diff --git a/alu/mul/columns.lisp b/alu/mul/columns.lisp
index aaf8cb9d..5002e064 100644
--- a/alu/mul/columns.lisp
+++ b/alu/mul/columns.lisp
@@ -1,46 +1,46 @@
 (module mul)
 
 (defcolumns
-    (MUL_STAMP :i32)
-    (COUNTER :byte)
-    (OLI :binary@prove)
-    (TINY_BASE          :binary@prove)
-    (TINY_EXPONENT      :binary@prove)
-    (RESULT_VANISHES    :binary@prove)
-    (INSTRUCTION :byte :display :opcode)
-    (ARG_1_HI  :i128)
-    (ARG_1_LO :i128)
-    (ARG_2_HI :i128)
-    (ARG_2_LO :i128)
-    (RES_HI :i128)
-    (RES_LO :i128)
-    (BITS :binary@prove)
+    (MUL_STAMP         :i32)
+    (COUNTER           :byte)
+    (OLI               :binary@prove)
+    (TINY_BASE         :binary@prove)
+    (TINY_EXPONENT     :binary@prove)
+    (RESULT_VANISHES   :binary@prove)
+    (INSTRUCTION       :byte :display :opcode)
+    (ARG_1_HI          :i128)
+    (ARG_1_LO          :i128)
+    (ARG_2_HI          :i128)
+    (ARG_2_LO          :i128)
+    (RES_HI            :i128)
+    (RES_LO            :i128)
+    (BITS              :binary@prove)
     ;==========================
-    (BYTE_A_3 :byte@prove)    ACC_A_3
-    (BYTE_A_2 :byte@prove)    ACC_A_2
-    (BYTE_A_1 :byte@prove)    ACC_A_1
-    (BYTE_A_0 :byte@prove)    ACC_A_0
+    (BYTE_A_3 :byte@prove)    (ACC_A_3 :i64)
+    (BYTE_A_2 :byte@prove)    (ACC_A_2 :i64)
+    (BYTE_A_1 :byte@prove)    (ACC_A_1 :i64)
+    (BYTE_A_0 :byte@prove)    (ACC_A_0 :i64)
     ;==========================
-    (BYTE_B_3 :byte@prove)    ACC_B_3
-    (BYTE_B_2 :byte@prove)    ACC_B_2
-    (BYTE_B_1 :byte@prove)    ACC_B_1
-    (BYTE_B_0 :byte@prove)    ACC_B_0
+    (BYTE_B_3 :byte@prove)    (ACC_B_3 :i64)
+    (BYTE_B_2 :byte@prove)    (ACC_B_2 :i64)
+    (BYTE_B_1 :byte@prove)    (ACC_B_1 :i64)
+    (BYTE_B_0 :byte@prove)    (ACC_B_0 :i64)
     ;==========================
-    (BYTE_C_3 :byte@prove)    ACC_C_3
-    (BYTE_C_2 :byte@prove)    ACC_C_2
-    (BYTE_C_1 :byte@prove)    ACC_C_1
-    (BYTE_C_0 :byte@prove)    ACC_C_0
+    (BYTE_C_3 :byte@prove)    (ACC_C_3 :i64)
+    (BYTE_C_2 :byte@prove)    (ACC_C_2 :i64)
+    (BYTE_C_1 :byte@prove)    (ACC_C_1 :i64)
+    (BYTE_C_0 :byte@prove)    (ACC_C_0 :i64)
     ;==========================
-    (BYTE_H_3 :byte@prove)    ACC_H_3
-    (BYTE_H_2 :byte@prove)    ACC_H_2
-    (BYTE_H_1 :byte@prove)    ACC_H_1
-    (BYTE_H_0 :byte@prove)    ACC_H_0
+    (BYTE_H_3 :byte@prove)    (ACC_H_3 :i64)
+    (BYTE_H_2 :byte@prove)    (ACC_H_2 :i64)
+    (BYTE_H_1 :byte@prove)    (ACC_H_1 :i64)
+    (BYTE_H_0 :byte@prove)    (ACC_H_0 :i64)
     ;==========================
     (EXPONENT_BIT               :binary@prove)
-    EXPONENT_BIT_ACCUMULATOR
+    (EXPONENT_BIT_ACCUMULATOR   :i8)
     (EXPONENT_BIT_SOURCE        :binary@prove)
     (SQUARE_AND_MULTIPLY        :binary@prove)
-    (BIT_NUM :byte)
+    (BIT_NUM                    :i8)
 )
 
 (defalias