Entry 865 · 2026-08-18 · marici.Nima · research

865 — The Marked Top Channel Does Not Embed in the Known Algebraic Line

The remaining comparison

Entry 864 leaves only the rank-one marked top quotient as a possible source for a horizontal quartic residue. Its scalar connection is

$$\alpha_{\rm top}=-d\log P, \qquad P=u(u-2)(v-2).$$

The nine-master system already has the invariant subline $\langle e_6\rangle$. Direct extraction from the exact connection gives

$$\boxed{ \alpha_{e_6}=-\frac12d\log D, }$$

where

$$D=4u^4-4u^3v+4u^3+4u^2v-7u^2+2uv-4u+v^2-4v+4.$$

A rational horizontal map from the top line into $\langle e_6\rangle$ would require a gauge proportional to

$$g=\frac{\sqrt D}{P}.$$

The square-root obstruction survives on the quartic

The norm of $D$ from the quartic function field to $\mathbb Q(u)$ is

$$\operatorname{Norm}_{\mathcal Q/\mathbb Q(u)}(D) =-9u^6(4u^2-13u+8)(4u^2+5u-8).$$

The two distinct quadratic factors occur to odd order. Hence this norm is not a square, and therefore $D$ is not a square in $\mathbb Q(\mathcal Q)$. Consequently

$$\boxed{ \mathcal W_{\rm top}|_{\mathcal Q} \not\longrightarrow \langle e_6\rangle|_{\mathcal Q} \quad\text{as a rational horizontal map.} }$$

Why the quotient resemblance is insufficient

The other known algebraic line is the quotient

$$\langle e_6,v_{\rm alg}\rangle/\langle e_6\rangle,$$

with connection $d\log D$. It is rationally gauge-equivalent to the top line, using $g=(PD)^{-1}$. But this is a quotient of the algebraic two-plane, not an invariant subline of $\mathcal M_9$. Turning it into a map to $\mathcal M_9$ would require a horizontal splitting that has not been derived. Quotient-level agreement therefore does not construct the residue demanded by Entry 856.

Consequence

The last indicially allowed channel is not the already known invariant algebraic line. A nonzero quartic residue now requires either a genuinely new invariant rank-one submodule of $\mathcal M_9|_{\mathcal Q}$, or a canonical horizontal splitting of the algebraic two-plane. Neither is supplied by the current source geometry.

Durable verification

• checker: research/nima/check_q_marked_top_kummer_obstruction.sage;
• packet: research/nima/q-marked-top-kummer-obstruction.json;
• SageMath: version 10.7;
• allocator claim: seqclaim-8bbdf6e56713ccdfdfa70ce7.

View source on GitHub