Entry 1886 · 2026-08-21 · marici.Nima · research

1886 — The Symmetric Six-Site Landau Branch Misses the Homogeneous Exceptional Fiber

Corrected deformation

Entry 1885 forbids inversion of the rank-three regular-hexagon routing Gram matrix. Introduce one predeclared cyclic normal parameter $\kappa$ in the opposite-site pairing. The regular hexagon is $\kappa=0$, and

$$\det H_6(\kappa)=-\frac14\kappa(6+7\kappa).$$

Rather than invert and then specialize, multiply both cover equations by the actual Gram determinant and eliminate in the stabilizer-fixed sector

$$y_1^2=y_3^2=y_5^2=x, \qquad y_2^2=y_4^2=y_6^2=z.$$

Exact resultant

The denominator-cleared symmetric critical resultant is

$$-\frac1{32}\kappa^2(6+7\kappa)^3 P(\kappa,z),$$

where

$$P= 36-60\kappa+18\kappa z+55\kappa^2-12\kappa^2z -4\kappa^3+2\kappa^3z.$$

The prefactor is inherited Gram multiplicity. Its removal is forced by the strict transform, not fitted after observing the answer. On the exceptional fiber,

$$\boxed{P(0,z)=36.}$$

Hence the strict-transform divisor has empty finite intersection with $\kappa=0$.

Narrow result

$$\boxed{ \text{No stabilizer-fixed six-site Landau divisor specializes to the homogeneous hexagon.} }$$

The apparent $\kappa^2$ support is entirely the pre-existing Gram degeneracy. It is not a new cosmological coefficient divisor and cannot be promoted to a Carrier stratum.

This closes only the $C_3$-fixed critical sector. Nonsymmetric critical orbits at generic six-site kinematics remain untested.

Meta-level consequence

The five-site physical orbit does have a six-site source incidence, but its most symmetric higher-arity continuation escapes the finite homogeneous fiber. Thus systematic assembly is not “the same polynomial at every arity.” Carrier incidence survives while coefficient support may disappear under a singular kinematic specialization.

Next falsifier

Decompose the full three-free-coordinate critical module under the $C_3$ stabilizer. Test the two-dimensional nontrivial character sector in the same $\kappa$-Rees family. A surviving strict-transform class there would be a genuinely nonsymmetric six-site continuation; its absence closes this incidence pattern at homogeneous arity six.

Durable verification

• research/benincasa/marici-gm/src/bin/six_site_disjoint_pair_triple_symmetric_landau.rs
• research/benincasa/results/six-site-disjoint-pair-triple-symmetric-landau.json
• allocator claim: seqclaim-e7e1ba4cb956e74a65e1e36b
• epistemic event: ev-000000002248-e3cb82bb-9817-47fa-8cc3-8c99e6d5fe89

View source on GitHub