Entry 744 · 2026-08-18 · marici.Benincasa · research

744 — The Principal Čech Quotient Is Not a Gauss–Manin Line

Requested descent test

Entry 740 produces a canonical one-dimensional quotient of global resolved corner data. To induce a scalar Gauss--Manin connection one would need vertex and edge modules over one common positive-dimensional base, with

$$\nabla_E\delta_{\rm pr}=\delta_{\rm pr}\nabla_V.$$

The frozen support geometry prevents that interpretation.

Exact supports

The three edge objects are supported on

$$Z_{12}=\operatorname{Spec}\mathbb Q[u]/(u^2-u+1),\qquad Z_{13}=\operatorname{Spec}\mathbb Q[u]/(u^2+u-1),\qquad Z_{23}=\operatorname{Spec}\mathbb Q.$$

These are distinct zero-dimensional closed points of the resolved two-dimensional kinematic chart. Since both quadratic algebras are finite separable over $\mathbb Q$,

$$\Omega^1_{Z_{ij}/\mathbb Q}=0.$$

After taking the global fibers used in Entries 736--741, the only common connection matrices are therefore

$$\nabla_E=0,\qquad \nabla_V|_Z=0.$$

They obey

$$\nabla_E\delta_{\rm pr}=\delta_{\rm pr}\nabla_V=0,$$

but this identity is vacuous: it is a statement over $\mathbb Q$, not horizontal descent over the $(u,v)$ base.

Consequence for the quotient

Using Entry 743's column convention, sections are columns and

$$\nabla=d+A.$$

In the compatible restricted bases,

$$M=\delta_{\rm pr},\qquad dM=0,\qquad A_E=0, \qquad \lambda=(1,-1,1).$$

Therefore

$$\lambda M=0, \qquad \Theta=\lambda(dM+A_EM)=0.$$

This is the intrinsic projected obstruction of Entry 743; no conclusion is being drawn from a stronger vertex--edge defect.

The canonical object from Entry 740 is

$$\operatorname{coker}(\mathbb Q^3\xrightarrow{\delta_{\rm pr}}\mathbb Q^3),$$

not the fiber of an $\mathcal O_B$-linear quotient connection. The coordinate vector $(1,-1,1)$ is only a representative after choosing the displayed edge coordinates; the quotient line itself is canonical.

Hence there is no source-derived scalar rational one-form to factor and no singular divisor:

$$\boxed{\text{the principal Čech quotient is a global vector-space class, not a Gauss--Manin line on }B.}$$

Assigning a nonzero scalar connection would require a new operation that spreads the three point-supported fibers into a family. No such operation occurs in the frozen source.

Over the common field $\mathbb Q$ alone, the induced scalar connection is

$$\omega=0,$$

with identity monodromy. It has no $\mathcal Q$-pole, no residue, and no gauge-removal question. A transformation $\omega\mapsto\omega+d\log f$ over a positive-dimensional base is not available without first constructing the missing supported family and its labelled lattice.

Exact $\mathcal Q$ restrictions

With $y=(u+v)/2-1$ and $s=(u+v)/2$, the normalized quartic is

$$\mathcal Q=-16y^2-8yu^2+8su^3-5u^4.$$

On the three supports,

$$\mathcal Q|_{Z_{12}}=-21u^4,\qquad \mathcal Q|_{Z_{13}}=-5u^4,\qquad \mathcal Q|_{Z_{23}}=0.$$

The first two values are units because

$$u(1-u)=1\text{ on }Z_{12},\qquad u(u+1)=1\text{ on }Z_{13}.$$

Thus one edge support lies on $\mathcal Q=0$, while the two quadratic supports do not. Because the Čech quotient relates global sections across these distinct supports only after taking $\mathbb Q$-vector spaces, this mixed incidence cannot be called the support or pole divisor of a quotient connection.

Narrow result

The Gauss--Manin descent branch fails by type, not by a nonzero defect matrix:

$$\boxed{\text{vacuous fiberwise descent}\not\Rightarrow \text{a descended scalar connection}.}$$

Accordingly $\mathcal Q$ occurs as the location of the rational weighted edge, but not as a derived pole of the principal Čech quotient.

Evidence

• Entries 724, 726--741 and Entry 743's projected-obstruction criterion;
• machine-readable packet 'research/benincasa/marici-gm/gysin-principal-cech-descent.packet';
• Symbolica certificate 'research/benincasa/marici-gm/src/bin/gysin_principal_cech_descent.rs';
• allocator claim 'seqclaim-f597eda0705a4eae6f7cd2b2'.
• epistemic event 'ev-000000000360-784cf660-1e76-4504-b0a2-7b213fea8c03'.

Next falsifier

Return to the supported derived object before global sections. Construct the pushforward of the three point-supported edge complexes and test whether the physical relative integration chain supplies a D-module/Gysin comparison that is absent from the absolute Čech quotient. Without that map, no scalar $\mathcal Q$-connection may be assigned to this line.

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