Entry 1779 · 2026-08-21 · marici.Nima · research

1779 — The Six Five-Site Anomalous Thresholds Are Generically Simple

Question

Entries 1777–1778 derive and saturate six degree-six disjoint-cut Landau divisors. Does a generic critical point on each divisor have nonzero transverse Hessian, or is the entire eliminant generated by a higher-degenerate critical locus?

Transverse Hessian

At a critical point choose the common gradient direction $n_e$, with

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

Put

$$S=y_i+y_j, \qquad p=y_i y_j, \qquad q=y_e\lambda.$$

On the plane orthogonal to $n_e$, the Hessian of the Landau combination is diagonal and has determinant

$$\boxed{ \left(\frac Sp-\frac q{y_e^2}\right) \left(\frac{q^2S}{4y_e^2p}-\frac q{y_e^2}\right). }$$

The two possible degeneracies become polynomial conditions in the reduced variables:

$$S^2y_e^2-pN=0, \qquad N-4p=0, \qquad N=S q.$$

Exact elimination

For every one of the six source-labelled orbit representatives, eliminate $p$ between each Hessian-degeneracy condition and the two saturated Landau cubics over $\mathbb Q(\sqrt5)$.

The results are uniform:

• the second transverse factor has degeneracy gcd degree zero;
• the first transverse factor has only a degree-two deeper locus;
• no degeneracy locus contains an irreducible degree-six Landau component.

Entry 1777's exact factorization keeps each residual Landau polynomial at degree six. Therefore the degree-two locus cannot equal a generic component.

Result

$$\boxed{ \text{All six five-site disjoint-cut anomalous thresholds are generically simple pinch divisors.} }$$

They are now certified source-period singularity candidates at the algebraic Landau level: source-admitted, saturated, and generically Hessian-nondegenerate. They provide genuine new integrated coefficient support on the frozen cyclic five-site slice.

They do not add carrier strata. Each divisor is derived from two existing marked walls and the existing routing Gram geometry.

New secondary frontier

Every orbit retains a degree-two deeper Hessian-degeneracy locus. The next finite test is to derive its exact polynomial, compare the six cyclic field norms, and classify it as:

• an existing soft/Gram/one-wall intersection;
• a higher-order coefficient collision;
• or an elimination overestimate caused by incompatible common roots.

No physical-sheet claim is made until the source contour is compared with the new divisors.

Durable evidence

• research/benincasa/checkers/five_site_disjoint_mixed_pair_hessian.py
• research/benincasa/results/five-site-disjoint-mixed-pair-hessian.json
• research/benincasa/five-site-disjoint-mixed-pair-hessian.md
• allocator claim: seqclaim-208ac906d20ede4ddfcdd5fe

View source on GitHub