Equation 677 Database

Magma e228a62fd3d3…

magma e228a62fd3d3
Size
80
Isomorphism class hash
e228a62fd3d3559bbd638181926874f3c67a2b4f285092004a05f3d875b1001a
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
yes
Submitted by
dwrensha
Submitted at
2026-06-10 12:53:20
Display reorder
37,77,45,61,65,33,41,53,73,69,3,57,25,49,29,21,22,46,26,78,70,66,0,54,62,30,38,50,34,74,42,58,52,48,2,36,64,28,68,60,76,56,24,40,44,72,20,32,67,51,23,43,59,47,39,71,63,1,75,19,27,35,55,31,18,10,12,15,11,6,13,17,8,14,9,7,5,4,79,16 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

Size-80 twisted fiber-bundle magma satisfying Eq 677 (and Eq 255). Carrier Z/5 × F_16 with operation (x, s) ◇ (y, t) = ( x + 4·(y − x) mod 5 , α_d·s + β_d·t in F_16 ), 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_16 = F_2[a]/(a^4+a+1), element i encodes its base-2 digits i0+2*i1+4*i2+8*i3 <-> i0 + i1*a + i2*a^2 + i3*a^3. The fiber coefficient pair (α_d, β_d) depends on the base difference d: d = 0: (α, β) = (2, 3) d = 1: (α, β) = (4, 6) d = 2: (α, β) = (12, 7) d = 3: (α, β) = (13, 2) d = 4: (α, β) = (14, 10) 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; every element is idempotent; it satisfies Eq 255 (consistent with the 677 ⇒ 255 finite conjecture). [text written by Claude]

last edited by dwrensha at 2026-06-10 12:53:21 · history