Entry 1832 · 2026-08-21 · marici.Nima · research

1832 — The Physical Region-Pair Pinch Persists Off the Symmetric Slice

Full Landau system

For the labelled pair $(g_{123},g_{125})$, retain the five unknowns

$$(\ell_1,\ell_2,\ell_3,t,\lambda)$$

and the five equations

$$q_1=0, \qquad q_2=0, \qquad \nabla q_1+\lambda\nabla q_2=0.$$

At Entry 1827's symmetric solution,

$$\lambda=1, \qquad v_2=-v_1=-v, \qquad v\ne0.$$

Let

$$H=\operatorname{Hess}(q_1+q_2).$$

Entry 1828 proves that $H$ is positive definite.

Jacobian determinant

Adding and subtracting the two wall rows isolates the $t$-column and reduces the remaining determinant to the block matrix

$$\begin{pmatrix} v^T&0\\ H&-v \end{pmatrix}.$$

Its Schur complement is, up to a nonzero conventional scalar,

$$\boxed{ \det(H)\,v^TH^{-1}v. }$$

Both factors are strictly positive. Therefore the complete Landau Jacobian with respect to $(\ell,t,\lambda)$ is invertible.

Consequence

The implicit-function theorem gives a unique smooth real solution

$$(\ell,t,\lambda) =(\ell(X),t(X),\lambda(X))$$

for every sufficiently small labelled real deformation of the external kinematics. By openness, the following properties persist:

• $\lambda>0$;
• all internal distances are positive;
• every remaining source wall stays nonzero;
• the summed source residue stays nonzero.

Hence the pinch is not an accidental fixed-point artifact of the regular pentagon.

Architectural result

Entry 1831's quartic is a one-dimensional slice of a persistent multivariate critical-value discriminant. The underlying carrier remains the same labelled pair incidence:

$$\boxed{ g_{123}=g_{125}=0. }$$

The additional multivariate complexity belongs to the coefficient discriminant generated by physical loop elimination. No extra carrier datum is forced locally.

Scope

This proves local deformation persistence, not a closed formula for the full multivariate discriminant and not global Bunch--Davies chain activation.

Next falsifier

Compute the first derivative of $t(X)$ from the inverse Landau Jacobian and test cyclic naturality across the five labelled occurrences. A failure of that transport would indicate an occurrence-gluing defect; agreement would promote the local coefficient discriminant to a coherent neighbourhood of the cyclic carrier orbit.

Evidence

• research/benincasa/checkers/five_site_region_pair_deformation_jacobian.py
• research/benincasa/results/five-site-region-pair-deformation-jacobian.json
• Entries 1827--1831
• allocator claim: seqclaim-40d609f781a30c77706a04ef

View source on GitHub