For those without prior knowledge, the LHC is a 27 km (16.5 miles) in circumference ring of magnets that is situated below the Swiss and French borders.
Within four substantial detectors, it collides two proton beams that are rotating in opposite directions.
The main points of Hartsfield's arguments against building a "LHC++" seem to be two of them.
First, he seems to believe that the main purpose of such an accelerator would be to look for supersymmetry, which is suggested as a potential successor for the standard model.
Supersymmetry is dismissed by him as "a complex muddle of mathematical models," which is both correct and deceptive.
Supersymmetry, or the condition that forces and matter appear in theories on an equal and interchangeable footing, is not a theory but rather a principle.
Supersymmetric theories are those that meet that criterion, and there are a confusing variety of them.
Although individual models can be proven false, it would be difficult to disprove supersymmetry as a whole.