Equation 677 Database

Size-127 idempotent right-cancellative magma satisfying Eq 677 and Eq 255. Tao Type II piecewise-linear construction on F_127 (127 prime): x ◇ y = x + f(y − x) in F_127, f(0) = 0, f(d) = 58·d when d is a quadratic residue mod 127, f(d) = 29·d when d is a non-residue. Why piecewise is required: 127 is prime but |F_127*| = 126 = 2·3²·7 is not divisible by 10, so F_127 contains no primitive 10th roots of unity. The polynomial Φ_10(α) = α⁴ − α³ + α² − α + 1 — whose roots give the multipliers for the simple-linear idempotent Eq 677 magmas x ◇ y = (1 − α)x + α y — has no roots in F_127. So no single-multiplier linear construction works at this size; the piecewise pairing (α_QR, α_NQR) = (58, 29) is needed. Note: 58 = 2·29 in F_127, and 2 is a quadratic residue mod 127 (since 127 ≡ −1 (mod 8)), so the two multipliers are related by a QR scalar. The display reorder presents elements as the orbit 0, τ(0), τ²(0), … of an order-127 fix-free magma automorphism τ (the hidden additive translation by 1 in F_127), making the cyclic F_127 translation symmetry visible: every row of the rendered Cayley table is a horizontal shift of row 0 = f. [text written by Claude]