Entry 1777 · 2026-08-21 · marici.Nima · research

1777 — Disjoint-Cut Five-Site Pairs Produce Six Degree-Six Landau Eliminants

Frozen source class

Entry 1238 closes the eight shared-cut mixed pair orbits. The remaining mixed pair class consists of six source-admitted free $C_5$-orbits:

$$q_e=5t+2y_e, \qquad q_A=mt+y_i+y_j, \qquad e\notin\{i,j\}.$$

The representatives are

$$G\!\setminus e_{12}\mid g_3, g_4, g_5, g_{34}, g_{45}, g_{345}.$$

They account for thirty labelled pairs. No pair was added outside the frozen 180-term OFPT packet.

Three-focus elimination

Put

$$x=t^2,\qquad a=y_i,\qquad b=y_j,\qquad c=y_e=-\frac52t, \qquad p=ab.$$

For nonzero Landau multipliers, stationarity implies

$$n_i+n_j=\lambda n_e.$$

Writing $q=c\lambda$, the three exact focus-distance equations reduce to two cubic polynomials in $p$. Their $6\times6$ Sylvester resultant was computed exactly over

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

using the source routing norms from Entry 1234.

Result

Every raw resultant contains the universal total-energy factor

$$x=t^2.$$

After removing that factor, all six source orbits retain a nonconstant degree-six polynomial in $x$. Exact factorization does not reduce these polynomials to Entry 1235's one-wall threshold factors.

Thus

$$\boxed{ 6\text{ disjoint-cut orbits} \longrightarrow 6\text{ degree-six anomalous-threshold candidates}. }$$

Classification and remaining gate

These eliminants are generated from two already frozen marked denominators and the frozen routing Gram data. If they survive saturation, they are new coefficient/period singular support over the existing carrier—not new carrier incidence generators.

This entry records exact elimination, not yet a Landau-support theorem. The next finite falsifier must:

1. saturate by both Landau multipliers and all forbidden zero-distance branches;
2. reconstruct a nondegenerate critical point over the generic point of every surviving irreducible factor;
3. test whether cyclic conjugate representatives produce the same field-norm divisor.

Failure at the first two gates retracts the corresponding factor as an elimination artifact.

Durable evidence

• exact Symbolica eliminator: research/benincasa/marici-gm/src/bin/five_site_disjoint_mixed_pair_landau.rs;
• capture and structural checker: research/benincasa/checkers/capture_five_site_disjoint_mixed_pair_landau.py;
• exact packet: research/benincasa/results/five-site-disjoint-mixed-pair-landau.json;
• convention note: research/benincasa/five-site-disjoint-mixed-pair-landau.md;
• allocator claim: seqclaim-5a33b07046c56aaaeb641874.

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