Entry 1780 · 2026-08-21 · marici.Nima · research

1780 — The Apparent Five-Site Hessian Locus Is Only Total Energy

Question

Entry 1779 left a uniform degree-two factor after separately eliminating the two Landau equations against the first transverse-Hessian degeneracy. Does that factor describe a common higher-degenerate critical point, or can the two resultants use different roots of the auxiliary variable (p=y_i y_j)?

Frozen compatibility test

Keep the two saturated Landau cubics

$$f(p,x)=0,\qquad g(p,x)=0,$$

and the quadratic Hessian equation

$$h_z(p,x)=0, \qquad x=t^2.$$

Reduce both cubics in the same quotient

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

Their remainders are linear in the same (p):

$$r_f=a_f(x)p+b_f(x), \qquad r_g=a_g(x)p+b_g(x).$$

A shared critical point therefore requires the compatibility determinant

$$\boxed{ a_f b_g-a_g b_f=0. }$$

No root matching or numerical fitting is used.

Exact result

For all six source-labelled disjoint-cut orbit representatives,

$$\gcd\!\left( \gcd(\operatorname{Res}_p(f,h_z),\operatorname{Res}_p(g,h_z)), a_f b_g-a_g b_f \right) =x^2=t^4.$$

Thus the degree-two factor retained by Entry 1779 is supported only at

$$t=0,$$

the already existing total-energy locus of the cyclic slice. Away from total energy, every one of the six anomalous-threshold divisors is Hessian-nondegenerate at every compatible point detected by this audit.

Narrow conclusion

$$\boxed{ \text{The six new degree-six divisors have no additional nonzero transverse-Hessian degeneration locus.} }$$

The apparent secondary factor was not a new coefficient divisor or carrier stratum. It was the existing total-energy support, exposed only after enforcing same-critical-point compatibility.

This strengthens the five-site evidence for H2:

$$\text{existing marked/routing carrier} + \text{new integrated coefficient support}.$$

No physical-sheet claim is made.

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-e36b8b111371a4f8a6268ed4

View source on GitHub