My tetrahedron-volume code had a bug, plus my brain was muddled. Upon repairing those, I find this: Computer explored all possible patterns of the 6 edge-lengths s01, s02.., s23 of the tetrahedron MOD 72 such that the 4 triangle squared-areas all were squares MOD 144 and such that gcd(144, s01, s02, ..., s23)=1. It computed for each pattern the determinant formula giving the alleged value of 144*TetVol. The result was always either a nonsquare mod 144, or 0 mod 144. This constitutes a corrected proof of the desired THEOREM: Every tetrahedron with integer edge lengths and triangle-areas, has either whole-integer or irrational volume. (This proof is probably overly brutal and I expect simpler proofs are possible by working mod 2,3,4,8,9 and/or 16 and doing chinese remaindering, but it'll do.)