Positive Cubic Laplacians: Exact Far-Field Anisotropy and a Minimal Stencil Isotropic Through Degree Six

We distinguish local symbol isotropy from the far-field anisotropy of three-dimensional lattice Green functions. For positive cubic stencils supported in \( [-1,1]^3 \), quartic isotropy defines a one-parameter family containing D3Q15, D3Q19, and D3Q27. We compute its leading off-source \( a^4/r^5 \) correction, show that degree-six isotropy is impossible in this box, and find the unique spherical least-squares optimum with exact rational weights. For complete cubic orbits with steps in \( [-2,2]^3 \), strictly positive weights, and lattice connectivity, an exact enumeration of 61 candidate supports proves that 33 points are necessary and sufficient for symbol isotropy through degree six. The connected stencil uses orbit weights $4/15$, $2/15$, $1/30$, and $1/60$ on $100$, $110$, $111$, and $200$. Its first noncontact Green artifact occurs at order \( a^6/r^7 \). We distinguish this moment condition from sixth-order finite-difference consistency and give the corresponding polynomial correction, which need not retain positive edge weights. Exact rational certificates and independent moment and spectral tests accompany the result. The 33-point minimality is restricted to the stated orbit, positivity, radius, and connectivity class.