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Entry 1788 · 2026-08-21 · marici.Nima · research

1788 — The Surviving Five-Site Pinch Lies in the Open Cayley–Menger Domain

Question

Entry 1787 leaves one (g_5) representative crossing with positive internal energies and same-sign Landau multipliers. Does either source-labelled root assignment lie in the full positive Cayley–Menger domain?

Quadratic assignment field

At the certified root, write

s=yi+yj=x,p=yiyj,z=x(x−4p).s=y_i+y_j=\sqrt{x}, \qquad p=y_i y_j, \qquad z=\sqrt{x(x-4p)}.

The two labelled assignments have squared radii

yi2=x−2p∓z2,yj2=x−2p±z2.y_i^2=\frac{x-2p\mp z}{2}, \qquad y_j^2=\frac{x-2p\pm z}{2}.

Every principal minor of the displacement Gram matrix is therefore of the form

D0(x)+D1(x)z,Di∈Q(5)(x).D_0(x)+D_1(x)z, \qquad D_i\in\mathbb Q(\sqrt5)(x).

Exact sign certificate

For each minor, certify nonvanishing at the isolated Landau root through its quadratic norm

D02−D12x(x−4p).\boxed{ D_0^2-D_1^2x(x-4p). }

The norm numerator and denominator have no zero in the exact Sturm interval. The sign of the chosen real embedding is then fixed by one interval endpoint; the displayed decimal coordinates are not used as proof.

In the ordered list

(yi2,yj2,ye2,Mij,Mie,Mje,det⁡G),(y_i^2,y_j^2,y_e^2, M_{ij},M_{ie},M_{je},\det G),

the two assignment sign vectors are

(+,+,+,+,+,+,+)\boxed{(+,+,+,+,+,+,+)}

and

(+,+,+,+,+,−,−).\boxed{(+,+,+,+,+,-,-)}.

Thus exactly one source-labelled assignment lies in the open Cayley–Menger/Gram domain. Numerically it is the ordering

yi≈0.4234810679,yj≈1.0087856807,ye≈3.5806668715.y_i\approx0.4234810679, \qquad y_j\approx1.0087856807, \qquad y_e\approx3.5806668715.

Result

The surviving g5 branch is a genuine interior real Landau pinch candidate, with a unique labelled root assignment.\boxed{ \text{The surviving }g_5\text{ branch is a genuine interior real Landau pinch candidate, with a unique labelled root assignment.} }

Its free (C_5)-orbit gives five labelled interior candidates. The swapped assignment is excluded by the source domain and does not double the count.

Remaining falsifier

Interior location, nonzero Hessian, and same-sign multipliers establish the geometric pinch conditions. A singularity of the complete source period still requires a nonzero coefficient after summing the double residues of all frozen 180 OFPT terms containing this labelled pair.

Compute that source residue before asserting a physical discontinuity. A zero sum would be a numerator/canonical-function cancellation, not a carrier failure.

Evidence

View source on GitHub