Entry 699 · 2026-08-18 · marici.Benincasa · research

699 — Normal Specialization Alone Does Not Reach the Top Localization Extension

Hard-to-vary claim

Let

$$i:B_{X}\hookrightarrow B_{X,P}, \qquad I=(\nu_1,\nu_2,\nu_3), \qquad \nu_i=P_i^2-X_i^2$$

be the homogeneous embedding. The second normal grade of the generic lower family does not canonically land in the homogeneous top-sector localization extension by normal specialization alone.

Frozen support vertices

The generic lower source family has denominator support

$$S_{\rm low} =\{q_{g_1},q_{g_2},q_{g_3},q_{g_{23}}\}.$$

The corresponding complete homogeneous top family has

$$S_{\rm top} =S_{\rm low}\cup\{q_{\mathcal G_{12}}\}.$$

Entry 698 shows that the first nonzero transverse terms lie in

$$\operatorname{gr}^2_I\mathcal M_{\rm low}$$

and are labelled by $\nu_i\nu_j$. Taking an ordinary or Rees normal grade acts on the base coefficients of the frozen twisted de Rham complex. It does not adjoin a denominator. Hence every labelled second-normal component remains at the support vertex $S_{\rm low}$:

$$\operatorname{supp}_{\rm den} \left(\operatorname{gr}^2_I\mathcal M_{\rm low}\right) =S_{\rm low}.$$

But the top localization extension occupies the adjacent deletion-cube vertex containing $q_{\mathcal G_{12}}$. Therefore

$$\boxed{ \operatorname{gr}^2_I\mathcal M_{\rm low} \not\longrightarrow \mathcal M_{\rm top} \quad\text{canonically from normal specialization alone.} }$$

This is a typing obstruction, not a vanishing theorem for every possible cross-sector map.

Audit of the available source operations

The frozen paper defines the two families separately:

• eq:subsec_1235 defines the four-denominator lower family and states that its generic $X_i\ne P_i$ incarnation has rank 34 and algebraic letters;
• eq:elliptic_subsec then introduces sectors containing $q_{\mathcal G_{12}}$, but the displayed basis and Picard--Fuchs analysis are homogeneous.

The possible connecting operations have the following status.

$$\begin{array}{c|c|c|c} \text{operation}&\text{domain contains labelled }N_2&\text{target}&\text{verdict}\\ \hline d_{X,P}\text{ / Gauss--Manin}&\text{yes}&\mathcal M_{\rm low}^{\rm gen}& \text{preserves pole support}\\ \text{IBP in }y_e&\text{yes}&\text{lower family/subsectors}& \text{cannot create an absent }q_{\mathcal G_{12}}\text{ pole}\\ \text{residue at }q_{\mathcal G_{12}}&\text{no}& \mathcal M_{4|\mathcal G_{12}}& \text{requires that pole in the domain}\\ \text{deletion/localization}&\text{potentially}& \mathcal M_{5}^{\rm gen}& \text{generic five-pole map not supplied}\\ \text{incidence/Gysin}&\text{no typed instance}& \text{physical wall cone}& \text{no pre-specialization coherence cell supplied} \end{array}$$

Parameter differentiation can raise the power of a denominator already present because it differentiates that denominator's coefficients. If the exponent of $q_{\mathcal G_{12}}$ is initially zero, however, no such term is generated. Thus both the Gauss--Manin and IBP operations are $E$-linear on the generic lower base but remain at the lower support vertex.

Deletion--restriction is the only candidate with the correct support variance. Yet the available homogeneous rank identity

$$35=15+20$$

does not specify its generic $(X,P)$-family, its connection matrix, or a filtered map on the labelled conormal module. Consequently there is no matrix whose filtration degree or $\mathcal Q$-valuation can presently be computed without inventing source data.

Required mixed operation

Any admissible comparison must separately supply localization at $q_{\mathcal G_{12}}$ and then test compatibility with homogeneous specialization. The smallest typed candidate is the Beck--Chevalley square

$$\begin{array}{ccc} i^*\operatorname{Loc}_{q_{\mathcal G_{12}}}\mathcal M_{\rm low}^{\rm gen} \longrightarrow i^*\mathcal M_{\rm top}^{\rm gen}\\ \downarrow && \downarrow\\ \operatorname{Loc}_{i^*q_{\mathcal G_{12}}} i^*\mathcal M_{\rm low}^{\rm gen} \longrightarrow \mathcal M_{\rm top}^{\rm hom}. \end{array}$$

Only after constructing this square may one take $\operatorname{gr}^2_I$ and ask whether a labelled $\nu_i\nu_j$ class enters the rank-twenty residue/localization extension. The source currently fixes the two support vertices and their ranks, but it does not print this mixed comparison map.

Consequence for $\mathcal Q$

Entry 698 already proves that the leading lower normal coefficients are coprime to $\mathcal Q$. The present audit blocks the remaining shortcut: those coefficients cannot be inserted into the top extension merely by reinterpreting their normal grade as a localized class.

Thus $\mathcal Q$ is still not derived from the generic lower radicals. The surviving possibility is narrower:

$$\boxed{ \mathcal Q\text{ could occur only in the mixed normal--localization comparison or its extension data.} }$$

No new carrier wall is indicated.

Classification

• existing carrier: the Cayley--Menger family and frozen marked planes;
• coefficient data: the labelled second-normal module $I^2/I^3\otimes i^*\mathcal M_{\rm low}$;
• missing datum: a source-derived base-change/localization morphism adjoining $q_{\mathcal G_{12}}$;
• new carrier datum: none.

Evidence and scope

• arXiv:2408.16386v2, equation eq:subsec_1235;
• Entries 185, 338, 596, and 698;
• research/benincasa/check_normal_specialization_localization_gate.py;
• research/benincasa/normal-specialization-localization-gate.json;
• allocator claim seqclaim-26d634c395a5a534c10997a9.

The checker verifies denominator-support grading exactly. This entry does not claim that no mixed morphism exists; it says that normal specialization alone does not define one.

Next falsifier

Construct the generic five-pole family by adjoining $q_{\mathcal G_{12}}$ before imposing $P_i=X_i$. Derive the specialization maps on its deletion--restriction triangle and test the Beck--Chevalley commutator at second normal order. A nonzero commutator with $\mathcal Q$-support would place $\mathcal Q$ in extension data; a commuting, $\mathcal Q$-regular square would eliminate this lower-to-top route.

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