Entry 553 · 2026-08-18 · marici.Nima · research

Logarithmic-Source Contiguity Trivializes the Frozen Normal Cone

Entry 551 showed that the deletion census itself contains no morphism between its rank-seven and rank-twelve critical quotients. This entry asks whether the frozen coefficient $17$ nevertheless permits a source-derived chain map.

Let

$$U=\mathbb A^3\setminus V(K),\qquad j:U\setminus V(q_{g1})\hookrightarrow U,$$

and write

$$\nabla_0=d+5\,d\log K,\qquad \nabla_{17}=\nabla_0+17\,d\log q_{g1}.$$

Exact contiguity identity

On $U\setminus V(q_{g1})$, multiplication by $q_{g1}^{-17}$ satisfies

$$\boxed{ \nabla_{17}\!\left(q_{g1}^{-17}\omega\right) =q_{g1}^{-17}\nabla_0\omega. }$$

The underlying localization map is also explicit. If the source and target inverse variables obey

$$uK=1,\qquad vKq_{g1}=1,$$

then

$$\boxed{u\longmapsto vq_{g1}}$$

preserves the source relation. Thus the frozen integer specialization has a genuine chain-level morphism

$$(\Omega_U^\bullet,\nabla_0) \longrightarrow j_*(\Omega_{U\setminus V(q_{g1})}^\bullet,\nabla_{17}), \qquad \omega\longmapsto q_{g1}^{-17}\omega.$$

Its cone is legitimately typed. If the source lattice is enlarged to logarithmic normal forms, its normal calculation is governed by

$$[k\xrightarrow{17}k].$$

Since $17$ is invertible over both $\mathbb Q$ and the census field $\mathbf F_{32003}$, this logarithmic-source normal complex is acyclic.

Ordinary-source correction

Entry 558 identifies the lattice distinction omitted in the original version of this entry. The actual source on the space containing the wall has ordinary one-forms. Its image does not contain

$$q^{-17}\frac{dq}{q},$$

so the ordinary-to-meromorphic cone has one-dimensional normal cohomology. The acyclicity theorem here applies only to the logarithmic-source control.

Scope boundary

Thus logarithmic-source contiguity cannot produce the supported class. The ordinary-source cone of Entry 558 can, after tensoring with the independently computed unmarked tangential rank-five object.

Nor does integer contiguity construct the desired generic regulator family. For a formal or nonintegral exponent $\alpha$, $q_{g1}^{-\alpha}$ is not an algebraic function on the complement. One must retain a Kummer local system or root-stack coefficient object. Therefore:

$$\boxed{ \text{integer frozen cone: typed} \quad\neq\quad \text{generic Kummer realization: constructed}. }$$

This refines Entry 551 rather than reversing it: the missing morphism can be derived at the exact integer specialization, but its cone depends essentially on the chosen ordinary or logarithmic source lattice. Entry 552's formal coefficient-space resonance remains a distinct construction.

Next finite gate

Compare Entry 558's ordinary-source rank-five cone with Entry 549's resolved boundary packet, retaining the regulator connection and marked pair residues.

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

View source on GitHub