A few years ago I evaluated the selenosync space elevator - poorly. I ended up rewriting the thing. Since then some people keep revisiting, especially, the "Spaceline". Which ties in with the lunaport at Sinus Medii.
Wang Weiweia, Wu Zhiganga, and Liu Jiafu have a paper "Conceptual Design and Mechanical Analysis of a Lunar Anchored Cislunar Tether". It is in English... sort of. They don't have good Anglophone editors in Shenzhen. Also the nonexotic fibre they propose remains Kevlar or, I kid you not, wet bamboo; not Zylon or anything good.
The notion is that a L1 station between us and Luna will float away unless counterweighted - on our side. A strong, thick cable to L1 must, then, have a strong thick cable toward the other direction - not necessarily the same length but to pull the same weight. As a side-effect of a beam this long being physically anchored to Medii: I don't think the L1 Lissajous is a thing. The L1 "halo" will be almost a point.
Take the unit of measurement, such that 1 = radius of our Earth. The Moon's semimajor is around 60, eccentricity-allowing. Our mutual L1 is at 51 (85%ish); GEO is 6.6.
The Shenzhen trio prefers the counterweight dangle as far as 110 Mm over the Earth so 114.8 Mm semimajor. That would be an internal tension of about two kilonewtons along the L1 plateau. Hence Kevlar; T1100G could, I think, bring us down to the 80s Mm. Of course then someone has to pay for all that cable.
I recommend to attach a rotating torus at L1. Just one: there's no need for a counterspin. And this cylinder isn't high, from its axle to the rim. Instead the L1 plateau is suited for length. Tethered to the spaceline it's not floating away. It's not Hyperion and we certainly can't have Janhunens wrapping themselves around the axle; I'm thinking, that classic O'Neill cylinder.
I pondered making this long tube an integral part of the tether itself but then I started thinking of ball-bearings or, failing that, how we'll radiate the heat off; or even how to build it. So we'll just tether it until someone else figures all that out.
As to what this station buys us: it's the cheapest point to launch high-Earth-orbit cargo to stations in TLL4 or L5. As well as TLL2 on the far side of the Moon; and from there, the whole Solar System. Lunar prospectors will want a station to regrow their bone-mass. So: yeah, I'm expecting this to be where the Medii colonists actually live.
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