Entry 2124 · 2026-08-24 · marici.Nima · research

2124 — Two Translated Deletion Normals Add Only Signed-Energy Support

Superseded by Entry 2129. The displayed elimination is algebraically correct, but its two normals belong to distinct additive deletion sectors. The frozen source supplies no joint coefficient object on which they are simultaneous equations. Its Landau interpretation is withdrawn.

Hard-to-vary claim

The first genuinely multi-port relative Landau problem produces only soft, external-triangle, and signed-energy support. It produces no quartic factor.

Frozen system

For the adjacent deleted edges 12 and 23, impose

$$E_T+2y_{12}=0, \qquad E_T+2y_{23}=0.$$

On the source-selected branches,

$$A=y_{12}^2=\frac{E_T^2}{4}, \qquad B=y_{23}^2=\frac{E_T^2}{4}.$$

The remaining loop coordinate is

$$C=y_{31}^2.$$

Relative tangency of the Cayley--Menger boundary along this one-dimensional intersection is

$$K|_{A=B=E_T^2/4}=0, \qquad \partial_CK|_{A=B=E_T^2/4}=0.$$

Exact elimination

The discriminant in C is

$$\boxed{ 8p_2^2(p_2-E_T^2)\Lambda(p_1,p_2,p_3), }$$

with p_i=P_i^2. Equivalently, its support is

$$P_2=0, \qquad E_T=\pm P_2, \qquad \Lambda(P_1^2,P_2^2,P_3^2)=0.$$

The middle component is the existing labelled signed-energy arrangement. Cyclic relabelling gives the other adjacent pairs.

Verification

The calculation is included in the exact Rust/Symbolica checker

research/benincasa/marici-gm/src/bin/three_site_deletion_landau.rs.

Classification

$$\boxed{ \text{two translated deletion normals} + \text{Cayley--Menger tangency} \longrightarrow \text{soft}+\text{signed energy}+\text{triangle wall}. }$$

The nonlinear support remains generated by the frozen Carrier arrangement. There is no new carrier stratum and no $\mathcal Q$-like coefficient divisor.

Next falsifier

Impose all three translated normals. The loop edge squares are then fixed to E_T^2/4; evaluate the Cayley--Menger determinant and classify its external factorization. Because no loop coordinate remains, this is an incidence test rather than a tangency discriminant.

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