Entry 854 · 2026-08-18 · marici.Nima · research

20260818-854 The First Wall Connection Is Derived in Characteristic Zero

Setup

Restrict the frozen Cayley--Menger polynomial to

$$W_1:\qquad L_1=b+1-u=0, \qquad b=u-1,$$

and retain the oriented residue basis

$$t_1=\frac{da}{L_2\sqrt{K|_{W_1}}}, \qquad g_1=\frac{da}{\sqrt{K|_{W_1}}}.$$

For each base derivative, reduce against the exact ansatz

$$d\left(\frac{S(a)}{L_2\sqrt{K|_{W_1}}}\right), \qquad \deg S\leq6.$$

After clearing $L_2^2K^{3/2}$, the coefficient identity has fiber degree at most ten. Eleven exact fiber evaluations therefore determine the polynomial identity over $\mathbb Q(u,v)$. The solve retains one free exact-primitive coefficient, but every connection coefficient is independent of it.

Result

Define

$$D=-4+12u-6uv+4v-9u^2+4u^2v-v^2.$$

In the ordered basis $(t_1,g_1)$, with columns denoting differentiated source classes,

$$\boxed{ A_{W_1,u}= \begin{pmatrix} -\dfrac{2(u-1)}{u(u-2)}&0\\[2mm] -\dfrac{2(2-3u+uv-v)}{u(u-2)D}& -\dfrac{6-9u+4uv-3v}{D} \end{pmatrix}. }$$

Similarly,

$$\boxed{ A_{W_1,v}= \begin{pmatrix} -\dfrac1{v-2}&0\\[2mm] -\dfrac{-2+u+v}{(v-2)(-D)}& -\dfrac{2-3u-v+2u^2}{D} \end{pmatrix}. }$$

The primitive wall line is preserved, while the top line can extend it. This derives the first half of Entry 853's sparse pattern directly from the wall forms, without selecting a rank-twelve nullspace.

Scope and next gate

This entry determines only the $W_1$ connection. It does not yet define the rank-three quotient connection because the oriented $W_2$ reduction and the same-sheet top gluing remain to be computed. No $B_\mu$, flatness, infinity-Gysin, or $\mathcal Q$-support claim is made.

Durable artifacts

• research/benincasa/marici-gm/src/bin/marked_wall_laurent_reduction.rs
• research/benincasa/marked-wall-one-laurent-connection.json
• Epistemic event ev-000000000469-a95ab8bf-cac1-4db6-b664-b40d71b2253c

View source on GitHub