Over the given sidereal month of any eccentric system, any spaceline releases and pulls its cable. Earth/Luna marks this in the megameters: on the Lunar side of L1, 60-to-61.5. Suppose the L1 station isn't pulling back the cable and releasing it; that's a lot of work and at the cable's sheer scale (and thickness!), I doubt the work is needed. So doubts everyone else who is treating this push-and-pull as a rounding-error ... at L1. They haven't considered the Sinus Medii groundbase. There, they must consider... elasticity.
The gravity forces nonequal tension applying mostly to the L1 plateau; but we're likely saving cost, so that the Lunar side of the cable be thinner. Therefore assume constant elongation throughout. L1-to-Luna elongation needs minimum 2.5%, or we have to add surplus cable. Kevlar-brand aramids give us this minimum, as do Zylon and Dyneema. Most mass will cluster the L1 tension plateau. Also - as noted - overdangling Earth GEO, is our strongest material and there we don't care about elasticity. On our side, is where Toray's T1100G.
Lunaside is the problem. T1100G is only 2.2% (I'm further suspicious about Jensen's anhydrous glass). So the 61.5 Mm apogee (tightest) is extended from 60.17 Mm perigee (loosest). Allowing for redundance (e.g. braiding the cable into rope) will mean more excess on the Lunar side - so the extra cable dump on the Moon is a minimum. By Kepler this slacker-week is shorter than apogee week, but - still.
That means at perigee the loose cable has left 170 km on the moon. At 1 cm2 cross-section: 1700 cubic meters if melted into a glob; at 1790 kgm-3 this is 3043 tonnes. Multiply that if braided.
Coil it, box it, unscroll it - what is the base supposed to do with all this? At the highest apogee, is potential-energy. That's the integral mg(h)dh, but this is a humble blog so we'll assume the 170km is negligible compared to 60+Mm as to treat g as a constant. Highschool lab mgh: 3043 kkg x 1.62 x 170000 ≤ 838,042,200 kilojoules.
I feel like for half this month, which is not quite the month of daylight, the drop of this cable is a steady force - which Loonies can harness. (L1 may or may not be harnessing the central elasticity, for its own part.) Pulling up is energy working on... pulling up, but this too can be harnessed - if we warn L1 we're doing that.
On the leadup to the Lunar perigee when only 60 Mm, when that cable is making its largest pile, is when the trip to/from L1 will be shortest. Maybe not always cheapest because Luna's side of energy depends on the 28-and-a-quarter month we see on Earth.
Under that; I want some structure as can funnel the 1700 m3 to some convenient depository away from useful work. We need to place it somehow/-where so we don't knot it and doesn't get METEORED. The depo would be large and/or deep but - we'll discuss that later. Assume the depo is dustfree anyway. (If we're banning lander-rockets, that should help.)
Traffic with L1 can disconnect at the top, thence to go up and down the internals of our structure (given airlocking); or, roll down to the surface.
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