Equation 677 Database

Size 181 is prime, so every Eq 677 magma here lives on a cyclic additive group of order 181 (or fails to even be translation-invariant, see below). The DB currently lists 16 magmas: 8 simple-linear and 8 Tao-Type-II piecewise-linear. All 16 satisfy Eq 255. 12 are idempotent; 4 are non-idempotent. 1. Simple linear (single multiplier). x ◇ y = α x + β y on F_181 for fixed α, β. Eight DB entries (all dwrensha submissions). 1a. Idempotent (α + β = 1, equivalently x ◇ y = x + α(y − x)). Eq 677 forces Φ_10(α) ≡ α⁴ − α³ + α² − α + 1 ≡ 0 (mod 181), giving four valid α ∈ {46, 56, 122, 139} (the four primitive 10th roots of unity in F_181). All four are in the DB: • magma#355635ab (α = 139) • magma#e8378e60 (α = 122) • magma#5c952978 (α = 56) • magma#d836a209 (α = 46) 1b. Non-idempotent (α + β ≠ 1). Four DB entries — magma#5c97e821, magma#6960b95b, magma#df073ec9, magma#45b4632c — with various (α, β). These are NOT translation-invariant under the additive group of F_181 (the obstruction (α + β − 1)·a ≡ 0 mod 181 forces a = 0); their natural symmetry is multiplicative scaling x ↦ λx for λ ∈ F_181*. 2. Tao Type II piecewise-linear (two multipliers, keyed to QR/NQR). x ◇ y = x + f(y − x) on F_181, where f(d) = α_QR · d when d is a quadratic residue mod 181 and f(d) = α_NQR · d when d is a non-residue (and f(0) = 0). All 8 b-reinke size-181 magmas have this form. (α_QR, α_NQR) pairs found: • magma#2c0a619d (RC): (122, 46) • magma#7580f29f (RC): (139, 122) • magma#8c3de954 (RC): (56, 139) • magma#f14f808f (RC): (139, 46) • magma#70249bf6 (RC): (56, 122) • magma#ddf8aaa4 (RC): (56, 46) • magma#2f965789 (non-RC): (139, 176) • magma#3beeddee (non-RC): (48, 139) The α-values appearing across the piecewise family are {46, 48, 56, 122, 139, 176}. The four α with Φ_10(α) ≡ 0 mod 181 (46, 56, 122, 139) also appear singly in family 1a; α = 48 and α = 176 only arise as one half of a piecewise pair. Display reorders for all 8 piecewise magmas have been set to the orbit of 0 under an order-181 fix-free magma automorphism (the hidden additive translation by 1 in F_181). After the reorder, the Cayley table is fully translation-invariant: every row is a horizontal shift of row 0 = the function f, and the rendered image shows clean diagonal banding across the entire 181×181 grid. [text written by Claude]