Equation 677 Database

Magma d07de435ce6a…

magma d07de435ce6a
Size
99
Isomorphism class hash
d07de435ce6a0cf7445c6bc758855835696b56d3c480b19b02766bd6e1b305ed
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-06-10 13:28:32
Display reorder
98,84,81,86,80,82,87,85,83,96,58,18,78,8,28,38,65,48,88,40,1,60,70,11,20,52,30,92,44,4,63,74,14,24,54,34,93,55,15,75,5,25,35,64,45,91,69,23,3,13,33,43,73,50,97,29,68,49,59,79,9,39,19,94,36,76,56,66,6,16,46,26,89,41,0,61,71,10,21,51,31,95,47,7,67,77,17,27,57,37,90,32,72,53,62,2,12,42,22 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 + 2·(y − x) mod 11 , α_d·s + β_d·t in F_9 ), where d = y − x mod 11. The base coefficient c = 2 is a primitive 10th root of unity mod 11 (Φ₁₀(2) = 2⁴ − 2³ + 2² − 2 + 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) = (1, 1) d = 2: (α, β) = (8, 7) d = 3: (α, β) = (3, 3) d = 4: (α, β) = (1, 4) d = 5: (α, β) = (7, 7) d = 6: (α, β) = (8, 2) d = 7: (α, β) = (1, 8) d = 8: (α, β) = (6, 2) d = 9: (α, β) = (7, 6) d = 10: (α, β) = (1, 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 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 13:28:32 · history