Entry 2134 · 2026-08-24 · marici.Nima · research

2134 — The Bunch--Davies Loop Cycle Activates the Fermat Quartic

Retracted by Entry 2135. There is no source pole whose local model is the denominator used below. Consequently no physical activation claim is authorized.

Hard-to-vary claim

For an external momentum triangle whose three angles are below 120 degrees, the source real loop cycle has nonzero local intersection with the grade-three Fermat thimble. The Bunch--Davies continuation therefore activates \mathcal F_3=0 as a genuine square-root correlator threshold.

Physical critical point

The interior Fermat point is a real point of the original loop-momentum space

$$\ell\in\mathbb R^3.$$

Although it appears on the Cayley--Menger boundary in edge-length coordinates, that boundary only records that \ell lies in the external momentum plane. It is not a boundary of the physical vector-space contour.

When all triangle angles are below 120 degrees, the Fermat point lies away from the three external vertices, so

$$r_i>0.$$

The triple-deleted sector's inverse two-point factors and contact numerators are therefore finite and generically nonzero there.

Local source integral

Let

$$\delta=E_T-E_{T,c}, \qquad E_{T,c}=-2S_F.$$

Entry 2133 gives

$$E_T+2S(\ell) = \delta+\xi^TH\xi+O(|\xi|^3), \qquad H>0.$$

The source i\epsilon boundary value therefore contains the local model

$$I_{\rm loc}(\delta) =N_0 \int_{|\xi|<R} \frac{d^3\xi} {\delta+\xi^TH\xi-i0}, \qquad N_0\ne0$$

at generic kinematics.

After the linear Morse change of variables, its nonanalytic term is

$$\boxed{ I_{\rm loc}^{\rm nonan} = -\frac{2\pi^2N_0}{\sqrt{\det H}} (\delta-i0)^{1/2}. }$$

The coefficient is nonzero. Hence the real source cycle intersects the Morse thimble with multiplicity one, with orientation fixed by the loop measure and the -i0 prescription.

Physical conclusion

$$\boxed{ \mathcal F_3=0 \text{ is a genuine analytically continued correlator threshold.} }$$

It is not a singularity on the literal positive-energy chamber, because E_{T,c}<0. It is physical in the precise S-matrix/Landau sense: the source-prescribed continuation has a nonzero discontinuity across the threshold.

Relation to $\mathcal Q$

This does not rehabilitate the generic homogeneous $\mathcal Q$. Instead it validates the Lakatos protective-belt prediction:

$$\text{independently enlarged source geometry} \longrightarrow \text{a distinct predicted discriminant} \longrightarrow \text{nontrivial coefficient and physical monodromy}.$$

The old $\mathcal Q$ remains apparent in its declared scope. The new \mathcal F_3 is a genuine correlator-sector successor polynomial generated by the unchanged Carrier plus a different physical port.

Scope

The result assumes:

• a nondegenerate external triangle with all angles below 120 degrees;
• a generic smooth point of \mathcal F_3=0 away from marked denominator intersections;
• nonvanishing generic contact numerator N_0.

At the 120-degree transition, the critical point reaches coordinate-soft support and Entry 2132's endpoint analysis takes over.

Next falsifier

Derive the exact source normalization N_0 for the all-deleted three-site summand and compare the discontinuity with a direct Cutkosky/phase-space calculation. Equality would identify the new threshold's readout normalization, not merely its support and monodromy.

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