diff --git a/engine-proto/gram-test/Engine.jl b/engine-proto/gram-test/Engine.jl
index 22f5914..1eb72f5 100644
--- a/engine-proto/gram-test/Engine.jl
+++ b/engine-proto/gram-test/Engine.jl
@@ -8,7 +8,8 @@ using Optim
 
 export
   rand_on_shell, Q, DescentHistory,
-  realize_gram_gradient, realize_gram_newton, realize_gram_optim, realize_gram
+  realize_gram_gradient, realize_gram_newton, realize_gram_optim,
+  realize_gram_alt_proj, realize_gram
 
 # === guessing ===
 
@@ -143,7 +144,7 @@ function realize_gram_gradient(
       break
     end
     
-    # find negative gradient of loss function
+    # find the negative gradient of the loss function
     neg_grad = 4*Q*L*Δ_proj
     slope = norm(neg_grad)
     dir = neg_grad / slope
@@ -232,7 +233,7 @@ function realize_gram_newton(
       break
     end
     
-    # find the negative gradient of loss function
+    # find the negative gradient of the loss function
     neg_grad = 4*Q*L*Δ_proj
     
     # find the negative Hessian of the loss function
@@ -313,6 +314,130 @@ function realize_gram_optim(
   )
 end
 
+# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
+# explicit entry of `gram`. use gradient descent starting from `guess`, with an
+# alternate technique for finding the projected base step from the unprojected
+# Hessian
+function realize_gram_alt_proj(
+  gram::SparseMatrixCSC{T, <:Any},
+  guess::Matrix{T},
+  frozen = CartesianIndex[];
+  scaled_tol = 1e-30,
+  min_efficiency = 0.5,
+  init_rate = 1.0,
+  backoff = 0.9,
+  reg_scale = 1.1,
+  max_descent_steps = 200,
+  max_backoff_steps = 110
+) where T <: Number
+  # start history
+  history = DescentHistory{T}()
+  
+  # find the dimension of the search space
+  dims = size(guess)
+  element_dim, construction_dim = dims
+  total_dim = element_dim * construction_dim
+  
+  # list the constrained entries of the gram matrix
+  J, K, _ = findnz(gram)
+  constrained = zip(J, K)
+  
+  # scale the tolerance
+  scale_adjustment = sqrt(T(length(constrained)))
+  tol = scale_adjustment * scaled_tol
+  
+  # convert the frozen indices to stacked format
+  frozen_stacked = [(index[2]-1)*element_dim + index[1] for index in frozen]
+  
+  # initialize variables
+  grad_rate = init_rate
+  L = copy(guess)
+  
+  # use Newton's method with backtracking and gradient descent backup
+  Δ_proj = proj_diff(gram, L'*Q*L)
+  loss = dot(Δ_proj, Δ_proj)
+  for step in 1:max_descent_steps
+    # stop if the loss is tolerably low
+    if loss < tol
+      break
+    end
+    
+    # find the negative gradient of the loss function
+    neg_grad = 4*Q*L*Δ_proj
+    
+    # find the negative Hessian of the loss function
+    hess = Matrix{T}(undef, total_dim, total_dim)
+    indices = [(j, k) for k in 1:construction_dim for j in 1:element_dim]
+    for (j, k) in indices
+      basis_mat = basis_matrix(T, j, k, dims)
+      neg_dΔ = basis_mat'*Q*L + L'*Q*basis_mat
+      neg_dΔ_proj = proj_to_entries(neg_dΔ, constrained)
+      deriv_grad = 4*Q*(-basis_mat*Δ_proj + L*neg_dΔ_proj)
+      hess[:, (k-1)*element_dim + j] = reshape(deriv_grad, total_dim)
+    end
+    hess_sym = Hermitian(hess)
+    push!(history.hess, hess_sym)
+    
+    # regularize the Hessian
+    min_eigval = minimum(eigvals(hess_sym))
+    push!(history.positive, min_eigval > 0)
+    if min_eigval <= 0
+      hess -= reg_scale * min_eigval * I
+    end
+    
+    # compute the Newton step
+    neg_grad_stacked = reshape(neg_grad, total_dim)
+    for k in frozen_stacked
+      neg_grad_stacked[k] = 0
+      hess[k, :] .= 0
+      hess[:, k] .= 0
+      hess[k, k] = 1
+    end
+    base_step_stacked = Hermitian(hess) \ neg_grad_stacked
+    base_step = reshape(base_step_stacked, dims)
+    push!(history.base_step, base_step)
+    
+    # store the current position, loss, and slope
+    L_last = L
+    loss_last = loss
+    push!(history.scaled_loss, loss / scale_adjustment)
+    push!(history.neg_grad, neg_grad)
+    push!(history.slope, norm(neg_grad))
+    
+    # find a good step size using backtracking line search
+    push!(history.stepsize, 0)
+    push!(history.backoff_steps, max_backoff_steps)
+    empty!(history.last_line_L)
+    empty!(history.last_line_loss)
+    rate = one(T)
+    step_success = false
+    for backoff_steps in 0:max_backoff_steps
+      history.stepsize[end] = rate
+      L = L_last + rate * base_step
+      Δ_proj = proj_diff(gram, L'*Q*L)
+      loss = dot(Δ_proj, Δ_proj)
+      improvement = loss_last - loss
+      push!(history.last_line_L, L)
+      push!(history.last_line_loss, loss / scale_adjustment)
+      if improvement >= min_efficiency * rate * dot(neg_grad, base_step)
+        history.backoff_steps[end] = backoff_steps
+        step_success = true
+        break
+      end
+      rate *= backoff
+    end
+    
+    # if we've hit a wall, quit
+    if !step_success
+      return L_last, false, history
+    end
+  end
+  
+  # return the factorization and its history
+  push!(history.scaled_loss, loss / scale_adjustment)
+  L, loss < tol, history
+end
+
 # seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
 # explicit entry of `gram`. use gradient descent starting from `guess`
 function realize_gram(
@@ -365,7 +490,7 @@ function realize_gram(
       break
     end
     
