diff --git a/engine-proto/ConstructionViewer.jl b/engine-proto/ConstructionViewer.jl
index 8cfa632..c9b0b7a 100644
--- a/engine-proto/ConstructionViewer.jl
+++ b/engine-proto/ConstructionViewer.jl
@@ -213,8 +213,8 @@ elements = begin
     1    1   -1   -1    0
     1   -1    1   -1    0
     1   -1   -1    1    0
-   -0.5 -0.5 -0.5 -0.5 -a-1
-    0.5  0.5  0.5  0.5 -a+1
+    0.5  0.5  0.5  0.5  1+a
+    0.5  0.5  0.5  0.5  1-a
   ]
 end
 
diff --git a/engine-proto/gram-test/Engine.jl b/engine-proto/gram-test/Engine.jl
index bedda00..73c5c31 100644
--- a/engine-proto/gram-test/Engine.jl
+++ b/engine-proto/gram-test/Engine.jl
@@ -39,7 +39,7 @@ rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), she
 
 # === elements ===
 
-point(pos) = [pos; -1; 0.25 * dot(pos, pos)]
+point(pos) = [pos; 1; 0.25 * dot(pos, pos)]
 
 plane(normal, offset) = [normal; 0; offset]
 
@@ -47,7 +47,7 @@ function sphere(center, radius)
   dist_sq = dot(center, center)
   [
     center / radius;
-    -0.5 / radius;
+    0.5 / radius;
     0.5 * (dist_sq / radius - radius)
   ]
 end
@@ -55,12 +55,12 @@ end
 # === Gram matrix realization ===
 
 # basis changes
-nullmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [1 -1; 1 1]//2]
-unmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [1 1; -1 1]]
+nullmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [-1 1; 1 1]//2]
+unmix = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [-1 1; 1 1]]
 
 # the Lorentz form
 ## [old] Q = diagm([1, 1, 1, 1, -1])
-Q = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [0 2; 2 0]]
+Q = [Matrix{Int64}(I, 3, 3) zeros(Int64, 3, 2); zeros(Int64, 2, 3) [0 -2; -2 0]]
 
 # project a matrix onto the subspace of matrices whose entries vanish at the
 # given indices
diff --git a/engine-proto/gram-test/overlapping-pyramids.jl b/engine-proto/gram-test/overlapping-pyramids.jl
index 8edb981..0d1f018 100644
--- a/engine-proto/gram-test/overlapping-pyramids.jl
+++ b/engine-proto/gram-test/overlapping-pyramids.jl
@@ -43,8 +43,8 @@ guess = begin
       1    1   -1   -1    0
       1   -1    1   -1    0
       1   -1   -1    1    0
-     -0.5 -0.5 -0.5 -0.5 -a-1
-      0.5  0.5  0.5  0.5 -a+1
+      0.5  0.5  0.5  0.5  1+a
+      0.5  0.5  0.5  0.5  1-a
     ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
     Engine.rand_on_shell(fill(BigFloat(-1), 2))
   )
diff --git a/engine-proto/gram-test/sphere-in-tetrahedron.jl b/engine-proto/gram-test/sphere-in-tetrahedron.jl
index 1c0dda8..631f0e5 100644
--- a/engine-proto/gram-test/sphere-in-tetrahedron.jl
+++ b/engine-proto/gram-test/sphere-in-tetrahedron.jl
@@ -46,7 +46,7 @@ guess = hcat(
     1  1 -1 -1  0
     1 -1  1 -1  0
     1 -1 -1  1  0
-    0  0  0  0 -1.5
+    0  0  0  0  1.5
     1  1  1  1 -0.5
   ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
   BigFloat[0, 0, 0, 0, 1]
diff --git a/engine-proto/gram-test/tetrahedron-radius-ratio.jl b/engine-proto/gram-test/tetrahedron-radius-ratio.jl
index 4218cb7..ed3ceb0 100644
--- a/engine-proto/gram-test/tetrahedron-radius-ratio.jl
+++ b/engine-proto/gram-test/tetrahedron-radius-ratio.jl
@@ -63,7 +63,7 @@ guess = hcat(
     1  1 -1 -1  0    0
     1 -1  1 -1  0    0
     1 -1 -1  1  0    0
-    0  0  0  0 -1.5 -0.5
+    0  0  0  0  1.5  0.5
     1  1  1  1 -0.5 -1.5
   ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 6)),
   Engine.point([-1, -1, -1] + 0.3*randn(3)),