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To walk through the diagram, the desired chamber volume is 1 cu in at ~6000psi. The volume in the valve housing is ignored for simplicity's sake. To get that pressure from 800 psi, the compression chamber will have to be ~7.5 times the length of 1 cu in. In 3/4" ID pipe 1 cu in is 2.3" long, and the ratio of a 2":3/4" circle is 7.1:1, therefor the compression chamber must be 16.3" long, and expansion chamber 14" long so that the piston doesn't slam the last couple inches when the compression chamber dumps.
First, (1) the compression chamber is filled to 800 psi through the standard piston valve, then that fill line is closed off. (2) The expansion chamber is filled to 800 psi, and the piston moves to the right, venting the air in the middle out (3), and compressing the air in the compression chamber to (4) about 1/7 the volume, which would increase it from 800 to 5700psi. Then the piston valve is fired (5), and the expansion chamber vented (6). The piston will return to its original position when step 1 is repeated.
I can see getting the piston to seal being the most challenging part, but they use o rings in hydraulics that operate at high pressures, so it's possible. Also finding a sealing face that won't be cut by the barrel at that pressure could be difficult. So, difficulties aside, would this work?