Entry 2130 · 2026-08-24 · marici.Nima · research

2130 — The Grade-Two Deletion Sector Produces a Double-Momentum Threshold

Superseded by Entry 2135. The grade-two graph is disconnected. Its source integrand has separate component-energy poles, not the global sum used below. The elimination is algebraically correct for an absent pole and has no cosmological Landau authority.

Hard-to-vary claim

The correctly typed grade-two deletion summand has a relative Landau discriminant supported only on a labelled soft divisor, the external triangle wall, and the source-derived threshold

$$E_T=\pm2P_2.$$

It produces no quartic factor.

Frozen sector

For S={12,23}, the summand has the single translated normal

$$E_T^{(S)}=E_T+2(y_{12}+y_{23}).$$

Set

$$a=y_{12}, \qquad y_{23}=-\frac{E_T}{2}-a, \qquad C=y_{31}^2.$$

Restrict the frozen Cayley--Menger determinant to this hyperplane. The generic critical system is

$$K_S(a,C)=0, \qquad \partial_aK_S=0, \qquad \partial_CK_S=0.$$

Exact elimination

Eliminating a,C gives

$$\boxed{ 32p_2^2(4p_2-E_T^2)^3 \Lambda(p_1,p_2,p_3)^3, \qquad p_i=P_i^2. }$$

Thus the discriminant support is

$$P_2=0, \qquad E_T=\pm2P_2, \qquad \Lambda(P_1^2,P_2^2,P_3^2)=0.$$

The momentum label 2 is the shared endpoint of the two deleted edges. Cyclic transport gives the other grade-two sectors.

Classification

The factor 4P_2^2-E_T^2 is a genuine source-derived pushforward threshold for this correlator summand. It is generated from the existing translated-normal and Cayley--Menger geometry, so it needs no new Carrier cell.

It is not the homogeneous algebraic-letter quartic $\mathcal Q$.

Hence

$$\boxed{ \text{grade-two deletion port} \longrightarrow \text{labelled double-momentum threshold coefficient support}. }$$

Verification

The exact calculation is the grade2_relative_landau branch of

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

It uses the single source normal, differentiates the restricted determinant, and performs exact Sylvester elimination.

Next falsifier

Compute the correctly typed grade-three summand with its one normal

$$E_T+2(y_{12}+y_{23}+y_{31})=0.$$

Restrict the Cayley--Menger determinant to this hyperplane, solve its two-dimensional critical system, and classify the resulting external discriminant. No simultaneous lower-grade normals may be inserted.

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