 pair therefore satisfying the standard linear-677 conditions α₀β₀(1 + β₀²) = 1, α₀ + α₀²β₀² + β₀³ = 0. This family strictly generalizes the quadratic-residue-class bundles searched earlier (where (α_d, β_d) was constant on QR/non-QR classes): here the pair may vary with d arbitrarily. Found June 2026 by an exhaustive constraint solve over all per-d coefficient assignments; Eq 677 verified directly on the full Cayley table. The magma is right-cancellative; not idempotent; it satisfies Eq 255 (consistent with the 677 ⇒ 255 finite conjecture). [text written by Claude]