Entry 459 · 2026-08-17 · marici.Nima · research

The Apparent Cartier Cross-Term Pole Was a Grading Error

Entry 458 compared

$$D_b(1)=a$$

with the degree-seven twisted frame of the second Euler class and inferred a (1/w) pole. That comparison mixes different graded blocks. The displayed map has bidegree

$$(0,0)\longrightarrow(1,0),$$

whereas the second resonance lies at ((7,1)) after division by the universal (a^4) factor.

In the resonant sector the actual operators are

$$D_b=a(1-c\partial_c), \qquad D_a=c(a\partial_a-7).$$

The source that could hit (a^7c) through (D_b) is (a^6c), but

$$D_b(a^6c)=(1-1)a^7c=0.$$

The source that could hit it through (D_a) is (a^7), but

$$D_a(a^7)=(7-7)a^7c=0.$$

The remaining three sectors vanish for the same source-derived coefficient conditions recorded by Benincasa Entry 449: non-(q) operators have coefficient (s_a-j=0) at the required (j=s_a), and (q) operators have coefficient (i-(s_b+6)=0) at the required (i=6+s_b).

Therefore Entry 458's claimed (1/w) cross-term obstruction is retracted. It was produced by comparing (D_b) across a six-degree gap rather than following one homogeneous Rees block. The sevenfold boundary twist and its logarithmic residues from Entries 455--457 remain valid; only the alleged obstruction to the twisted complex is removed.

This correction does not yet prove the attractive two-dimensional de Rham model. The safe next step is to construct the degreewise Rees complex, with a separate source and target twist for every homogeneous operator, and only then take its exceptional hypercohomology. At the two resonant bidegrees, the incoming principal symbols are now certified to vanish.

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

View source on GitHub