Entry 2123 · 2026-08-24 · marici.Nima · research

2123 — A Translated Deletion Normal Produces Only Existing Relative Landau Support

Hard-to-vary claim

The relative Landau tangency between one source-translated deletion normal and the three-site Cayley--Menger boundary produces only existing total-energy, soft, and external-triangle support. It produces no quartic factor.

Frozen system

Take the labelled deleted edge 12 and set

$$E_T+2y_{12}=0, \qquad A=y_{12}^2=\frac{E_T^2}{4}.$$

Write

$$B=y_{23}^2, \qquad C=y_{31}^2, \qquad p_i=P_i^2.$$

Let K(A,B,C;p_1,p_2,p_3) be the frozen five-by-five Cayley--Menger determinant. Away from the coordinate-soft endpoints B C=0, generic tangency to the fixed-A hyperplane is governed by

$$K=0, \qquad \partial_BK=0, \qquad \partial_CK=0.$$

Exact elimination

Symbolica elimination of B,C gives

$$\boxed{ \operatorname{Elim}_{B,C} (K,K_B,K_C) = -128E_T^2p_1^2 \Lambda(p_1,p_2,p_3)^3, }$$

where

$$\Lambda(p_1,p_2,p_3) =p_1^2+p_2^2+p_3^2 -2p_1p_2-2p_2p_3-2p_3p_1.$$

Thus the support is

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

The cyclic images give the corresponding labelled soft factors for deletion of 23 and 31.

Verification

The exact Rust/Symbolica checker is

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

It constructs the Cayley--Menger determinant from the frozen distance matrix, differentiates before elimination, and computes the two-stage Sylvester resultants exactly.

Classification

$$\boxed{ \text{translated deletion normal} + \text{Cayley--Menger tangency} \longrightarrow \text{existing energy/soft/triangle carrier only}. }$$

No new carrier divisor and no $\mathcal Q$-like coefficient support occurs in this one-deletion relative Landau sector.

Scope

This closes the generic tangency for one deleted edge and its cyclic images. It does not yet close:

• simultaneous tangency involving two or three translated normals;
• coordinate-soft endpoint components excluded by the generic BC nonzero typing;
• marked denominator collisions beyond the Cayley--Menger boundary;
• physical relative-cycle activation after analytic continuation.

Next falsifier

Test the first genuinely multi-port locus using two translated normals, for example

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

together with the Cayley--Menger boundary and the correctly reduced tangency equations. Saturate coordinate-soft components before interpreting the elimination. A surviving factor must be classified against existing support before comparison with any successor quartic.

View source on GitHub