I don't think Gyri is worth playing, because by the time you have the tools to utilize the reduced hand size, they'll have a good code gate breaker out. Unless you mix it with yagura...or good breaker trashing. No advanceable traps means scoring anything that needs advanced will be super tough.

You can only play one Philotic

When I find my deck design starting to shape towards a flatline variant, there are a few steps that I follow to ensure a solid flatline design.

One thing to keep in mind is that a reliable flatline deck needs to be able to both entice and discourage the Runner from hitting certain targets on demand. This is manipulated primarily through how much information the Corporate player leaks to the Runner via face up and face down cards, what cards you allow the Runner to access, and is a skill that the Corporate player needs to develop.

Agenda

When I consider flatline grade Agendas, I'm having to balance the economic and damage resources provided against the density of Agendas in the deck. With flatline, you typically want to extend the game via low Agenda density. This ensures that you are seeing more of the structure you need for flatline when you draw, and helps draw the Runner into running against your servers.

Fetal AI is an excellent flatline Agenda, but I have doubts regarding House of Knives. It appears to give us additional damage, but at the expense of increased Agenda density with the limitation of only 1 point of damage per run. That 1 point may be a tipping point, but at the cost of slowing down how frequently the Runner initiates runs against you. The longer the Runner builds his Rig, then the fewer options are available for flatline. Nisei MK II also has a similar slow down effect, as the Runner strives for the credits to recover from a prematurely ended run.

Philotic Entanglement is limited to only 1 card per deck, as @Mark mentions, and is kind of gimmicky as well since it counts Agenda cards versus Agenda points. I would still include the single allowed copy of this card, but it's really hard to build a reliable strategy behind this card.

The Future Perfect is an odd utility card, as it dramatically decreases Agenda density while hosting a really robust defense (as long as you never try to score it). Depending on how much Economy you want to load into the Agendas, there is probably room to include TFP as a defensive measure.

However, none of your Agendas really assist in Economy. Some Corporate Warfare would help round out your Agenda functionality while decreasing Agenda density, and I really believe that a flat line deck can absorb the Bad Publicity introduced by Profiteering.

Asset

Snare! and Pyschic Field are excellent choices for a Jinteki flatline deck, but I'm concerned by Shock! and Private Contracts. One of the requirements for a good flatline deck is that you need to be able to entice the Runner into hitting your traps. Your current Asset choices will tip off the Runner around Phase 2 that the only advancable cards you play are Agendas, and that unrezzed and unadvanced cards are typically traps. To that end, Private Contracts actually hurts your objective despite the economy boost. This is easily replaced with Agenda based economy though.

Shock! is an unusual card in that it can harm the Runner from Archives. Aside from that benefit, a single point of damage suffers from similar limitations as House of Knives above. Once the Runner realizes that unrezzed and unadvanced cards are traps, it encourages the Runner to stay away from unnecessary runs.

To that end, I would look towards introducing Project Junebug and either Aggressive Secretary/Cerebral Overwriter or GRNDL Refinery depending on your economic need (personally, I'm more inclined to GRNDL, but that's a personal preference). The trap cards are Phase 2 and Phase 3 solutions to be played once the Runner builds the false confidence that advanced cards are Agendas. If the Runner grows cautious after hitting a trap or two, then that just leads to the environment for free pursuit of Agendas.

Keep in mind that when Mushin no Shin is released, it significantly changes the dynamic of non-advancable trap cards by allowing the placement of advancement tokens on cards that normally cannot be advanced (this can be the subject of a much more extended discussion by itself). By the same token, when Plan B is released the dynamic changes as well, especially in the hands of a skilled shell game Corporate player.

Operation

With Honor and Profit cards, I really feel that Neural EMP is on the road to obsolescence. Considering your Punitive Counterstrike presence, I would look at Cerebral Cast instead. The Runner either takes Brain Damage or a tag to lead into Punitive Counterstrike. This also opens the door for other options for exploiting tags.

With that said, I'm also not certain if Punitive Counterstrike is the best choice for your influence. For the influence cost it is excellent bang for the buck, but it suffers the limitation of needing to allow the Runner to take Agendas the previous turn, successfully pay for a trace, and then is limited in how much damage is actually done. Despite the influence cost, Scorched Earth would be much more consistant. Don't discount Ronin for a Phase 3 end game solution. By Phase 3 the Runner should believe that they have the Corporate tempo identified in regards to trapped and untrapped assets. Advancing a Ronin in Phase 2 and leaving it alone will lull the Runner into believing that it's a trap, which allows for a surprise final strike in Phase 3.

I'm on the fence with categorizing Celebrity Gift as economy in a trap heavy flatline deck. Normally I love the card because it allows you to entice the Runner with the promise of Agendas to come. However, with the sheer number of traps in a flatline build you will likely be scaring the Runner after a full hand reveal into a more cautious approach (which is bad as described above). Celebrity Gift can be a great tool in our entice/discourage dynamic, but you'll rarely want to reveal a full hand and this limits how many credits it generates. Despite the fact that it gifts the Runner additional credits, I'd look into Medical Research Fundraiser or possible build a strategy to consistantly use Restructure over Celebrity Gift in order to properly support the economy, or move more economy to Agendas (and GRNDL Refinery) as described above.

ICE

Flatline deck ICE is unusual. We actually prefer porous ICE on our remotes, because we want the Runner to be heavily taxed to help our traps land.

Dydra has an pretty nice description for why Wall of Thorns is a poor Barrier in the Replicating Kills v1.0 deck, and I recommend that you read it.

I have to side with x3r0h0ur on Gyri. I played around a bit with Gyri a couple of weeks ago, and just couldn't find a reliable way to leverage Gyri. The few times were Gyri actually affected the results were pretty unusual.

Archer is an excellent piece of ICE, but I have to ask why we are going outside of Jinteki for any ICE to begin with (this also applies to Gyri)? We want to tax the Runner, preferably with as much damage as possible. Jinteki ICE has just about everything we need to make that happen. Himitsu-Bako definitely has a role in protecting the Central servers alongside Tsurugi, and Komainu is like a second Psychic Field. Shinobi has the potential to flatline an unsuspecting Runner by itself if rezzed at the right time.

One thing to keep in mind with cards like Komainu and Shinobi are that they are really only effective on the turn that they are rezzed. Due to their extremely high impact against the Runner's hand, the Runner will never run a server hosting those ICE unless they have a solid defense against them. The Corporate player needs to be confident that they have a flatline situation when they actually rez those pieces of ICE.

Of course, this all changes if Inazuma is introduced into the ICE mixture. An Inazuma in front of a Komainu or Shinobi can introduce the Runner to a world of hurt.

Upgrades

I can't say this enough, Nisei Capice needs to be in just about every Jinteki deck that there is. In a Jinteki flatline deck, Nisei needs to protect your R&D. With only a 1 in 9 chance of getting through Nisei, it is a very effective block against all of the R&D shenanigans that the Runner can through at Jinteki.

There will be people who say that Hokusai Grid needs to be present in a flatline deck. Honestly, the only place I would ever add a Hokusai Grid is HQ and/or R&D. Those are the only places where I do not want the Runner to consistantly run against.

Corporate Troubleshooter is another Upgrade that needs to be seriously evaluated, especially in a Shinobi environment. Being able to force Shinobi to stick in a Phase 2/3 environment may be disruptive enough lead to a flatline.