-    # find the negative gradient of loss function
+    # find the negative gradient of the loss function
     neg_grad = 4*Q*L*Δ_proj
     
     # find the negative Hessian of the loss function
diff --git a/engine-proto/gram-test/irisawa-hexlet.jl b/engine-proto/gram-test/irisawa-hexlet.jl
index 67def8c..607db61 100644
--- a/engine-proto/gram-test/irisawa-hexlet.jl
+++ b/engine-proto/gram-test/irisawa-hexlet.jl
@@ -74,4 +74,13 @@ if success
   for k in 5:9
     println("  ", 1 / L[4,k], " sun")
   end
-end
\ No newline at end of file
+end
+
+# test an alternate technique for finding the projected base step from the
+# unprojected Hessian
+L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
+completed_gram_alt = L_alt'*Engine.Q*L_alt
+println("\nDifference in result using alternate projection:\n")
+display(completed_gram_alt - completed_gram)
+println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
+println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
\ No newline at end of file
diff --git a/engine-proto/gram-test/sphere-in-tetrahedron.jl b/engine-proto/gram-test/sphere-in-tetrahedron.jl
index 97f0720..5d479cf 100644
--- a/engine-proto/gram-test/sphere-in-tetrahedron.jl
+++ b/engine-proto/gram-test/sphere-in-tetrahedron.jl
@@ -64,4 +64,13 @@ else
   println("\nFailed to reach target accuracy")
 end
 println("Steps: ", size(history.scaled_loss, 1))
-println("Loss: ", history.scaled_loss[end], "\n")
\ No newline at end of file
+println("Loss: ", history.scaled_loss[end], "\n")
+
+# test an alternate technique for finding the projected base step from the
+# unprojected Hessian
+L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
+completed_gram_alt = L_alt'*Engine.Q*L_alt
+println("\nDifference in result using alternate projection:\n")
+display(completed_gram_alt - completed_gram)
+println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
+println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
\ No newline at end of file
diff --git a/engine-proto/gram-test/tetrahedron-radius-ratio.jl b/engine-proto/gram-test/tetrahedron-radius-ratio.jl
index 7ceb794..9fec28e 100644
--- a/engine-proto/gram-test/tetrahedron-radius-ratio.jl
+++ b/engine-proto/gram-test/tetrahedron-radius-ratio.jl
@@ -93,4 +93,13 @@ if success
   infty = BigFloat[0, 0, 0, 0, 1]
   radius_ratio = dot(infty, Engine.Q * L[:,5]) / dot(infty, Engine.Q * L[:,6])
   println("\nCircumradius / inradius: ", radius_ratio)
-end
\ No newline at end of file
+end
+
+# test an alternate technique for finding the projected base step from the
+# unprojected Hessian
+L_alt, success_alt, history_alt = Engine.realize_gram_alt_proj(gram, guess, frozen)
+completed_gram_alt = L_alt'*Engine.Q*L_alt
+println("\nDifference in result using alternate projection:\n")
+display(completed_gram_alt - completed_gram)
+println("\nDifference in steps: ", size(history_alt.scaled_loss, 1) - size(history.scaled_loss, 1))
+println("Difference in loss: ", history_alt.scaled_loss[end] - history.scaled_loss[end], "\n")
\ No newline at end of file