A facet-exchange counterexample to the fixed-mass regular-simplex conjecture

We give a written counterexample to the arbitrary-positive-mass statement of Heilman 2019 Conjecture 1.16. For every 0<p<1/4, the mass vector (p,(1-p)/3,(1-p)/3,(1-p)/3) has a strict non-regular improvement. The equal-mass tetrahedral problem is untouched. The record contains the frozen PDF, source package, exact Q(sqrt(2)) diagnostics, partial Lean algebra, and a targeted prior-work comparison. No external peer review or worldwide priority claim is made.