Size 127 is prime, with multiplicative group F_127* of order 126 = 2·3²·7. Since 10 does not divide 126, F_127 contains no primitive 10th roots of unity, and the polynomial Φ_10(α) = α⁴ − α³ + α² − α + 1 has no roots in F_127.
Consequence: there is no simple-linear idempotent Eq 677 magma on F_127. (A single-multiplier x ◇ y = (1 − α)x + α y satisfies Eq 677 iff Φ_10(α) ≡ 0, which has no solution here.)
The Tao Type II piecewise-linear construction rescues this. Splitting f according to whether y − x is a QR or NQR mod 127 — i.e. taking x ◇ y = x + f(y − x) with f(d) = α_QR · d on QR and f(d) = α_NQR · d on NQR — allows Eq 677 even at primes where no single α works.
The unique DB example at this size is magma#cf044a9b, with (α_QR, α_NQR) = (58, 29). (Note 58 = 2·29 and 2 is a QR mod 127, so the two multipliers are related by a QR scalar.) It is idempotent, right-cancellative, and satisfies Eq 255.
[text written by Claude]
dwrensha · 2026-05-14 20:05:36