The Full Weighted Strict Transform Is a Trivial Double Section
Entry 460 isolates the remaining nearby-cycle question. Before transporting the exact complex, the full Cayley--Menger carrier itself admits an exact factorization on the weighted (u)-chart.
Use (a^2=ut), divide the full relation by its exceptional factor (u^2), and put
Direct coefficient comparison gives
Thus the polynomial translation
identifies the complete strict-transform family with
over the (u)-base. The doubled section does not split, collide, or acquire nontrivial carrier monodromy: it is algebraically constant after translation. In particular, the reduced section has identity monodromy.
This sharply locates the remaining difficulty. Ordinary topological nearby cycles are insensitive to nilpotent thickening, so the carrier hypersurface alone cannot distinguish the two Euler classes or generate their transport. Any nontrivial specialization datum must reside in the transformed exact-form complex--its sector lattices, differential, or support framing--not in the Cayley--Menger equation.
The result also explains why the Newton initial form was a perfect square: it was the special fiber of a perfect square throughout the weighted chart, not merely an accidental leading-order degeneracy.
The next gate is to conjugate all transformed exact operators by the same translation (z=\psi-5u/4+u^2/2), then test whether their two-class cokernel is constant, has a nilpotent extension, or still mixes with the quartic tail.
The executable audit is research/voevodsky/check_soft_axis_strict_transform_square.py.