So, my opinion is mixed.

On one hand, the atmosphere here is fascinating. You've crafted a pretty original object-level setting (gargantuan tower-city that ceaselessly expands outwards, now managed by a flawed algorithm), intercepted with some very compelling Kafkaesque/surreal horror (the trial at the beginning with an absent defendant, C's inability to get through to anyone, the secret police agent (?) following her that everyone pretends not to see, the RAT KINGDOM lurking at the edges of everything...). I've read all public entries to the contest, and on that metric, it's one of the most intriguing.

On the other hand:

• The message is a bit muddled. What the Tower and the algorithm symbolize is obvious, but what are all the rats about?  I'd suspect "rat = rationalist", but it doesn't really map on... The mystery is very compelling, but the answer is either confusing or  unsatisfying. And the swerve to recommending the Precipice at the end seems awkward.
• On the technical level, the prose is... lacking.  Though that's naturally fixable with practice.

Overall, an intriguing read. Especially loved the ‘Gustav, I need to know about the rats’ refrain, it was so delightfully weird.

Warning: I'm an unrepentant perfectionist, incredibly nitpicky, and way more suited for describing flaws in things than for celebrating the good parts. I think I'm getting better at the latter, but, uh, don't be too discouraged.

• To start off, your command of prose is at a fairly competent level.
• I find the general concept quite interesting. Communication between different versions of the same person from different timelines, circumstances that ensure it's only ever a community of different instances of that same person and not some general cross-temporal organization, that they can't transfer resources to each other, etc.
• The concept is well-explored, as well. The logistics of Ls communicating with their n-year groups, the statistical observations, the Eternal Birthday Club for Ls who don't want to return to a doomed world, interesting background details like an L who'd spend 6000 years in the dream and the ISIS!L — you'd clearly thought it out pretty well.
• And herein lies the first flaw: the story places too much focus on said logistical details. The first part of the story, especially, is way too technical, and does very little to progress the plot or demonstrate the characterization. It's interesting, in an abstract way, but it's not engaging. In my opinion, it's generally unnecessary to lay all the details out explicitly, especially when writing for a crowd pre-selected for intelligence.  I'd advice to focus on the story and only ever mention background technicalities when and if they come up, trusting your readers to piece things together on their own (and they may actually have more fun doing that).
• The mystery of Ls' statistically anomalous deaths was intriguing, especially the "something is killing Ls" line — really injected a sense of urgency. The answer was, to me, obvious, but mainly because the contest's theme gave it away. General idea is solid.
• To return to the concept, it's well-suited for the contest's theme, too. To show off concrete effects of tail risks by engineering a context in which their sufferers would have a presence while still making it clear what a deceptively small presence they would have — very clever, I wish I'd thought of that.
• The actual speech about existential risks, though, I was less impressed by. It felt too... generic, not connected to all the other weird things the story was doing. I know expositing on this sort of thing is squarely the contest's goal, but IMO it didn't flow well. (Or maybe I was just tired from these sorts of speeches, as I'd spent the day reading all the other entries.)
• The main flaw, though, is the lack of stakes. It's established early on that birthday!Ls will retain no memory, and would have no control over what actually happens in the future. As such, it's difficult to care about what happens in the story, since by fiat none of it is going to actually matter. 
• As a suggestion, it could've been a point of tension: Ls becoming desperate to figure out some way to communicate with the outside, or despairing over their inability to protect the future, thereby perhaps making the readers acutely aware of their comparatively high ability to affect the world.
• Or, simpler, perhaps there could've been some small way of affecting Ls' timelines, and the story could've been about an L deciding how to best utilize it. And you do introduce a vague mechanism of maybe-doing-that at the end, but by that point it's too little too late, the reader has already spent most of the story only barely engaged.

To sum up: very interesting and well-explored concept, but the narrative focus is misaimed, and the management of readers' expectations needs some work.

No payment necessary: your willingness to pay serves as a strong enough signal of your interest to  improve for me to take the time to respond.

Lovely! Ontological mysteries are always fun, and the narrator's voice is extremely engaging, with this combination of sheer enthusiasm and simple vocabulary being used to describe complex thoughts. A bit disappointed it didn't speed up towards some sort of resolution, though: the story works as an exploration of the joys of learning, but it's a bit too long for its message, started to drag on for me in the last third.

Still, very fun. Have you written anything else?