What can marked ruler and compass construct? Septics and octics with Conway’s “extended instrument”

We use J. H. Conway’s extension of the marked ruler, in which a second rigid pair of points, a divider of fixed opening k, slides along the ruler line independently of the two marks one unit apart. In one reflection-symmetric configuration (circle, line, circle, line) the stop condition of a single placement is a septic in cos α; the configuration is found from one quartic, square roots and rational operations, and with rational transport and the sextic theorem of [1] every root of every separable real septic with 3, 5 or 7 real roots is constructible. Angle septisection needs no transport, and the regular 29-gon and 49-gon are given explicitly. The main new result concerns octics. Two real critical points of an octic are roots of a septic, and when that septic has 3, 5 or 7 real roots they are already constructible; normalizing them to ±1 turns the setup of a threeline, one-circle configuration into a cubic and square roots. With four rational references and transport, every root of every separable real octic having a real root is constructible, including the real and imaginary parts of its nonreal roots. Padding a septic with one real root by a known linear factor then settles the remaining septic case: every root of every separable real septic is constructible, and in particular every real seventh root. The construction of 7√2 is given explicitly, with no transport of the octic. All regular n-gons with φ(n) free of prime factors above 7 are constructible; below 100 this leaves exactly 23 , 46, 47, 53, 59, 67, 69, 79, 83, 89, 92, 94 undecided.