Ain't nobody got time for that

I think you are trusting in Jinteki Replicating Perfection's ability a little too much. Keep in mind that the common Runner strategy adopted against Jinteki RP is to initiate a run against a central server, then jack out after the first piece of ICE. This satisfies the requirements to run against remotes.

I was going to comment on your economy a bit, but then you posted this revision. It's satisfies my economy concerns, but I'm a little worried about Melange with so little ICE. Let's talk about ICE first, then I'll poke at the economy.

First of all, I'll admit that I include Chum and Wall of Thorns in my Finger Snap deck. However, I've learned more since I originally devised that deck. Chum is useless by itself, so you've effectively got 9 pieces of ICE that can be used out of the gate. This gives you a 19% chance of drawing early usable ICE, a little less than 1 ICE per 5 cards. That means you'll likely only have a single piece of usable ICE in your initial draw, and there will be times when you get no usable ICE in your initial draw. This is mollified a bit by your 9 upgrades, but I'm not sure if the upgrade makes up for the lack of ICE. I love Caprice, but she's Unique so you'll only be able to protect a single server with her. Strongbox is decent, but won't provide much protection against Phase 1 blind runs. Ruhr Valley combined with Strongbox and Jinteki RP provides for a pretty taxing run, but is too expensive for Phase 1 usage. All together, you don't have much Phase 1 protection in this deck, and that's going to make it difficult to get a Phase 2 economy operating to pay for the protection that is in this deck.

In addition, all of your low cost ICE is porous. WoT and Tsurugi are the only ETR ICE, and they cost 6 or 8 credits to rez each! In addition, if you take a look at Dydra's deck Replicating Kills v1.0 you'll see a nice write up about why Wall of Thorns is a poor choice for ICE.

Currently your economy consists of Melange Mining Corp, Sundew, and Celebrity Gift. Sundew and Celebrity Gift are very strong cards here (assuming the ICE is increased to protect Sundew), and Melange is a powerful card in it's own right. However, I don't think there is enough protection in this deck to keep Melange for more than a turn or two after rez. There is also Gila Hands Arcology, but it's a one off and gimmicky in this build so it's hard to give it any weight here. Considering the half life for Melange, I would recommend reviewing GRNDL Refinery coupled with Mushin no Shin. Another card to review in a Jinteki Replicating Perfection build is Successful Demonstration to synergize with the Jack Out Runner strategy I explained above.

I'm not too certain about the Agenda mix in this deck. You've got the one off of Gila Hands Arcology, but then you have three Clone Retirements. What is the purpose of these one point Agendas? Fetal AI makes this not appear to be a Rush deck. I think a choice needs to be made regarding how much Rush you want this deck to be, either replacing Fetal AI with NAPD Contract or Medical Breakthrough (when available), or take out the Rush Agendas and substitute more robust Agendas such as The Future Perfect or NAPD Contract.

With the current build, Shock! and Snare! really don't feel like they should be present in this density. Honestly, if it were me I'd remove them entirely and replace them with a single or double Project Junebug plus a full set of Mushin no Shin.

I'm new to Jinteki. I'm glad my deck made you theorize about the identity and it's strategies :)