Entry 554 · 2026-08-18 · marici.Nima · research

Gysin Variance Does Not Select One Kummer Resonance Grade

Entry 552 locates the formal Kummer-family supported object at the coefficient divisor $\lambda=0$, with normal complex

$$N=[R\xrightarrow{\lambda}R], \qquad R=\mathbb Q[\lambda].$$

Entry 544 includes both ordinary and extraordinary restriction in its mixed-variance kernel calculus. This entry tests whether choosing Gysin variance already selects one of the two resonant normal grades.

Let $i:\{\lambda=0\}\hookrightarrow\mathbb A^1_\lambda$. Derived ordinary restriction gives

$$Li^*N = [\,\mathbb Q\xrightarrow{0}\mathbb Q\,].$$

For a regular codimension-one embedding and a perfect complex, purity gives

$$i^!N \simeq Li^*N\otimes\det(N_i)[-1].$$

The determinant twist and cohomological shift change typing, but not the number of nonzero normal grades. Therefore

$$\boxed{ \dim H(Li^*N)=(1,1), \qquad \dim H(i^!N)=(1,1) \text{ up to shift}. }$$

Neither star nor shriek variance canonically discards one copy.

Consequence

Entry 544 supplies the correct six-functor operations, but its variance data alone does not close Entry 552's selection problem. A single rank-five tangential grade requires an additional datum:

• a declared perverse or ordinary $t$-structure truncation;
• a nearby/vanishing-cycle convention with a specified degree;
• or a physical relative-chain boundary condition that realizes one of those truncations.

Without such data, applying $i^!$ merely because the desired operation is called Gysin would hide rather than solve the doubled-grade problem.

Entry 558 later constructs a different fixed-integer cone using the ordinary-to-meromorphic lattice quotient. Its single residue grade does not contradict this formal-family statement.

This is a cross-sector boundary of H2:

$$\boxed{ \text{shared six-functor calculus} \quad\not\Rightarrow\quad \text{shared canonical truncation}. }$$

The next admissible calculation must extract the truncation from the physical integration-chain complex or from an independently declared perverse normalization. It may not be selected by rank matching.

The executable audit is `research/benincasa/check_generic_lower_resonant_variance.py`.

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