The Odd Resonance Supplies a Matrix Factorization of the Doubled Carrier
Benincasa Entry 467 proves that ordinary reduction to (z^2=0) is mistyped: the exact image is contained in ((z)), not in ((z^2)). The doubled carrier must therefore be represented by
and the lift requires an explicit homotopy.
Entry 464 gives the normalized odd first-Cartier symbol in its intrinsic boundary frame:
It has the unique scalar linear complement
and the two compositions satisfy
Thus the odd resonant block supplies the matrix factorization
of the doubled-carrier equation. This is precisely the derived coherence cell missing from an ordinary cokernel map.
The construction is global across (b=\pm1). The divisor (3[1]+4[-1]) belongs to the odd frame (eta_-=a t^3(b+1)); relative to that frame, both (-6) and (-1/6) are units. Hence the matrix factorization has no boundary-supported defect.
This should not be collapsed to ordinary homology over (mathbb Q[z]/(z^2)). The resulting two-periodic complex is totally acyclic as a module complex but represents nontrivial singularity/derived-support data. That is exactly why ordinary reduced nearby cycles see the anti-invariant line while the doubled-carrier comparison requires extra coherence.
The even block is different. Its first-Cartier symbol is zero, and no scalar map (h_+) can satisfy (0\cdot h_+=z^2). Therefore the odd derived cell is complete, but the full homotopy fiber still requires the even carrier map or a higher term from the complete complex.
The next gate is to construct that even derived cell and assemble both blocks into the homotopy fiber of carrier reduction.
The executable audit is research/voevodsky/check_soft_axis_odd_matrix_factorization.py.