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Entry 1887 · 2026-08-21 · marici.Nima · research

1887 — The Nonsymmetric Hexagon Sector Is a Gram-Supported Excess Line

Remaining sector

Entry 1886 closes the stabilizer-fixed six-site critical sector. Retain all three free squared loop coordinates

x=y12,v=y32,w=y52x=y_1^2,\qquad v=y_3^2,\qquad w=y_5^2

and specialize the denominator-cleared κ\kappa-Rees cover to the regular hexagon exceptional fiber.

Exact exceptional system

The two strict-transform cover equations are

FE=−132L2,GE=−14M,F_E=-\frac1{32}L^2, \qquad G_E=-\frac14M,

where

L=−2+x−6z+4v+w,M=−6+2x−6z+5v−w.L=-2+x-6z+4v+w, \qquad M=-6+2x-6z+5v-w.

All three 2×22\times2 critical Jacobian minors are nonzero scalar multiples of the same linear form LL. Thus the critical scheme is exactly

L=M=0.L=M=0.

Solving gives the affine line

x=83+4z−3v,w=−23+2z−v,\boxed{ x=\frac83+4z-3v, \qquad w=-\frac23+2z-v, }

with vv free. It lies over every zz and contains no symmetric point x=v=wx=v=w.

Narrow result

The nontrivial stabilizer-character sector survives only as an excess critical line supported on the pre-existing Gram exceptional divisor. It does not project to a finite discriminant in zz:

no homogeneous six-site z-divisor arises from this incidence.\boxed{ \text{no homogeneous six-site }z\text{-divisor arises from this incidence.} }

The doubled equation L2=0L^2=0 may carry a derived Cartier/excess coefficient object. That object has not been computed and must not be replaced by the reduced line. Nevertheless, neither the reduced nor doubled support is a new Carrier stratum; both live on the frozen Gram-degeneracy support.

Meta-level consequence

The six-site test separates three notions sharply:

  1. the Carrier incidence is source-admitted;
  2. the generic Gram-normal family has a symmetric Landau divisor away from the exceptional fiber;
  3. the homogeneous specialization retains only Gram-supported excess, not a physical energy discriminant.

This is another positive H2 pattern: shared incidence and excess calculus, with coefficient support changing under kinematic specialization.

Next falsifier

Compute the derived excess object of FE=L2F_E=L^2, GE=MG_E=M: its Koszul homology, Cartier length, and compatibility with the nine source-term restriction maps. The test is whether it is exhausted by the existing Gram Gysin/excess calculus or requires an undeclared incidence operation.

Durable verification

View source on GitHub