Equation 677 Database

Magma 91bcf0577a3d…

magma 91bcf0577a3d
Size
95
Isomorphism class hash
91bcf0577a3d1b6ecb69ddc3909104c6e79eb964f00ff4418db38b77046c0539
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-06-10 12:53:36
Display reorder
94,86,78,75,87,85,84,80,79,81,77,83,76,72,89,74,88,82,73,90,9,45,68,13,5,1,20,48,24,41,32,37,52,56,64,16,28,60,92,71,3,27,39,67,63,47,7,51,35,59,31,11,15,23,43,55,19,93,22,58,46,26,18,14,34,62,2,54,10,50,66,70,42,30,6,38,91,17,53,40,21,12,8,29,57,33,49,4,44,61,65,36,25,0,69 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, 4) d = 1: (α, β) = (7, 2) d = 2: (α, β) = (11, 13) d = 3: (α, β) = (11, 5) d = 4: (α, β) = (7, 13) 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 12:53:41 · history