Entry 1868 · 2026-08-21 · marici.Nima · research

1868 — Every Surviving Triple Wall Is a Square Kummer Complete Intersection

Source-normalized compiler

Entry 1867 leaves 96 nonzero dihedral triple-wall representatives on the cyclic physical slice. Each representative is now compiled on the frozen rank-three multi-Kummer cover with fiber variables

$$(u_1,u_2,u_3,y_1,y_2,y_3,y_4,y_5).$$

The coefficient field is $\mathbb Q(\sqrt5)$. The routing Gram matrix, its determinant, the coordinates of the fifth routing center, the five Kummer equations, and every labelled wall equation are derived from the source conventions rather than selected per example.

Rank theorem

For each triple, form the $3\times5$ matrix of coefficients of $(y_1,\ldots,y_5)$ in its three wall equations. Exact integer elimination gives

$$\boxed{ \operatorname{rank}=3 \quad\text{for all }96\text{ representatives}. }$$

Thus no surviving case has a rank-deficient wall pullback. The five Kummer cover equations plus the three wall equations form a square system of eight constraints in eight fiber variables at fixed $t$.

Canonical critical ideal

For every representative the unsaturated critical ideal is therefore

$$I_{\rm crit} = \left( K_1,\ldots,K_5, q_{a},q_b,q_c, \det J_{\rm fib} \right),$$

where $J_{\rm fib}$ is the $8\times8$ Jacobian of the five Kummer and three wall equations with respect to the eight fiber variables.

The determinant is canonical in the frozen ordering up to a nonzero orientation sign. A genuine triple-wall eliminant must still saturate away:

• edge-soft factors $y_1\cdots y_5$;
• lower-wall solutions where a Landau multiplier vanishes;
• coincident-focus or vanishing-gradient components;
• the already inverted routing Gram determinant.

Narrow result

The 96-case problem now has one uniform exact ideal template. No exceptional rank-deficient complex or fitted critical equation is required. This is computational infrastructure, not evidence that all 96 ideals have nonempty saturated support.

Next falsifier

Run saturation and elimination for one representative in each source wall type/profile, beginning with the profiles of smallest support overlap. Verify that representatives sharing a proposed template have identical eliminants under their exact dihedral transition before transporting the result.

Evidence

• research/benincasa/marici-gm/src/bin/five_site_cyclic_triple_landau_ideal_compiler.rs
• research/benincasa/results/five-site-cyclic-triple-landau-ideal-compiler.json
• research/benincasa/results/five-site-compatible-landau-subsets.json
• Entries 1215--1217 and 1867
• allocator claim: seqclaim-28a78800ad392e8130ec88ce

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