A 12.5 km/s, 4.32 kg/m rotor
Expanded from the old 14km/s argument below. A velocity-transformer track increases efficiency and launch rate, and a somewhat slower rotor allows for more mass in the rotor. This also reduces maximum launch speed, but the loop can still launch smaller vehicles beyond escape velocity.
The Earth's standard gravitational parameter (ignoring J_2 oblateness effects) is 398600.44 km³/s², so gravity at 80 km altitude (6458 km radius) is 9.55744 m/s² plus the J₂ oblateness term of 0.01514 m/s (proportional to 1/r⁴), for a total of 9.5728 m/s². The track is rotating at 72.9211509 microradians per second with the Earth, so that adds 471 m/s to the velocity and a centrifugal acceleration of -0.03434 m/s at altitude to the gravity, or 9.5834 m/s² acting on the fixed track.
The rotor moves 12.5 km/s relative to the rotating-with-the-Earth track at altitude, so it moves at 12,971 m/s in the fixed frame. The centrifugal force on the rotor is ρV²/R = 112.55 N/m outwards. The gravitational force on the rotor is 4.32 kg/m × 9.5728 m/s² or 41.35 N/m, with a net 71.2 N/m to support the ( 71.2 N/m / 9.5834 ) or 7.43 kg track and suspension cables. Total track and cable mass for the 2000 km acceleration path between east and west stations is about 15,000 tonnes.
The new design transformer track converts rotor energy to payload velocity with nearly 100% efficiency (system efficiency less because of power dissipated in power conversion, residual gas drag, and coil resistance losses). Assuming 400 kg/s maximum payload rate, 10 km/s launch velocity, 5 tonne payloads, and 1 tonne launch sleds, then the loop emits 20 GW of launch power into the payloads, and temporarily puts an additional 4 GW into the sleds (mostly recovered at the east end).
Escape velocity from 6458 km altitude is √( 2 μ / r ) = 11.11 km/s, or 10.64 km/s relative to the rotating track. Launches to Earth orbits will be slower; for example, vehicles to a high, slow ConstructionPort cargo orbit will launch at (10.51-0.47) or 10.04 km/s relative to the rotating track. Large interplanetary missions will assemble (and test) in a Construction orbit, then add extra delta V at apogee with high Isp electric-thrust rockets for trips to Mars and the asteroids.
Launching vehicles removes energy from the rotor. Assuming 98% energy efficiency, 400 tonnes per hour, and an average launch velocity of 10.3 km/s, that is 21.65 TJ/hr or 6 GW removed from the rotor. Rotor mass flow rate is conserved; it is 4.32 kg/m × 12500 m/s or 54 tonnes per second coming in and going out. Launch removes an average of 111111 Joules per kilogram of rotor; the rotor speed (in the fixed frame) drops from 12971 m/s to 12962.4 m/s, and the rotor density increases by 0.066%, presumably by reducing the distance between 10 meter rotor bolts by 6.6 millimeters.
The rotor will stretch much more during the descent to the ground. Ignoring J₂, the gravitational energy difference is μ ( 1/R0 - 1/R80 ) = 398600.44 ( 1/6378 - 1/6458 ) = 0.774186 MJ/Kg = 774186 J/kg. The rotor speed in the fixed frame increases from 12962.4 m/s to 13022.0 m/s, so the rotor density must decrease, stretching by 0.46%, or 4.6 centimeters for a 10 meter bolt.
The stretch is helpful! The rotor must be disassembled into individual bolts for deflection and sorting on the surface. The rotor bolts will be held together longitudinally by long magnets connected to compliant springs; the 4.6 centimeter stretch will pull the bolts apart so they can be separated. The specific mechanism hasn't been defined, but we can exploit the stretch during normal system operation.
The uneven mass stream plus a resonator may help separate the bolts radially. The bolts must be separated and rotated for proper orientation into the deflection magnets; this may occur during the last few hundred meters of descent. The reverse will occur after the bolts pass through the motors and turnaround and the inspect/sort/replace switchyard.
At east station, after the east end curve downwards from 100 km altitude to 50 km altitude, the launch sleds slow to a stop and are lowered back down to the surface with elevators, and shipped back to west surface station (as air or ocean cargo) for refurbishment and reuse. The method chosen will depend on sled cost (mostly magnet cost) and interest rates; hopefully cheap iron-nitride supermagnets will make sleds cheap, and fast container ship transit cost-effective. Either way, the sleds must disassemble and fit into a standard 40 foot shipping container, constraining (folded?) length to 12 meters. Details of sled design are a different topic.
From 0 to 12.6 km/s: Starting a Launch Loop
A pre-deployment launch loop stretched out near the ocean's surface will need external power for the levitation magnets and to start moving the rotor. Without rotor movement past generator coils, this power must be provided externally, at frequent intervals along the 2000 km length of the launch and return tracks.