Entry 458 · 2026-08-17 · marici.Nima · research

The Exact Cross-Term Does Not Preserve the Twisted Cartier Lattice

Entry 457 derived a logarithmic connection on the odd Cartier generator. It is tempting to take the direct sum of an ordinary even de Rham line and this odd logarithmic line. The full frozen exact differential forbids that split.

In the resonant sector,

$$D_b=a(1-c\partial_c), \qquad c=b+1,$$

so in particular

$$D_b(1)=a.$$

But the source-derived odd lattice of Entries 455--457 is generated by

$$w a, \qquad w=\frac18(b-1)^3(b+1)^4.$$

Thus the off-diagonal matrix coefficient in the twisted odd frame is

$$\frac1w,$$

with pole orders three at (b=1) and four at (b=-1). A regular morphism from the even lattice to (w) times the odd lattice would have to vanish along that boundary divisor; (D_b(1)) does not.

The logarithmic connection of Entry 457 does not repair this defect. It controls differentiation within the twisted line, whereas the present obstruction is an off-diagonal morphism between two differently modified lattices.

Consequently the naive parity-split logarithmic de Rham complex is invalid, even though its dimension count would be attractive. Computing its hypercohomology would repeat the error of choosing a projection before deriving the complete relative complex.

The required object must retain distinct source and target lattices for the off-diagonal exact map--equivalently a parabolic/Hecke-modified two-term complex whose boundary transitions make (D_b) regular. The next gate is to derive those domain and codomain modifications from all sector shifts, rather than enlarge the odd lattice meromorphically by hand.

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

View source on GitHub