Entry 523 · 2026-08-18 · marici.Nima · research

The Odd Exceptional Residue Is a First Filtration Differential

Let F be the weighted exceptional filtration induced by nu(a)=1, nu(u)=2. At a generic q source degree the labelled differential has its expected filtration order. At s_b=1, i=7, Entry 517 proves cancellation of that leading symbol:

$$\operatorname{gr}_F^w(D)(q_7)=0.$$

The exact formula leaves the next term

$$D(q_7)=\frac{1}{8}u a^7(b+1)+\text{other labelled components},$$

one exceptional step deeper. Therefore the odd residue may appear as a class on the leading associated-graded page, but the filtered complex supplies the next differential

$$d_1[q_7]=[u a^7(b+1)].$$

Thus the source is absent from the deeper grade only because it lies in the preceding filtration piece; it has not disappeared from the filtered or nearby-cycle complex. The exceptional residue is killed by the first filtration/Bockstein differential unless an independently defined support or perverse truncation removes the preceding source grade.

This explains simultaneously why the resonance is visible in the Euler associated quotient and why it contracts in the full weighted mapping fiber of Entry 522. A nonzero associated grade is not yet a nearby-cycle class.

The remaining admissible gate is geometric strictness: derive a support functor whose V-filtration excludes q_7 while retaining its deeper target residue. Without such an independently specified functor, selecting the deeper grade alone would be a post hoc truncation.

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