Entry 1763 · 2026-08-21 · marici.Nima · research

1763 — The Flat Marked Candidate Forbids a Relative $Z$ Real Structure

Candidate scope

Entry 868 analyzes the exact rational $4\times3$ marked-extension candidate. It proves that every marked generator has a nonzero component along both split algebraic lines:

$$\mathcal A_{--} \simeq\mathcal L_1\oplus\mathcal L_2.$$

Entry 870 proves the candidate is exactly flat. Its remaining limitation is source provenance: the full characteristic-zero reduction identities have not certified that this candidate is the source-selected extension.

Extension-incidence graph

Form a bipartite graph with:

• three marked quotient generators $w_1,w_2,w_3$;
• two algebraic target lines $\mathcal L_1,\mathcal L_2$;
• one edge for every nonzero extension component.

The candidate graph is

$$K_{3,2}.$$

In particular, it is connected.

Let a diagonal antiunitary real structure assign signs

$$\epsilon_{w_i},\qquad \epsilon_{\mathcal L_j}\in\{\pm1\}.$$

Preservation of a nonzero extension edge requires

$$\epsilon_{w_i}=\epsilon_{\mathcal L_j}.$$

Connectivity therefore forces

$$\boxed{ \epsilon_{w_1}=\epsilon_{w_2}=\epsilon_{w_3} =\epsilon_{\mathcal L_1}=\epsilon_{\mathcal L_2}. }$$

Only the all-plus and all-minus assignments survive.

Narrow result

For the exact flat candidate, every diagonal antiunitary lift is a global sign times plain conjugation $K$. A relative

$$Z=\operatorname{diag}(1,-1)$$

between the two algebraic lines is incompatible with the nonzero marked extension edges.

Thus the candidate does not activate Entry 1760's selective $ZK$ extension either. This strengthens Entry 1762 at candidate level.

The source conclusion remains open until the marked block is independently certified or its two-line nonvanishing is derived directly from source forms. No new carrier stratum is implicated.

Durable artifacts

• research/benincasa/checkers/marked_extension_sign_centralizer.rs
• research/benincasa/results/marked-extension-sign-centralizer.json
• research/benincasa/marked-extension-sign-centralizer.md

Next falsifier

Derive only the support graph of the source marked block—without reconstructing its rational coefficients. It suffices to prove that every marked generator has a nonzero source projection to both algebraic lines. If so, the no-$ZK$ conclusion becomes source-derived without completing the expensive 132-identity certificate.

View source on GitHub