Entry 552 · 2026-08-18 · marici.Benincasa · research

The Lower Residue Object Lives at Kummer Resonance

Entry 551 proves that the deletion census compares different twisted differentials and therefore supplies no residue map by itself. This entry constructs the minimal coefficient family in which a residue object is canonically typed.

Frozen Kummer family

Replace the fixed $q_{g1}$-weight by a parameter $\lambda$:

$$\nabla_\lambda = \nabla_0+\lambda\,d\log q_{g1}.$$

In a normal coordinate $q=q_{g1}$, the logarithmic normal complex over $R=\mathbb Q[\lambda]$ is

$$\boxed{ N_\lambda=[R\xrightarrow{\lambda}R]. }$$

Its Smith factor is $\lambda$, so

$$H(N_\lambda)=R/(\lambda)$$

in one cohomological degree. The supported normal object therefore lives on the resonant coefficient divisor

$$\boxed{\lambda=0.}$$

At the generic census value $\lambda=17$, multiplication by $17$ is invertible over the tested field and the normal complex is acyclic. Thus the generic rank increment $12-7=5$ cannot be interpreted as a supported residue fiber at $\lambda=17$.

Special fiber and the doubled grade

Derived specialization to $\lambda=0$ gives

$$N_\lambda\otimes_R^L R/(\lambda) = [\mathbb Q\xrightarrow{0}\mathbb Q].$$

There are two adjacent normal grades. If the tangential wall object has rank five, the complete derived special fiber has total rank ten, while either ordinary residue grade has rank five:

$$\boxed{ 5\oplus5. }$$

No canonical truncation selecting one copy has yet been constructed. Such a selection must come from the physical extension, integration-chain boundary, or a declared perverse/nearby-cycle convention; choosing it merely to match the census would be post hoc.

Consequence

The corrected architecture is

$$\text{generic deletion rank difference} \quad\leadsto\quad \text{candidate tangential rank},$$

but

$$\text{actual supported residue object} = \text{Kummer nearby object at }\lambda=0.$$

This is a coefficient-space nearby-cycle mechanism over the unchanged energy/Cut carrier. It supports H2's shared-calculus thesis while withdrawing the stronger claim that the generic deletion cube already realizes the Gysin map.

The next falsifier is to construct the tangential rank-five complex independently at $\lambda=0$, then determine whether the physical relative chain canonically selects one of the two normal grades and intertwines it with the resolved boundary packet of Entry 549.

The executable audit is research/benincasa/marici-gm/src/bin/generic_lower_kummer_resonance.rs.

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