Entry 2121 · 2026-08-24 · marici.Nima · research

2121 — The Translated Deletion Normals Miss the Literal Positive Chamber

Hard-to-vary claim

At physical positive kinematics, none of the eight translated total-energy normals from Entry 2119 meets the literal Cayley--Menger loop contour.

Proof

The source chamber has

$$E_T>0, \qquad y_{12},y_{23},y_{31}\ge0.$$

For every deletion subset S,

$$E_T^{(S)}=E_T+2\sum_{e\in S}y_e \ge E_T>0.$$

Therefore

$$\boxed{ \Gamma_{ m CM}^{+} \cap \{E_T^{(S)}=0\} =\varnothing \quad\text{for all }S. }$$

No subdivision term introduces a finite physical pole or contour pinch through these normals on the undeformed positive chamber.

Verification

research/benincasa/checkers/deletion_normals_positive_chamber.rs enumerates all eight labelled sectors over three exact positive loci and verifies the inequality. The general result is the displayed order argument.

Consequence

The translated-normal pushforward cannot make $\mathcal Q$, or any successor polynomial, into a singularity of the literal positive correlator merely by adding deletion sectors.

Any nontrivial discriminant from this mechanism requires analytic continuation in external energies and a genuine Landau pinch involving:

• a translated normal;
• the Cayley--Menger boundary or another frozen integration boundary;
• the stationary/tangency equations of the loop pushforward.

Thus the admissible problem is not the unconstrained resultant of the eight linear normals. It is their relative Landau discriminant with the source contour boundary.

Next falsifier

For one cyclic representative, solve

$$E_T+2y_{12}=0, \qquad CM(y,P)=0,$$

together with tangency of the two hypersurfaces in loop-edge space. Eliminate the loop variables and classify every factor against existing soft, Gram, signed-energy, and Landau support before comparing with $\mathcal Q$.

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