http://www.oldcomputers.arcula.co.uk/files/images/pdp8104.jp...
It's fairly easy to design a computer like this.
Later TTL/CMOS designs replaced the packaged modules with much smaller 74xx/40xx ICs.
You can make basic logic gates with just diodes and resistors, but you need transistors for inversion, buffering, and a usable flip flop.
That's probably the minimum level for useful computing/calculating. If civilisation has ended and you have no transistors you probably don't have the resources to make glass valves either, so that's going to be a problem.
Of course there's always clockwork...
(I googled this to make sure I wasn't misremembering what I read 40 years ago in an already outdated book at the library and I was suddenly filled with a sense memory of the smell of the interiors of old electric appliances loaded with tubes and dust.)
http://www.righto.com/2018/01/examining-1954-ibm-mainframes-...
They were shockingly unreliable and incredibly expensive. Tubes have a very low mean time between failure, so any design that uses tubes exclusively can't work for more than short periods without breaking down - possibly minutes, maybe hours, probably not days, and absolutely not months or years.
And each failure means a cycle of fault finding, which can take hours or days in turn.
As a technology it sort of works in a prototype way - you can get some work done until you can't. But the unreliability means it's qualitatively different to a modern laptop or server farm.
The wonderful thing about integration on silicon is that it's the opposite - it's incredibly reliable, as long as you keep the thermals reasonable.
Tube reliability improved radically over the 15-20 years tube computers were a thing; it had too. And just like you point out about silicon, reasonable thermal management became recognized as important to tube reliability and designs changed accordingly. MTBF was lower than a modern computer, but they certainly ran for days or weeks and more. And debugging was usually fairly quick as you ran some diags that pinpointed the module (not single tube) that failed and replaced the whole thing.
I have an acquaintance with a Bendix G15 that still runs. Admittedly, the G15 is much simpler than an IBM 700, but it's a nearly 65 year old tube machine.
When sci-fi writers need to create some plot tension around getting a computer either up and running or down and disarmed, the characters will inevitably be plugging/unplugging colorful modules at some point.
edit: didn’t pay attention that you were talking about the manga. That makes more sense. Sounds highly readable!
But yeah, it is a fun "what if".
"What if a super genius with the entirety of wikipedia in his brain were sent back to the stone age? Could he rebuild modern society?"
I don't doubt that they could have made glass plates or something, but they start turning out vacuum tubes and borosilicate beakers next to each other like it was all a matter of knowing the recipe.
Yes. Silicon wafers are cut from a monocrystalline boule, a single flawless silicon crystal with no defects or inclusions. A big chunk of silicon atoms, nothing else at all. (Doping happens later) To the extent any physical object can be called "perfect", a semiconductor wafer is perfect.
(Of course after manufacturing it will start picking up embedded hydrogen and helium atoms from cosmic rays and alpha particle background radiation.)
Sometimes insects or moths would get stuck in the relays which would screw up the system. This is the origin of the word "bug."
Prior to incorporating logic into electronics, computing machines were hand cranked or motor cranked gear machines. See: https://www.youtube.com/watch?v=fhUfRIeRSZE. The YouTube video literally is a hand cranked portable calculator.
The world you envision has already existed.
It can even do square roots?! That's amazing. And it fits the palm of your hand!
Now we're down to specks of sand calculating so fast they melt without cooling. Seriously wtf.
You need a source of problems to solve, and until you've bootstrapped the rest of society at least to the point where something like high-resolution trigonometric tables, desktop publishing, high-speed accounting, (for example) are needed, the effort isn't going to keep you fed...
As far as mechanical devices, mechanical calculating machines didn't arise until the late 1600s and weren't reliable for many years. It's unlikely that you'd be capable of building a mechanical computer until the industrial revolution. Note that Babbage was unsuccessful in building his machines even in the late 1800s.
If your goal is to build a Turing-complete machine of some sort, even if totally impractical, you could push the date back a lot. But that would be more of a curiosity than a useful computer.
Some semi-random examples
https://web.cecs.pdx.edu/~harry/Relay/
Main issue is memory. Takes a lot of space to make any usable about of memory out of relays.
I used to play that same thought experiment with more basic utilities like my toaster with it's various settings and electronic controllers. Then I was given a Dualit. No more philosophical dilemmas!
Kidding aside, it's always staggering how far removed we really are from operating on (humanly) first principles. Humbling.
But if you read those blogs you still notice the mind boggling height of the giants whose shoulders the bloggers stand on. Having access to simple chemicals like acids or various salts for example is huge. I wouldn't even know where to start if I had to bootstrap a highschool chemistry kit starting with nothing but my hands and my knowledge.