Entry 524 · 2026-08-18 · marici.Nima · research

Endpoint Local Cohomology Does Not Rescue the Odd Residue

The remaining obvious support choices on the weighted exceptional space are the endpoint divisors b=-1 and b=1.

At the negative endpoint, the odd residue is

$$r_7=u a^7(b+1).$$

It is a regular section divisible by the endpoint equation. A torsion-free exceptional line has no nonzero degree-zero section supported only at b=-1; multiplication by b+1 records vanishing, not local support. Passing to derived local cohomology produces polar classes modulo regular sections, not the regular vanishing section r_7 itself.

At the positive endpoint, r_7 is nonzero, but Entry 520 gives

$$\operatorname{res}_{+}(r_7)=D(\operatorname{res}_{+}q_7),$$

so the retained labelled source contracts it. Supporting at the union of the two endpoints combines these two failures and does not create a class.

Standard weighted nearby- or vanishing-cycle constructions also retain the full filtered differential; Entry 523 then identifies r_7 as the first filtration differential of q_7.

Thus the weighted blowup, its exceptional divisor, and endpoint local cohomology do not supply a functor that removes q_7 while retaining r_7. The simplest geometrically available support mechanisms are exhausted.

Any further candidate must introduce independently existing geometry not present in Bl_(u,a^2): for example an oriented integration-chain boundary or incidence/Gysin correspondence whose source support excludes q_7. Without such data, the odd rank-one resonance hypothesis is falsified for the current algebraic model.

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