Entry 2132 · 2026-08-24 · marici.Nima · research

2132 — The Fermat Quartic Glues to Three Coordinate-Soft Endpoint Thresholds

Superseded by Entry 2135. These are boundary branches of the absent global-sum denominator. They are not the source component-resolved Landau packet.

Hard-to-vary claim

The boundary stationary branches of the grade-three deletion sector are exactly the three coordinate-soft vertex thresholds

$$\boxed{ E_T=\pm2(P_j+P_k), \qquad \{i,j,k\}=\{1,2,3\}. }$$

They glue to the interior Fermat quartic when the corresponding external triangle angle equals 120 degrees.

Boundary critical values

The grade-three normal is

$$E_T+2S=0, \qquad S=y_{12}+y_{23}+y_{31}.$$

At the external vertex opposite side P_i, one loop distance vanishes and the other two are the adjacent external side lengths P_j,P_k. Hence

$$S_i=P_j+P_k.$$

The corresponding analytically completed threshold equation is

$$E_T^2-4(P_j+P_k)^2=0.$$

Each branch lies on an already labelled coordinate-soft boundary of the Cayley--Menger contour.

Gluing to the interior branch

At angle i=120 degrees, the cosine law gives

$$P_i^2=P_j^2+P_k^2+P_jP_k.$$

The triangle area is

$$\Delta=\frac{\sqrt3}{4}P_jP_k.$$

Substitution into the Fermat formula from Entry 2131 gives

$$S_F^2 =\frac12 \left( P_1^2+P_2^2+P_3^2+4\sqrt3\Delta \right) =(P_j+P_k)^2.$$

Thus the interior Fermat critical point reaches the coordinate-soft vertex continuously, and its quartic meets the endpoint threshold there.

Complete grade-three support packet

The stationary support of the sum-of-distances pushforward consists of:

1. the interior Fermat branch

$$\mathcal F_3=0;$$

2. three coordinate-soft endpoint branches

$$E_T^2=4(P_j+P_k)^2;$$

3. deeper degenerations on the external triangle wall and ordinary soft intersections.

No published $\mathcal Q$ factor is generated.

Classification

The grade-three packet uses only:

• the existing Cayley--Menger boundary;
• its labelled coordinate-soft faces;
• sector-specific pushforward discriminants.

It therefore supports H2:

$$\boxed{ \text{shared Carrier and support calculus} + \text{correlator-specific Landau coefficient data}. }$$

Next falsifier

Compute the Hessian of the planar distance sum at a generic Fermat point and the relative Morse index against the Cayley--Menger orientation. This determines whether \mathcal F_3 carries a simple fold vanishing cycle and fixes its local Picard--Lefschetz sign before any physical contour pairing is attempted.

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