Equation 677 Database

Magma 0bacb28de22b…

magma 0bacb28de22b
Size
99
Isomorphism class hash
0bacb28de22b06c85662a25d81db81e19c98e1108c703424ce6d8031ab119fc8
Satisfies Equation 255
yes
Right-cancellative
no
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-06-10 15:12:23
Display reorder
98,84,6,86,5,7,87,85,83,90,64,43,74,78,48,54,68,59,96,39,20,10,15,25,28,1,33,91,65,45,75,80,50,55,70,60,89,63,44,73,79,49,53,69,58,88,67,47,77,82,52,57,72,62,97,41,21,11,16,26,31,2,36,93,42,22,12,17,27,32,3,37,95,40,19,8,13,23,30,0,35,92,66,46,76,81,51,56,71,61,94,38,18,9,14,24,29,4,34 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

Size-99 twisted fiber-bundle magma satisfying Eq 677 (and Eq 255). Carrier Z/11 × F_9 with operation (x, s) ◇ (y, t) = ( x + 6·(y − x) mod 11 , α_d·s + β_d·t in F_9 ), where d = y − x mod 11. The base coefficient c = 6 is a primitive 10th root of unity mod 11 (Φ₁₀(6) = 6⁴ − 6³ + 6² − 6 + 1 ≡ 0 mod 11), which is exactly what Eq 677 forces on the base operation x + c·(y − x). The fiber field is F_9 = F_3[a]/(a^2+1), element i encodes i0 + 3*i1 <-> i0 + i1*a. The fiber coefficient pair (α_d, β_d) depends on the base difference d: d = 0: (α, β) = (1, 5) d = 1: (α, β) = (0, 1) d = 2: (α, β) = (2, 2) d = 3: (α, β) = (0, 1) d = 4: (α, β) = (0, 1) d = 5: (α, β) = (0, 1) d = 6: (α, β) = (2, 2) d = 7: (α, β) = (2, 2) d = 8: (α, β) = (2, 2) d = 9: (α, β) = (0, 1) d = 10: (α, β) = (2, 2) 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 11): 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 NOT 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 15:12:23 · history