Entry 2133 · 2026-08-24 · marici.Nima · research

2133 — The Fermat Landau Branch Is a Nondegenerate Three-Dimensional Fold

Superseded by Entry 2135. The Morse calculation is mathematically correct for the sum-of-three-distances function, but that function is not a denominator of the all-deleted source integrand.

Hard-to-vary claim

At a generic interior Fermat point, the grade-three Landau critical point is a positive nondegenerate Morse point in the full three-dimensional loop space. Its local singularity is a simple fold with square-root character.

Hessian

Let u_i be the three unit vectors from the Fermat point to the external triangle vertices and let r_i>0 be their lengths. For

$$S(\ell)=\sum_{i=1}^3|\ell-v_i|,$$

the Hessian is

$$H=\sum_{i=1}^3\frac{I-u_iu_i^T}{r_i}.$$

The u_i lie in the external momentum plane, have pairwise angles 120 degrees, and sum to zero.

The eigenvalue normal to the plane is

$$\lambda_\perp=\sum_i\frac1{r_i}>0.$$

The planar determinant is

$$\det H_\parallel =\frac34\sum_{i<j}\frac1{r_ir_j}>0.$$

Therefore

$$\boxed{ \det H =\frac34 \left(\sum_i\frac1{r_i}\right) \left(\sum_{i<j}\frac1{r_ir_j}\right)>0, \qquad \operatorname{index}H=0. }$$

Why the third direction matters

In edge-length coordinates the Fermat point lies on the Cayley--Menger boundary and the measure contains a square-root Jacobian. On the physical loop-vector cover, that boundary normal is the ordinary out-of-plane displacement. Retaining it gives a three-dimensional Morse point.

Discarding the cover normal would incorrectly type the singularity as a two-dimensional logarithmic critical point.

Local coefficient character

For the simple translated total-energy denominator, the local model is

$$\int_{\mathbb R^3} \frac{d^3\xi}{\delta+\xi^TH\xi}.$$

After subtraction of analytic terms, its first nonanalytic contribution is proportional to

$$\delta^{1/2}.$$

Hence a loop around the generic Fermat discriminant acts by

$$\boxed{T_{\mathcal F_3}=-1}$$

on the rank-one local vanishing coefficient.

Verification

research/benincasa/checkers/fermat_hessian_fold.rs checks the exact determinant formula and positivity at three independent positive rational-weight packets. The displayed determinant identity proves the general result.

Classification

The grade-three port therefore carries a genuine sector-specific Kummer/fold coefficient along \mathcal F_3=0, over the unchanged Cayley--Menger Carrier.

This establishes coefficient monodromy. It does not establish physical activation by the source Bunch--Davies relative cycle.

Next falsifier

Continue the normalized positive loop cycle using the source i\epsilon prescription toward a generic smooth point of \mathcal F_3=0. Compute the oriented intersection with the Morse thimble.

Because the coefficient monodromy is nontrivial, the remaining decision is binary:

• zero intersection: physically invisible coefficient branch;
• nonzero intersection: genuine correlator threshold on the existing Carrier.

View source on GitHub