The science
Balance point. In the frame that rotates with the Earth a mass at radius r feels g_eff = GM/r² - ω² r. It is zero at r_GEO = (GM/ω²)^(1/3) = 42,164 km (35,786 km altitude). Below it the cable is pulled down, above it up, so the tension is largest at GEO.
Uniform-stress taper. For stress s the cable area satisfies s dA/dr = ρ A g_eff, so ln(A/A₀) = (ρ/s) [GM(1/R - 1/r) - ½ ω²(r² - R²)]. The working stress is breaking stress divided by the safety factor. The exponent is the energy to lift 1 kg to GEO (about 48 MJ/kg) divided by the material's specific strength s/ρ.
Climber. Lifting one kilogram from the surface to GEO costs 48 MJ (13 kWh) at the cable, independent of speed; at constant speed the power is P = m g_eff v, and the trip time is distance / speed. Release from the tip at radius r gives a speed ωr relative to inertial space; it equals the local escape speed √(2GM/r) at r = (2GM/ω²)^(1/3) = 53,100 km (46,700 km altitude).
Simplifications. Equatorial anchor, point-mass Earth (J2 ignored), no Moon or Sun, no cable dynamics (oscillations, climber-induced swing, Coriolis), no atmosphere, drag or debris, and a counterweight of whatever mass makes the tension balance. Real designs also taper the cable above GEO. Materials strengths are theoretical or lab values; no cable of this length has been made.