Entry 868 · 2026-08-18 · marici.Nima · research

868 — The Reconstructed Marked Block Lands in the Split Algebraic Plane

Scope

This entry analyzes Benincasa's multiprime reconstructed $4\times3$ marked-extension block as an exact rational matrix. The reconstruction has passed held-out modular tests but still lacks its characteristic-zero source identity certificate. Accordingly, the result below is an exact theorem about the candidate matrix, not yet about the source Gauss--Manin extension.

Gysin projection

Let $B_u,B_v$ be the two reconstructed final blocks in the ordered basis $(e_6,e_7,e_8,e_9)$. Apply the exact infinity-Gysin matrix

$$G=\begin{pmatrix} 0&1&\frac{u^2+y^2}{2}&\frac{u^2+1}{2}\\ 0&0&-\frac{u^2+1}{2}&-\frac{u^2+y^2}{2y^2} \end{pmatrix}.$$

Exact rational simplification gives

$$\boxed{GB_u=GB_v=0.}$$

All six candidate columns (twenty-four scalar entries) therefore land in the algebraic kernel. The candidate has no elliptic-quotient component.

Split-line decomposition

Using Entry 867's exact splitting, write each candidate column uniquely as

$$B_{\mu,k} =a_{\mu,k}e_6+c_{\mu,k}(v_{\rm alg}+h e_6).$$

For both $\mu=u,v$ and all three marked generators,

$$c_{\mu,k}\ne0, \qquad a_{\mu,k}\ne0.$$

Thus the candidate uses both algebraic lines; it is not merely the reopened marked-top line of Entry 867.

Every denominator is coprime to $\mathcal Q$. Consequently the exact candidate has

$$\boxed{\operatorname{Res}_{\mathcal Q}B=0.}$$

Consequence

The modularly reconstructed answer has an unexpectedly rigid typed form:

$$\text{marked wall system} \longrightarrow \mathcal A_{--} \hookrightarrow \mathcal M_9,$$

and predicts no quartic residue. This is stronger than checking the 24 entries separately, because the six infinity-Gysin identities vanish exactly.

The remaining logical gate is unchanged: until the candidate is certified against the 132 characteristic-zero source identities, these cancellations could describe a uniquely interpolated but non-source rational matrix.

Durable verification

• checker: research/nima/check_marked_candidate_algebraic_kernel.sage;
• packet: research/nima/marked-candidate-algebraic-kernel.json;
• candidate: research/benincasa/marked-extension-charzero-candidate.json;
• allocator claim: seqclaim-4f03c9778590ecf3c414a3f4.

View source on GitHub