Well, if it were to be in our dimension, it'd have to be far enough in the future for three things:
1. Rabbits become Bipedal creatures.
2. Chunks of land to leave earth yet be trapped in earths gravity well
3. Humanity inventing robots that weaponize incredibly hightech weaponry.
There's a small window (In perspective of thousands of years) where Humanity still has to be alive, yet there need to be drastic changes in Earth.
1. The problem with this, is the fact that basic evolutions take millions of years, let alone bipedal creatures developing language. This could be chalked up to mutation due to an outside factor, (gene splicing, small doses of radiation, forced natural selection due to new environmental circumstances, etc.) Yet that'd still take hundreds of years for the Mimiga to be a 100% sentient life-form.
This variable will be represented as M
Answer 1: Anywhere between 100-1500 years in the future natural evolution is discounted due to humanity probably being extinct before they naturally become sentient creatures with language.
2. I don't even know how to approach this, this idea seems so abstract scientifically, that an explanation is almost inconceivable. So I'm going to make assumptions:
Assumption 1: The chunk of land floating through the sky is part of the earth due to a mining related disaster to explain the water canals and cave systems that look man made. (+35 years)
Assumption 2: Judging from images of pre-Cave Story Island's orbit, it appears to be in the earth's atmosphere, fair enough as there wouldn't be any life on it if it weren't for Earth's atmosphere.
Assumption 3: To give a bit of lee-way, the island straddles the equator, at 42'000 feet.
Assumption 4: The Island orbit's earth in a perfect circular orbit at this 42'000 foot orbit,
Assumption 5: The island's acceleration is equal to 0.
So to acquire this orbit we'd have to use the equation v = root(G(M+m))/r Which equates to v=root(Mu)/radius, which leads to our next assumption:
Assumption 6: Mu, (GM+Gm) must equate to the standard gravitational parameter of earth, (398, 600.44189) coupled with the 42000 feet in the air, makes r= 20'945'520 ft. or 3966.9545m. or the number we'll use, to match Mu's use of metric unit that I'm too lazy to convert, 6384.1944km
This makes our new equation v=root(398,600.44189km(cubed)/s(to the power of -2)/6384.1944km) ... (v = (631.34811km(squared)/s(-2)/79.90115km) ... (v = 7.9016km/(s-2)) the equivalent of (0.01601km/s*3600 = 57 km/hr) fitting as things are sucked directly to the walls on the outside walls a bunch of math later I found the orbit of an object this size at this distance is 29 days.
Assumption 7: This 29 day orbit is 100% stable outside of Ballos and core.
Now, before the doctor there were 3 other kings, which leads to-
Assumption 8: There were three generations before (10 years ago.) Ballos era, King 1, and King 2. (+10)
Assumption 9: Each generation is equal to 1258 orbits, (Or 100 years in our time) due to the theory of Gravitational Time Dilation. (29*1258=100 years per generation.) (This is just a rough estimation, I'm not actually going to do the Time Dilation Equation) (+300) (+/-15 years for leeway)
This estimation will be represented as I
Answer: 330-360 years
3. a)Now, laser technology and robots are not far off as we speak, but from what I can tell, laser weaponry seems nigh-impossible, because if you missed, it would just keep going until it hit something, because it's light. And the amount of energy required to blast matter in such a concentration, that it becomes such that of pure energy or near energy, the energy required can be represented by this equation:
EIU<ER<Infinity
EIU = Total energy in universe at man's disposable use.
ER = Energy required for this sort of weapon.
Infinity = Uncalculatable value
Luckily, we can discount the spur and the snake as bullshit, the flamethrower as a flamethrower, the bubbler as a bubble blower, and the rocket launcher as a rocket launcher, in the real-life canon we make, the Snake and Spur don't exist.
"But guy, how do we know what the machine gun and polar star are?" I was getting to that, the polar star and machine gun are railguns (weapons that take a projectile made of metal, and use magnets to project the projectile at massive speeds.
Here's a diagram:
-|+-|+-|+-|+-|+-|+
+>><><><><><><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<|
-|+-|+-|+-|+-|+-|+
This same use of magnets is how scientists are trying to make a perpetual motion machine.
These all can be made today so (+0 years)
b) Robots, I already told this but there are three generations of robots seen in Cave Story
1) Massive, clumsy robots with an unknown purpose (Balrogs)
2) More humanoid, but still very primitive (Core room robots)
3) Quote and Curly. (Incredibly advanced machines that have passed the threshold into android territory.)
Now, just like with synthesizers and computers, in repeating gaps, you can see them get smaller and smaller, (1950's computer is massive, 1980's small box, 2010's they're in phones and tablets.) (1960's Moog Synth is this massive box with knobs, 1980's they're size of computers, 2000's they are on a small processor)
So to estimate the time I had to research robots.
1954: First fully autonomous programmable robot "UNIMATE" made.
2009: Numerous autonomous programmable humanoid robots showcased at IREX Tokyo
So at this rate of 55, let's keep this pattern going.
2054: Theodore Baxford unveils fully autonomous robot, the Baxford Locomotive Utilizing Realtime Programming (BALROG, for short) at IREX London
2109: United Emirates of America's Military is shown to be in possession of humanoid robots.
Answer: So, from this awesome timeline, it's safe to assume Quote and Curly were made (10 years ago.) However the shocking jump from programmed soldiers to androids makes me think there's multiple steps inbetween the Core Room Robots. regardless, this adds a baseline of 185 years in the future.
This estimation is represented by R.
So now here's the equation I'll use to represent this date.
When (T) requires humanity's survival to atleast (10 years ago) 185 years in the future, to account for R(obot) factor.
T + R > 2198 C.E.
However, the next factor, I(sland) factor, requires 331-359 years in the future, if it were to begin right now. This has to happen after the next factor M(imiga) as there's no other explanation as to why they're on the island.
It's also safe to assume the melatonin red flowers are made by man, unknowingly conflicting with the spliced mimiga.
So the new equation is T + R > I + M or T > I + M - R
T > (330>360) + (330<1500) - 185
As you can see, this creates an irrationality in I where the floor value is larger than the roof value in order for the value to be either floor or roof value, so I'll need to add +/- after, to stabilize.
T > 345 + (330<1500) - 185
Now I subtract R from M's floor and roof.
T> 345 [|(345<M<1315) + (<970)]
The addition of M is conditional, as the addition only happens if the value is above I.
This makes the Sum 330<T<1315
We still haven't added ten years. so +10 universal
340<T<1325
Now to make it a year
2'353<T<3'338
If the events leading to Cave Story happened now.
It'd be Cave Story O'Clock somewhere 2353 and 3338.