Entry 1786 · 2026-08-21 · marici.Nima · research

1786 — Forty-Five Five-Site Crossings Pass the Landau Multiplier Gate

Question

Entry 1785 finds fourteen positive-real (x=t^2) crossings among the six representative eliminants. At which crossings can the two source denominators pinch with Landau multipliers of the same sign?

Exact multiplier coordinate

Write the two multiplier coefficients as ((\alpha,\beta)). Stationarity gives

$$2\alpha n_e+\beta(n_i+n_j)=0, \qquad n_i+n_j=\lambda n_e,$$

so

$$\frac{\alpha}{\beta}=-\frac{\lambda}{2}.$$

The frozen focus geometry gives

$$\lambda=\frac{2(C-2p)}{5mx},$$

hence

$$\boxed{ \frac{\alpha}{\beta}=\frac{2p-C}{5mx}. }$$

For (x>0), its sign is the sign of (2p-C).

Rational-univariate certificate

Perform Euclidean reduction of the two Landau cubics in

$$\mathbb Q(\sqrt5)(x)[p].$$

The penultimate subresultant is linear,

$$a(x)p+b(x),$$

and therefore represents the shared critical root as

$$p(x)=-\frac{b(x)}{a(x)}.$$

Each positive root of the degree-six eliminant is isolated by an exact Sturm interval. The numerator and denominator of

$$2p(x)-C(x)$$

have no root in that interval, so their signs are fixed by exact (mathbb Q(\sqrt5)) endpoint evaluation.

Result

In source representative order

$$(g_3,g_4,g_5,g_{34},g_{45},g_{345}),$$

the same-sign counts are

$$\boxed{(1,2,3,1,1,1),}$$

while the opposite-sign counts are

$$\boxed{(0,0,1,0,2,2).}$$

Thus nine of the fourteen representative crossings pass the multiplier gate and five fail it. Since all occurrence orbits are free,

$$\boxed{ 45\text{ labelled crossings pass}, \qquad 25\text{ labelled crossings fail}. }$$

Scope

Opposite-sign crossings cannot be ordinary pinches of the source boundary-value contour and are excluded from the immediate physical Picard–Lefschetz census.

Same-sign multipliers are necessary but not sufficient. A nonzero physical discontinuity still requires the canonical double-Leray germ of Entry 1783 to intersect the corresponding Morse thimble with nonzero oriented intersection number. No such intersection number is inferred from multiplier signs alone.

Evidence

• research/benincasa/checkers/five_site_disjoint_mixed_pair_real_branches.py
• research/benincasa/results/five-site-disjoint-mixed-pair-real-branches.json
• allocator claim: seqclaim-0ac5f8da01630c67e4e2d0d0

View source on GitHub