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Entry 1787 · 2026-08-21 · marici.Nima · research

1787 — One Five-Site Orbit Passes Both Real Pinch Gates

Question

Entry 1786 leaves nine representative crossings with same-sign Landau multipliers. Which of them also place all three internal energies appearing in the two wall equations on the positive real contour after continuation to negative total energy?

Continued real sheet

Take

t=−x<0.t=-\sqrt{x}<0.

The two wall equations imply

ye=−52t>0,yi+yj=−mt=mx>0.y_e=-\frac52t>0, \qquad y_i+y_j=-mt=m\sqrt{x}>0.

Writing (p=y_i y_j), the two roots (y_i,y_j) are both real and positive exactly when

p>0,m2x−4p≥0.p>0, \qquad m^2x-4p\ge0.

Exact sign audit

Use Entry 1786's rational-univariate representation

p(x)=−b(x)a(x)p(x)=-\frac{b(x)}{a(x)}

on each exact Sturm interval. The numerator and denominator of both (p(x)) and (m^2x-4p(x)) have no zero in the relevant interval, so their signs are certified by exact endpoint evaluation in (mathbb Q(\sqrt5)).

For all fourteen positive-real crossings,

m2x−4p>0.m^2x-4p>0.

The product (p) is positive at five crossings, but four of those five have opposite-sign Landau multipliers. Exactly one crossing satisfies both gates:

g5,x≈2.051388039104848,p≈0.4272016373581.\boxed{ g_5, \qquad x\approx2.051388039104848, \qquad p\approx0.4272016373581. }

The decimals identify the certified isolating intervals; they are not used to establish the signs.

Labelled result

The (g_5) representative belongs to a free (C_5)-orbit. Therefore

5 labelled crossings pass both the multiplier and positive-internal- energy gates.\boxed{ 5\text{ labelled crossings pass both the multiplier and positive-internal- energy gates.} }

The remaining 65 positive-real labelled crossings fail at least one of these two necessary real-pinch conditions.

Scope

This is stronger than a Landau-eliminant census but remains one step short of a physical discontinuity theorem. The five surviving crossings must still satisfy the complete Cayley–Menger signed-minor domain and have nonzero oriented intersection between Entry 1783's canonical double-Leray germ and the local Morse thimble.

No new carrier stratum is indicated. The surviving candidate is one cyclic orbit inside the existing rank-30 coefficient object.

Evidence

View source on GitHub