diff --git a/engine-proto/Engine.jl b/engine-proto/Engine.jl
index 38ed672..546bf21 100644
--- a/engine-proto/Engine.jl
+++ b/engine-proto/Engine.jl
@@ -225,6 +225,8 @@ end
 
 # ~~~ sandbox setup ~~~
 
+using Random
+using Distributions
 using AbstractAlgebra
 using HomotopyContinuation
 
@@ -243,7 +245,8 @@ Engine.push!(ctx, b)
 Engine.push!(ctx, s)
 Engine.push!(ctx, b_on_s)
 ideal_ab_s, eqns_ab_s = Engine.realize(ctx)
-println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freedom")
+freedom = Engine.dimension(ideal_ab_s)
+println("Two points on a sphere: ", freedom, " degrees of freedom")
 
 ##spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
 ##tangencies = [
@@ -260,33 +263,67 @@ println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of f
 
 # --- test rational cut ---
 
+cut_coeffs = [
+  1 1 1 0 0 0 1 1 1 1 1;
+  2 1 1 0 0 0 1 2 1 1 1;
+  1 2 0 0 0 0 1 1 0 1 2
+]
 cut = [
-  sum(vcat([1, 1, 1] .* a.coords, [1, 1, 1, 1, 1] .* (s.coords - [0, 0, 0, 0, 1])))
-  sum(vcat([2, 1, 1] .* a.coords, [1, 2, 1, 1, 1] .* (s.coords - [0, 0, 0, 0, 1])))
-  sum(vcat([1, 2, 0] .* a.coords, [1, 1, 0, 1, 2] .* (s.coords - [0, 0, 0, 0, 1])))
+  sum(vcat(cf[1:3] .* a.coords, cf[4:6] .* b.coords, cf[7:end] .* (s.coords - [0, 0, 0, 0, 1])))
+  for cf in eachrow(cut_coeffs)
 ]
 cut_ideal_ab_s = Generic.Ideal(base_ring(ideal_ab_s), [gens(ideal_ab_s); cut])
-cut_dim = Engine.dimension(cut_ideal_ab_s)
-println("Two points on a sphere, after cut: ", cut_dim, " degrees of freedom")
-if cut_dim == 0
-  vbls = Variable.(symbols(base_ring(ideal_ab_s)))
+cut_freedom = Engine.dimension(cut_ideal_ab_s)
+println("Two points on a sphere, after cut: ", cut_freedom, " degrees of freedom")
+if cut_freedom == 0
+  coordring = base_ring(ideal_ab_s)
+  vbls = Variable.(symbols(coordring))
   cut_system = System([eqns_ab_s; cut], variables = vbls)
-  cut_result = HomotopyContinuation.solve(cut_system)
-  println("non-singular solutions:")
-  for soln in solutions(cut_result)
-    display(soln)
-  end
-  println("singular solutions:")
-  for sing in singular(cut_result)
-    display(sing.solution)
-  end
+  ##cut_result = HomotopyContinuation.solve(cut_system)
+  ##println("non-singular solutions:")
+  ##for soln in solutions(cut_result)
+  ##  display(soln)
+  ##end
+  ##println("singular solutions:")
+  ##for sing in singular(cut_result)
+  ##  display(sing.solution)
+  ##end
   
-  # test corresponding witness set
-  cut_matrix = [1 1 1 1 0 1 1 0 1 1 0; 1 2 1 2 0 1 1 0 1 1 0; 1 1 0 1 0 1 2 0 2 0 0]
-  cut_subspace = LinearSubspace(cut_matrix, [1, 1, 2])
-  witness = witness_set(System(eqns_ab_s, variables = vbls), cut_subspace)
+  # test a random witness set
+  max_slope = 2
+  binom = Binomial(2max_slope, 1/2)
+  Random.seed!(6071)
+  samples = []
+  for _ in 1:3
+    cut_matrix = rand(binom, freedom, length(gens(coordring))) .- max_slope
+    ##cut_matrix = [
+    ##  1 1 1 1 0 1 1 0 1 1 0;
+    ##  1 2 1 2 0 1 1 0 1 1 0;
+    ##  1 1 0 1 0 1 2 0 2 0 0
+    ##]
+    sph_z_ind = indexin([sph.coords[5] for sph in ctx.spheres], gens(coordring))
+    cut_offset = [sum(cf[sph_z_ind]) for cf in eachrow(cut_matrix)]
+    println("sphere z variables: ", vbls[sph_z_ind])
+    display(cut_matrix)
+    display(cut_offset)
+    cut_subspace = LinearSubspace(cut_matrix, cut_offset)
+    wtns = witness_set(System(eqns_ab_s, variables = vbls), cut_subspace)
+    append!(samples, solution.(filter(isreal, results(wtns))))
+  end
   println("witness solutions:")
-  for wtns in solutions(witness)
-    display(wtns)
+  for soln in samples
+    display([vbls round.(soln, digits = 6)])
+    k_sq = abs2(soln[1])
+    if abs2(soln[end-2]) > 1e-12
+      if k_sq < 1e-12
+        println("center at infinity: z coordinates $(round(soln[end], digits = 6)) and $(round(soln[end-1], digits = 6))}")
+      else
+        sum_sq = soln[4]^2 + soln[7]^2 + soln[end-2]^2 / k_sq
+        println("center on z axis: r² = $(round(1/k_sq, digits = 6)), x² + y² + h² = $(round(sum_sq, digits = 6))")
+      end
+    else
+      sum_sq = sum(soln[[4, 7, 10]] .^ 2)
+      println("center at origin: r² = $(round(1/k_sq, digits = 6)); x² + y² + z² = $(round(sum_sq, digits = 6))")
+    end
   end
 end
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