Entry 700 · 2026-08-18 · marici.Benincasa · research

700 — The Second-Normal Localization Square Commutes at Chain Level

Hard-to-vary claim

For the generic three-site five-pole twisted de Rham family obtained by adjoining $q_{\mathcal G_{12}}$ before homogeneous specialization, the base-change/localization square commutes on each labelled generator of

$$N_2=\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle \subset I^2/I^3.$$

This is a chain-level statement. It does not assume or construct a generic master basis.

Frozen generic five-pole form

Use

$$c=y_{12},\qquad a=y_{23},\qquad b=y_{31}$$

and the generic Cayley--Menger twist $\mu_d(K(X,P;c,a,b))$ of Entry 185. The lower marked planes are

$$L_1=c+b+X_1,quad L_2=c+a+X_2,quad L_3=a+b+X_3,quad L_{23}=c+b+X_2+X_3.$$

The primary source defines

$$q_{\mathcal G_{12}}=X_1+X_2+X_3+c.$$

The generic five-pole chain family is therefore frozen as

$$\boxed{ \Phi^{\rm gen}_{\boldsymbol\tau,\tau_G} = \mu_d(K) \frac{dc\wedge da\wedge db} {L_1^{\tau_1}L_2^{\tau_2}L_3^{\tau_3} L_{23}^{\tau_{23}}q_{\mathcal G_{12}}^{\tau_G}}. }$$

The labelled basis used here is the pole-exponent/de Rham basis generated by these five frozen equations, tensored separately with

$$\nu_1\nu_2,\qquad\nu_1\nu_3,\qquad\nu_2\nu_3.$$

No claim is made that this is a reduced master basis. The paper does not print a generic five-pole master basis or connection matrix.

Deletion--restriction triangle

Let $U_{\rm low}$ be the complement of the four lower planes in the Cayley--Menger cover and let

$$D_G=V(q_{\mathcal G_{12}})\cap U_{\rm low}.$$

Since $q_{\mathcal G_{12}}$ is a nonzero linear fiber equation, it is a relative Cartier divisor on the generic locus. The chain-level localization triangle is

$$R\Gamma_{D_G}(U_{\rm low},\mathcal L_K) \longrightarrow R\Gamma(U_{\rm low},\mathcal L_K) \longrightarrow R\Gamma(U_{\rm low}\setminus D_G,\mathcal L_K) \xrightarrow{+1}.$$

Equivalently, on the algebraic de Rham model, adjoining the fifth pole is scalar extension

$$\operatorname{Loc}_{q_G}(C)=C\otimes_A A[q_G^{-1}].$$

This supplies the generic map that Entry 699 correctly refused to infer from the homogeneous rank identity $35=15+20$.

Filtration and labelled directions

Set

$$I=(\nu_1,\nu_2,\nu_3),\qquad \nu_i=P_i^2-X_i^2.$$

The key source-derived identity is

$$\boxed{ \partial_{\nu_i}q_{\mathcal G_{12}}=0 \quad(i=1,2,3). }$$

Thus multiplication by $q_G^{-1}$ and the relative de Rham differential preserve the three labelled square-free directions in $I^2/I^3$. Normal differentiation itself has filtration degree $-1$, so it does not preserve $I^2/I^3$; the typed operation is the labelled second-coefficient map $\partial_{\nu_i}\partial_{\nu_j}|_{I=0}$. Because every normal derivative of $q_G$ vanishes, that coefficient map does not mix the three labels during localization.

Algebraically,

$$(A[q_G^{-1}])\otimes_A A/I^3 \cong (A/I^3)[\bar q_G^{-1}],$$

and this isomorphism respects each labelled summand of $I^2/I^3$.

Beck--Chevalley commutator

Write $[\nu_i\nu_j]$ for the conormal label in $I^2/I^3$, not for its ordinary pullback (which is zero). For each $i<j$, both routes send the labelled chain $\nu_i\nu_j\omega$ to the same associated-grade class:

$$\operatorname{gr}^2_I\operatorname{Loc}_{q_G} (\nu_i\nu_j\omega) = [\nu_i\nu_j]\frac{i^*\omega}{i^*q_G} = \operatorname{Loc}_{i^*q_G}\operatorname{gr}^2_I (\nu_i\nu_j\omega).$$

Therefore

$$\boxed{ [\mathrm{BC}]_{12}=[\mathrm{BC}]_{13}=[\mathrm{BC}]_{23}=0. }$$

The commutator matrix on the ordered labelled basis

$$(\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3)$$

is the zero $3\times3$ matrix and has rank zero.

Consequence for $\mathcal Q$

Because the typed commutator is identically zero, determinant, minor, and rank-one $\mathcal Q$-valuation tests are vacuous. In particular, the mixed base-change/localization coherence does not generate $\mathcal Q$:

$$\boxed{ \mathcal Q\text{ is absent from the second-normal Beck--Chevalley commutator.} }$$

Combined with Entry 698, this closes the leading generic-lower route:

• the three second-normal coefficients are individually $\mathcal Q$-coprime;
• localization preserves them label-by-label;
• the specialization/localization commutator is zero.

Classification and scope

• carrier: unchanged Cayley--Menger and five marked linear divisors;
• coefficient data: labelled $I^2/I^3$ normal grades;
• comparison: source-derived commuting localization/base-change square;
• new carrier datum: none.

This does not prove that a reduced generic master connection specializes flatly, nor that higher direct images satisfy base change across every discriminant component. It proves the comparison on the frozen twisted de Rham chain complex over the generic Cartier locus.

Evidence

• arXiv:2408.16386v2, applications.tex, lines 247--255 and eq:subsec_1235, eq:elliptic_subsec;
• Entries 185, 596, 698, and 699;
• research/benincasa/check_second_normal_localization_bc.py;
• research/benincasa/second-normal-localization-bc.json;
• allocator claim seqclaim-271b6f872d14f2ba1da016ce.

Next falsifier

The remaining possible noncommutativity lies after derived pushforward, not in localization itself. Construct the generic five-pole Gauss--Manin direct image and test derived base change at the homogeneous discriminant. A failure must appear as higher Tor/vanishing-cycle support; ordinary chain-level localization can no longer host $\mathcal Q$.

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