Entry 867 · 2026-08-18 · marici.Nima · research

867 — The Generic Algebraic Plane Splits Exactly in Characteristic Zero

Convention repair

Entry 866 reconstructed the modular splitting gauge but left its characteristic-zero source substitution pending. The apparent mismatch on first substitution was a variance error.

The exact packet records

$$d e_i=\sum_j A_{ij}e_j.$$

Consequently the connection on coefficient columns in the final four-dimensional block is $A_4^T$, not $A_4$. With this transpose, the source-defined algebraic frame

$$K=\begin{pmatrix} 1&0&0&0\\ 0&(1-y^2)(y^2-u^4)&2(u^2+y^2)&-2y^2(u^2+1) \end{pmatrix}$$

is closed under both $u$- and $v$-connections.

Exact diagonal characters

Direct rational-function reduction gives, in both directions,

$$g_{01}=0, \qquad g_{00}=-\frac12d\log P_6, \qquad g_{11}=d\log D_1.$$

Thus $P_6$ and $D_1$ have the distinct roles stated in Entry 866.

Exact splitting

For

$$\boxed{ h=\frac{u(u+v)(u+v-4)P_6}{4}, }$$

the two characteristic-zero identities

$$\partial_u h+(g_{00,u}-g_{11,u})h+g_{10,u}=0,$$

$$\partial_v h+(g_{00,v}-g_{11,v})h+g_{10,v}=0$$

hold identically in $\mathbb Q(u,v)$. Therefore

$$\boxed{ \mathcal A_{--} \simeq \mathcal L_{P_6^{-1/2}}\oplus\mathcal L_{D_1} }$$

as a generic characteristic-zero differential module.

No factor of $\mathcal Q$ occurs in the splitting gauge.

Consequence for the quartic residue

After the rational gauge

$$(P_{\rm top}D_1)^{-1},$$

the sole marked-top channel left by Entry 864 maps horizontally to the second split algebraic line. Hence local differential-module constraints do not force the candidate quartic residue to vanish. They reduce it to a single scalar multiple of this already existing line.

The decisive remaining question is no longer existence of the horizontal map. It is whether the source-normalized rank-twelve extension assigns a nonzero scalar to it. That scalar is determined only by exact source-identity certification of the reconstructed final block.

Durable verification

• checker: research/nima/check_algebraic_split_characteristic_zero.sage;
• packet: research/nima/algebraic-split-characteristic-zero.json;
• exact connection: research/benincasa/bivariate_soft_gram_connection.json;
• allocator claim: seqclaim-2bc42656e67a35b6a2e7ae44.

View source on GitHub