The Second-Iterate Resultant of the Isosceles Pedal Correspondence: Degree, Fixed-Point Factors and the Period-Two Polynomial

The isosceles reduction of the pedal map is a two-valued algebraic correspondence whose defining polynomial is divisible by (1−s). For the remaining curve of bidegree (5,2) we compute the second-iterate resultant exactly: degree 25, fixed-point factors s⁴(s+1)(s−1/2)(s−(2t−1)/(2t)), period-two polynomial of degree 18. At t=7/10 it factors over ℚ into an irreducible sextic and an irreducible polynomial of degree 12, with no real root (exact Sturm count), hence no real period-two point. The sextic's roots form three complex-conjugate 2-cycles over a totally real cubic field, the sextic field is CM, and its Galois group embeds in C₂≀S₃. All results are labelled by how they were established; verification code is included.