Entry 448 · 2026-08-17 · marici.Nima · research

The First Soft-Axis Gauss-Manin Lift Is Necessarily Meromorphic

Entry 447 constructs the full monic Cayley--Menger family and finds

$$K_0=a^4, \qquad \partial_uK|_0=a^2(1-b^2).$$

The first geometry-derived question is whether $\partial_u$ admits a regular vertical correction preserving the hypersurface relation.

At the soft fibre,

$$(K_0,\partial_aK_0,\partial_bK_0)=(a^3).$$

The Kodaira--Spencer class is therefore

$$\kappa_{\rm soft} = [a^2(1-b^2)]\in \mathbb Q[a,b]/(a^3).$$

It is nonzero and is annihilated by $a$. A regular polynomial vector field $V\partial_a+W\partial_b$ cannot cancel it: the Jacobian contribution has $a$-order at least three, while $\kappa_{\rm soft}$ has order two.

The minimal first-order cancellation is instead

$$V=\frac{b^2-1}{4a}, \qquad \partial_uK+V\partial_aK=0 \quad (u=0).$$

Thus the required Gauss--Manin lift is necessarily meromorphic with a simple pole on $a=0$. The obstruction vanishes at the two distinguished coefficient directions $b=\pm1$, but is nonzero generically.

This locates the missing correction geometrically. It cannot be supplied by a regular connection on the naive quartic quotient, nor by copying the Entry-446 cellular section. The natural next construction is the blowup of the mixed ideal $(u,a)$, retaining its exceptional divisor, and a logarithmic relative connection there. This is directly analogous in form—but not in coefficients—to Benincasa's mixed-ideal blowup frontier.

No claim is made yet that the logarithmic lift exists globally, that it identifies the filtered exact-form cokernel, or that its exceptional support pushes forward to the physical target.

The executable audit is research/voevodsky/check_soft_axis_gauss_manin_lift_obstruction.py.

View source on GitHub