Equation 677 Database

Magma 1ebd82c0406f…

magma 1ebd82c0406f
Size
95
Isomorphism class hash
1ebd82c0406f91d35ea166c75cfa8c6e5f6fbec49909b55d7e548b7c0d0f64ef
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-06-10 12:54:06
Display reorder
94,85,80,77,81,82,86,88,75,89,76,83,72,79,74,73,84,87,78,90,9,48,36,25,29,13,21,64,0,68,33,52,44,60,56,5,17,40,92,39,14,2,55,59,43,51,30,67,34,63,18,10,26,22,71,47,6,93,38,15,3,54,58,42,50,31,66,35,62,19,11,27,23,70,46,7,91,8,49,37,24,28,12,20,65,1,69,32,53,45,61,57,4,16,41 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

Size-95 twisted fiber-bundle magma satisfying Eq 677 (and Eq 255). Carrier Z/5 × F_19 with operation (x, s) ◇ (y, t) = ( x + 4·(y − x) mod 5 , α_d·s + β_d·t in F_19 ), 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_19 = Z/19Z (prime field). The fiber coefficient pair (α_d, β_d) depends on the base difference d: d = 0: (α, β) = (7, 3) d = 1: (α, β) = (4, 18) d = 2: (α, β) = (14, 18) d = 3: (α, β) = (14, 18) d = 4: (α, β) = (4, 18) 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]

last edited by dwrensha at 2026-06-10 12:54:06 · history