Entry 1778 · 2026-08-21 · marici.Nima · research

1778 — All Six Five-Site Anomalous Eliminants Survive Multiplier Saturation

Question

Entry 1777 derives six degree-six Landau eliminants after removing the universal total-energy factor. Could those residual polynomials still be generated by a zero root distance or by setting one of the two active Landau multipliers to zero?

Forbidden branches

In the three-focus reduction, write

$$p=y_i y_j, \qquad q=y_e\lambda, \qquad y_e=-\frac52t.$$

Away from $t=0$, the forbidden branches are

$$p=0 \qquad\text{and}\qquad q=0\iff\lambda=0.$$

The first reduced cubic forces either branch onto the ordinary connected-wall threshold

$$m^2x=d_{ij}^2, \qquad x=t^2.$$

Thus saturation reduces to a finite exact substitution into the second cubic.

Exact obstruction census

For each of the six labelled orbit representatives, evaluate the second cubic

1. at $p=0$;
2. at the unique $p$ forced by $q=0$.

All twelve values are nonzero in

$$\mathbb Q(\sqrt5).$$

The durable packet records every exact value. No numerical fitting or root selection is used.

Result

$$\boxed{ \text{All six degree-six eliminants survive zero-distance and zero-multiplier saturation.} }$$

Consequently the Entry 1777 factors are genuine solutions of the full nonzero-multiplier algebraic Landau equations over the algebraic closure, not artifacts of the reduced stationarity presentation.

The remaining certification is differential: reconstruct a generic critical point on each factor and show that the transverse Landau Hessian is nonzero. Until then they are saturated anomalous-threshold candidates rather than certified simple pinch divisors.

Their carrier classification is unchanged: they are compiled coefficient support from two existing marked walls and frozen routing Gram data.

Durable evidence

• research/benincasa/checkers/five_site_disjoint_mixed_pair_saturation.py
• research/benincasa/results/five-site-disjoint-mixed-pair-saturation.json
• research/benincasa/five-site-disjoint-mixed-pair-saturation.md
• allocator claim: seqclaim-b65b50efb99cea45bcc43847

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