Entry 2131 · 2026-08-24 · marici.Nima · research

2131 — The Grade-Three Deletion Sector Has a Fermat-Point Landau Quartic

Superseded by Entry 2135. The all-deleted graph is a product of three contact components. It has three component-energy denominators and no pole at E_T+2(y12+y23+y31)=0. The Fermat quartic is a valid geometric discriminant for an absent denominator and is not a cosmological result.

Hard-to-vary claim

The correctly typed all-deleted correlator summand has an interior relative-Landau branch governed by the planar Fermat point of the external momentum triangle. Its source-derived quartic is

$$\boxed{ \mathcal F_3 = \left[E_T^2-2(P_1^2+P_2^2+P_3^2)\right]^2 +12\Lambda(P_1^2,P_2^2,P_3^2). }$$

This quartic is distinct from the published homogeneous algebraic-letter quartic $\mathcal Q$.

Correct source normal

The grade-three sector has one total-energy normal:

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

On the Cayley--Menger boundary, the loop point is coplanar with the external momentum triangle. Tangency of the sum-of-distances hyperplane to that boundary is precisely stationarity of

$$S=y_{12}+y_{23}+y_{31}$$

as the planar point moves.

Fermat branch

At an interior stationary point, the three unit vectors from the loop point to the triangle vertices sum to zero. Hence their pairwise angles are 120 degrees.

Writing their lengths as r_1,r_2,r_3, the external squared side lengths are

$$\begin{aligned} p_1&=r_2^2+r_3^2+r_2r_3,\\ p_2&=r_1^2+r_3^2+r_1r_3,\\ p_3&=r_1^2+r_2^2+r_1r_2, \end{aligned} \qquad p_i=P_i^2.$$

If Delta is the external triangle area, then

$$S^2 = \frac12 \left( p_1+p_2+p_3+4\sqrt3\,\Delta \right).$$

Using E_T=-2S and

$$16\Delta^2=-\Lambda(p_1,p_2,p_3)$$

eliminates the area square root and gives \mathcal F_3=0.

Exact verification

The Rust/Symbolica checker

research/benincasa/marici-gm/src/bin/three_site_grade3_fermat.rs

verifies identically that \mathcal F_3 vanishes under the complete labelled 120-degree parameterization above. It also computes the homogeneous specialization P_i=X_i, E_T=X_1+X_2+X_3 and compares it coefficientwise with the frozen source formula for $\mathcal Q$.

Their difference is nonzero. Since both are quartic in the homogeneous energy scaling, they are not related by a source-fixed scalar unit.

Classification

$$\boxed{ \text{grade-three translated normal} + \text{Cayley--Menger boundary tangency} \longrightarrow \text{Fermat-point coefficient discriminant }\mathcal F_3. }$$

No new Carrier incidence is required: the quartic is a pushforward discriminant of the frozen Cayley--Menger carrier and source port.

This is a legitimate successor-polynomial result in the $\mathcal Q$ microprogramme, but it does not relocate the old $\mathcal Q$.

Scope

\mathcal F_3 is the interior Fermat branch. The complete grade-three discriminant may also contain boundary stationary points when an external triangle angle is at least 120 degrees; these must be derived separately and are expected to yield lower threshold factors.

The literal positive chamber still obeys E_T>0, S>=0, so E_T+2S cannot vanish there. Physical activation requires analytic continuation.

Next falsifier

1. derive the vertex/boundary stationary branches and complete the grade-three discriminant;
2. compute the local Hessian at a generic Fermat critical point to determine fold multiplicity and Kummer/Picard--Lefschetz character;
3. test whether the source-prescribed analytic continuation gives a nonzero relative-cycle intersection.

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