The Degreewise Rees Complex Geometrizes Both Euler Resonances
Entry 459 requires following each homogeneous block separately. For a divided target monomial (a^Ic^J), with (c=b+1), its restriction to the Cartier section carries the boundary divisor
at ((b=1,b=-1)). The first component counts tautological Rees factors (t=(b-1)(b+1)/2); the extra (J) in the second counts incidence factors (c=b+1).
This assignment makes the frozen exact operators regular degree by degree. For
the map into target degree ((I,J)) comes from ((I-1,J)). Its lattice increment is zero when (I) is odd and (operatorname{div}(t)=(1,1)) when (I) is even, exactly matching the conversion (a^2=ut). For
the map comes from ((I,J-1)) and its increment is always (operatorname{div}(c)=(0,1)).
The scalar coefficients into ((I,J)) are
Scanning the entire nonnegative bidegree grid, both incoming maps fail simultaneously only at
The first has (B(0,0)=(0,0)). The second has
recovering the sevenfold boundary divisor of Entries 455--457. The other three exact sectors also vanish at ((7,1)) by the source-degree coefficient test of Entries 449 and 459.
Thus the degreewise exceptional Rees cokernel has exactly two primitive homogeneous classes, with the same bidegrees and representatives as the Euler quotient. This is the first construction that simultaneously derives their dimension, grading, and boundary lattices from the weighted geometry.
It is not yet a nearby-cycle theorem. The construction identifies the exceptional associated graded cokernel. The next gate is to derive the specialization/monodromy operator and test whether these two classes survive in the relative nearby-cycle object rather than only its associated grade.
The executable audit is research/voevodsky/check_soft_axis_degreewise_rees_complex.py.