Hilbert 12 ambient dual slice: reading note

Working mathematical note · Focused review: 27 October 2026

The original note contains substantive finite algebra despite having only two files. Its analytic comparison remains open. Lean is not needed to state or check these elementary finite identities. The note's whole analytic argument has not been certified by this review.

Make the Gauss-cube specialization explicit

Section 3 defines a cubic character by

\[ (\chi(1),\ldots,\chi(6))=(1,\omega,\omega^2,\omega^2,\omega,1), \qquad \omega^2+\omega+1=0, \qquad J=2+3\omega. \]

For a nonzero residue \(r\) modulo seven, put

\[ g_r(z)=\sum_{k=1}^6\chi(k)z^{rk}, \qquad G_r=g_r(\zeta_7), \]

where exponents of \(z\) are reduced modulo seven and \(\zeta_7\) is a primitive seventh root of unity. The original checker obtains the coefficient vector \((6J,-J,-J,-J,-J,-J,-J)\) in the cyclic group ring. Thus the exact intermediate identity is

\[ g_r(z)^3=7J-J\Phi_7(z), \qquad \Phi_7(z)=1+z+\cdots+z^6, \qquad z^7=1. \]

After specializing to \(z=\zeta_7\), the relation \(\Phi_7(\zeta_7)=0\) gives \(G_r^3=7J\). The scalar identity is not an identity in the larger ring with only \(z^7=1\): evaluation at \(z=1\) gives zero on the left, while \(7J\) is nonzero. The original code already retains the correction term. This note makes the passage between the two rings explicit.

The new exact check covers all six nonzero residue classes. It neither selects a cube-root branch nor identifies an elliptic-gamma value with an algebraic one. The inherited divisor theorem, analytic period carries, and normalized special-value comparison require their own evidence.

Review boundary

The focused review records this clarification, the finite fixed-vector check, and the limited review provenance. The original source and its historical evidence remain intact. The work remains in active development with the existing completion and confidence labels.

Original checker · Recorded run · Research index