Update to search results: 4 vertices in 2 dimensions --- pushed up to max edge 50, finding 1579 distinct proper integer cases. Approx 450 are non-cyclic. None is ambiguous. 5 vertices in 2 dimensions --- 29 noncyclic cases found by gluing pairs of the above. [6 of these constitute an apparently promising set where each pair of members has 6 edge-lengths in common; sadly, attempting to glue these into a 6-tope fails.] 5 vertices in 3 dimensions --- pushed up to max edge 12, finding 217 cases; 5 conspheric, 2 ambiguous pairs. 6 vertices in 3 dimensions --- I earlier opined (along with MK)
Almost certainly a considerably more extensive search is required to find such an object, if it exists then went ahead and tried anyway. Mirabile dictu, gluing pairs produces just the single example (ta-DAH!) ---
[36, 33, 30, 27, 18, 15, 21, 30, 30, 27, 21, 18, 27, 18, 13] with the customary order of edges [AB,AC,BC,AD,BD,CD,AE,BE,CE,DE,AF,BF,CF,DF,EF]. I can't resist digressing on how regularly our loosely probabilistic intuitions about these objects have been confounded :--- (a) Apparently, concyclic charts should be in a small minority, since they satisfy an extra constraint. In 3-space this is indeed the case; but in 2-space the majority turn out to be concyclic [see Michael Reid, this thread Dec 12th]. (b) In 2-space, parallelograms and trapezia [two pairs of equal edges] ought also to be uncommon; but for similar reasons, they too actually preponderate [WFL, this thread Jan 3rd]. (c) The reasonable guess that "isosceles" 5-topes in 3-space [with 3 pairs equal] turns out to be similarly way off-target --- these arise in a straightforward fashion by gluing together equal tetrahedra [MK, this thread Jan 6th]. (d) Conversely, the more general "gluing" procedure, with which I optimistically set out to construct (proper) integer charts in n-space with more than n+1 vertices, produces hundreds of improper (or irrational) examples for every proper one --- despite the probability that a random n-tope has zero volume being vanishingly small. [In 2-space, these duds arise by combining a parallelogram with a triangle, so that one side of the triangle coincides with a diagonal of the p'gram, and the other vertex of the triangle lies on the other diagonal of the p'gram. If all the existing edges are integer, then the single new edge created by gluing is the difference of two integers --- but for the same reason, the new triangle it belongs to is linear!] In all the above, the reason for the dodgy insight is that some "special" case turns out far more frequent than the general. The correct approach here seems to be (somehow!) to enumerate and consider these special cases beforehand; and only risk a probabilistic argument once they have been eliminated. Fred Lunnon