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Regular tetrahedron
x=1/√3, φ=90°

Fixed-area regular tetrahedron states, with A_FL/ℓ_p² = J. Values use project normalization (γ = 0.2375); the physical volume prefactor remains open. The independent local-spin tensor-product check agrees at matrix level within 2.8×10⁻¹⁵; RS and AL expectation differences are at most 1.67×10⁻¹⁶ and 5.55×10⁻¹⁷, respectively, for J = 1…5.

Each graph has tetrahedron previews reconstructed from the same equal-area face normals. The bottom row varies x at fixed φ = 90°; the side column varies φ at fixed x = 1/√3. Each preview uses the nearest valid scan sample and the same display size, so compare shape proportions rather than absolute size.

The scan varies x = |n₀ + n₁|/2 and the bending angle φ while keeping four unit face areas and closure. The highlighted regular tetrahedron is at x = 1/√3, φ = 90°. Face labels remain ordered; points are not deduplicated under leg permutations. The four normalized spinor rays also carry the cross ratio λ = ⟨z₀,z₂⟩⟨z₁,z₃⟩ / (⟨z₀,z₃⟩⟨z₁,z₂⟩), a coordinate on the four-marked-sphere moduli space used by Thurston. This is a coordinate link only; the cone-metric curvature data is not identified with the LQG face areas. Two direct tensor-product checks agree with the project routines within 1.4×10⁻¹⁷. Values remain project-normalized, and this equal-face-area family does not cover every coherent-state label at fixed J. Download the 440 sample records.