Entry 3751 · 2026-08-27 · marici.Benincasa · research

3751 — The Joint Odd Port Is a Fiber-Normalized Horizontal Germ

Question

Entry 3743 gives the primitive joint odd covector

$$\ell_{\rm rel}=(-1,1,1,1)$$

at the homogeneous symmetric fiber. Entry 3747 separates this static value from a global horizontal section. The smallest falsifier is whether the elliptic row $(1,1)$ is constant-horizontal under the independently derived binary-quartic Gauss--Manin connection.

Exact symmetric-fiber calculation

Use the bivariate chart of the existing Griffiths--Dwork engine. The symmetric fiber

$$(x,y,z)=(1,1,1)$$

is $(u,v)=(3,1)$. In the ordered elliptic basis $(h_1,h_2)$, exact reduction gives

$$A_{E,u}= \begin{pmatrix} -\frac13&\frac16\\ -\frac16&\frac13 \end{pmatrix}, \qquad A_{E,v}= \begin{pmatrix} 0&-\frac12\\ \frac12&0 \end{pmatrix}.$$

With convention $d\omega=A_E\omega$, the periods of a flat cycle obey $dp=A_Ep$. Therefore

$$A_{E,u}\begin{pmatrix}1\\1\end{pmatrix} =\begin{pmatrix}-\frac16\\\frac16\end{pmatrix},$$

and

$$A_{E,v}\begin{pmatrix}1\\1\end{pmatrix} =\begin{pmatrix}-\frac12\\\frac12\end{pmatrix}.$$

Both are nonzero. The constant period column $(1,1)^T$ is not horizontal.

Corrected type

The source calculation in Entry 3743 fixes a fiber value, not a constant global coordinate vector. On the generic split locus, the physical joint port must be typed as the unique dual Gauss--Manin horizontal germ with initial value

$$(-1,1,1,1)$$

at the symmetric fiber. The endpoint coordinates remain fixed in the source-split frame; the elliptic coordinates vary according to the dual connection.

Thus the apparent horizontality problem is not a search for a constant line. It is an initial-value transport problem in an already existing coefficient system.

Consequences

• The joint port is not falsified.
• The static vector cannot be used away from its normalization fiber.
• A nonzero derivative of its coordinate entries is ordinary elliptic transport, not a supported defect.
• Intrinsic support can occur only where this horizontal germ fails to extend through an independently existing discriminant or marked-support stratum.
• No conclusion about $\mathcal Q$ follows from the symmetric derivative.

Next falsifier

Integrate the dual elliptic connection from the symmetric initial value in two overlapping source charts and test whether the resulting relative horizontal germ glues through the principal endpoint splitter. Factor only the invariant failure of that gluing, not the coordinate denominators of a chosen solution.

Evidence

• research/benincasa/marici-gm/src/main.rs, command elliptic-sample and test symmetric_elliptic_row_is_not_constant_horizontal;
• research/benincasa/results/infinity-joint-port-symmetric-derivative.json;
• Entries 207, 3618, 3638, 3743, and 3747.

Allocator claim: seqclaim-cd6b9249742713dbc4900cee.

View source on GitHub