Equation 677 Database

Magma aa78455ebb84…

magma aa78455ebb84
Size
80
Isomorphism class hash
aa78455ebb848599b7eed3f4eaf5b78527cf031421ada84aeadde26b952ffb1a
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-06-10 12:52:46
Display reorder
79,71,72,74,73,67,69,65,61,62,68,63,70,60,66,64,75,59,3,47,7,35,51,27,11,15,39,19,55,43,31,23,77,32,36,20,40,8,24,0,44,48,12,52,28,16,4,56,78,10,14,58,18,46,2,38,22,26,50,30,6,54,42,34,76,45,49,33,53,21,37,13,57,1,25,5,41,29,17,9 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) = (10, 4) d = 1: (α, β) = (4, 5) d = 2: (α, β) = (13, 3) d = 3: (α, β) = (11, 1) d = 4: (α, β) = (12, 8) 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 12:52:46 · history