91' Hallmark Bed Sag Fix

What you're trying to make is called a "torsion box". The skin must be rigidly attached to the frame tying it together as one structure.

Think of an "I" beam: the I shape (would be handy if this font had serifs on the I) is really about the two horizontal surfaces held together at a distance apart by the web. If you support both ends of a beam and add a load to the span, the beam tries to bend under the load. That means the top horizontal surface is in compression - it's trying to get shorter; and the bottom horizontal surface is in tension - it's being stretched. The further apart those two surfaces are, the stronger. The separation of the horizontal flanges influences strength more than their size - this is whey a 2x4 is stiffer on edge than laying flat.
The center of the web has no load at all on it - you'll sometimes even see I beams with large holes in the web to reduce material and therefor weight/ cost.

Ok, so for a TC cabover, you want the web as tall as possible, and you want the skin (plywood) glued/ screwed to the frame rigidly. Even foam insulation board in this scenario - if continuously attached to a skin on each side - adds significant structural strength. That's why airplanes, boats, etc. are often made of a tough, strong skin (carbon fiber) sandwiching some lightweight honeycomb material. It's an I beam in sheet form... like the floor of a TC cabover.

I worked the math out for a platform bed span I made years ago: Click for math.
I was frustrated by very seasoned pro woodworkers not really having any idea how to CALCULATE or ENGINEER the structure... they all just guessed and used their intuition, recommending a structure which would have been a LOT heavier and more expensive to make. From the date on that blog post, I can tell you that bed platform has been in service for 5 years now, no problems.