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To summarize, endochitin would be chitin that an insect uses for internal reinforcement as beams or some sort of network of structures that would grow along with their chitin exoskeleton. This would basically give them two skeletons, one internal and one external. To my knowledge there aren't any insects with this feature, but I don't think it's too far fetched for it to come about at some point if the pressures were right.

The point of endochitin would be that it might allow them to get around the square-cube law that prevents insects from becoming too big due to the inevitable structural failure of their exoskeleton. I'm aware that there are more things keeping bugs from becoming too big, like oxygen concentration of the atmosphere and the fact that they don't have that great of a respiratory/oxygenation system, but for the purposes of this question we'll assume that all other conditions are favorable such as when they were at their biggest in the carboniferous period.

Would endochitin allow insects, or in the more general referral as arthropods, to become even bigger than they ever were?

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It sounds like an unnecessary complication.

If you are going to have "beams" or whatever to sustain the body, you are practically creating an endoskeleton.

At that point it makes more sense to spend your resources on a single effort: either the exoskeleton or the endoskeleton.

Remember that resources are always limited, and the organism using them better has an advantage. If you are placing additional resources on a double skeleton, you are removing them from something else.

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The current limiting factor on insect size isn't the exoskeleton but oxygen concentration.

Without lungs oxygen needs to diffuse through tissue. There used to be dragonflies with 28 inch wingspans when oxygen concentrations were greater. The leading theory for why there are no insects that large today is reduced oxygen availability.

So even with a better structural system (See L.Dutch's post for more about why that's not so simple), you're still not going to get gigantic insects.

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Since the question is "bigger than they ever were" I am assuming the oxygen problem is fixed by higher oxygen concentrations and/or a unique pulmonary cardiovascular system.

The problem is the combination of muscle and skeleton. You can increase the volume of the skeleton/chitin, but that means you need more muscle power. You could keep the muscles as-is, but that means that just walking around now takes the energy of jogging and at even larger sizes its a constant olympic achievement to just walk around, its very exhausting and you are limited in your speed! The solution then is to increase the diameter of the muscle, muscle is heavy and requires space. So now you are in a rocket equation: more muscle means you weigh more and need more space within the chitin and more structural chitin which means you need more muscle...

If all you had to do was stand there and you had a rather unique pulmonary cardiovascular system you might be able to scale it up to an elephant, but it would not be able to hunt or eat or even walk around.

Now this is a slight exaggeration, you can scale up the insects a little before you get into trouble. How much? I dont know.

As for the idea to use an endo+exoskeleton, if done right it might give you a strong advantage. But only if you can manage the highly complex space distribution within the body where you still need all the flesh, nerves, blood vessles, underappreciated lymph system, fat etc to be present and able. The strenght comes from how you can locate and anchor muscles. A crab can pinch much harder than a human can because the muscle is in a highly efficient position meaning most of its pull is in the direction you want and it has a leverage, where all muscles in the human body have anti-leverage* and take more energy for the same movement. With your adaptations it might be possible to create more of such highly idealized muscle attachments and limit the amount of distance muscles have to bridge to their attachments, saving weight that is immediately swallowed by the extra internal chitin. Although that only works for non-multi-articular muscles.

*My life prevents a lot of sleep right now and finding the proper words is difficult.

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