diff --git a/docs/arithmetization/tables/cpu.tex b/docs/arithmetization/tables/cpu.tex
index b3f1be65a..590b95668 100644
--- a/docs/arithmetization/tables/cpu.tex
+++ b/docs/arithmetization/tables/cpu.tex
@@ -71,5 +71,5 @@ \subsubsection{CPU columns}
 before) the instruction. \texttt{stack\_len\_bounds\_ aux} is used to check that the stack doesn't overflow in user mode. We use the last four columns to prevent conflicts with the other general columns.
 See \ref{stackhandling} for more details.
     \label{push_general_view}
-    \item  \texttt{Push}: \texttt{push\_prover\_input\_not\_kernel} is used to skip range-checking the output of a PUSH operation when we are in privileged mode, as the kernel code is known and trusted.
+    \item  \texttt{Push}: \texttt{is\_not\_kernel} is used to skip range-checking the output of a PUSH operation when we are in privileged mode, as the kernel code is known and trusted.
 \end{itemize}
diff --git a/docs/arithmetization/zkevm.pdf b/docs/arithmetization/zkevm.pdf
index 5a9d91dd6..d1986f70d 100644
Binary files a/docs/arithmetization/zkevm.pdf and b/docs/arithmetization/zkevm.pdf differ
diff --git a/evm_arithmetization/src/cpu/columns/general.rs b/evm_arithmetization/src/cpu/columns/general.rs
index 8affe6f08..64b7e4ce5 100644
--- a/evm_arithmetization/src/cpu/columns/general.rs
+++ b/evm_arithmetization/src/cpu/columns/general.rs
@@ -193,15 +193,14 @@ pub(crate) struct CpuStackView<T: Copy> {
     pub(crate) stack_len_bounds_aux: T,
 }
 
-/// View of the first `CpuGeneralColumn` storing the product of the negated
-/// `is_kernel_mode` flag with the `push_prover_input` combined op flag, to
-/// filter out `PUSH` instructions from being range-checked when happening in
-/// the KERNEL context.
+/// View of the first `CpuGeneralColumn` storing the negation of
+/// `is_kernel_mode` flag, to filter out `PUSH` instructions from being
+/// range-checked when happening in the KERNEL context.
 #[repr(C)]
 #[derive(Copy, Clone)]
 pub(crate) struct CpuPushView<T: Copy> {
     /// Product of `push_prover_input` with the negated `is_kernel_mode` flag.
-    pub(crate) push_prover_input_not_kernel: T,
+    pub(crate) is_not_kernel: T,
     /// Reserve the unused columns.
     _padding_columns: [T; NUM_SHARED_COLUMNS - 1],
 }
diff --git a/evm_arithmetization/src/cpu/control_flow.rs b/evm_arithmetization/src/cpu/control_flow.rs
index 18aeb65ba..a88576456 100644
--- a/evm_arithmetization/src/cpu/control_flow.rs
+++ b/evm_arithmetization/src/cpu/control_flow.rs
@@ -82,8 +82,7 @@ pub(crate) fn eval_packed_generic<P: PackedField>(
     // it is set to 0 if the operation is a `PROVER_INPUT`, as the latter is a
     // kernel-only instruction. This is enforced in the `decode` module.
     yield_constr.constraint(
-        lv.op.push_prover_input
-            * ((lv.is_kernel_mode + lv.general.push().push_prover_input_not_kernel) - P::ONES),
+        lv.op.push_prover_input * ((lv.is_kernel_mode + lv.general.push().is_not_kernel) - P::ONES),
     );
 
     // If a non-CPU cycle row is followed by a CPU cycle row, then:
@@ -152,10 +151,7 @@ pub(crate) fn eval_ext_circuit<F: RichField + Extendable<D>, const D: usize>(
     // it is set to 0 if the operation is a `PROVER_INPUT`, as the latter is a
     // kernel-only instruction. This is enforced in the `decode` module.
     {
-        let temp = builder.add_extension(
-            lv.is_kernel_mode,
-            lv.general.push().push_prover_input_not_kernel,
-        );
+        let temp = builder.add_extension(lv.is_kernel_mode, lv.general.push().is_not_kernel);
         let constr =
             builder.mul_sub_extension(lv.op.push_prover_input, temp, lv.op.push_prover_input);
         yield_constr.constraint(builder, constr);
diff --git a/evm_arithmetization/src/cpu/cpu_stark.rs b/evm_arithmetization/src/cpu/cpu_stark.rs
index a30306afb..55a3bf2d4 100644
--- a/evm_arithmetization/src/cpu/cpu_stark.rs
+++ b/evm_arithmetization/src/cpu/cpu_stark.rs
@@ -265,7 +265,7 @@ pub(crate) fn ctl_data_byte_packing_push<F: Field>() -> Vec<Column<F>> {
 pub(crate) fn ctl_filter_byte_packing_push<F: Field>() -> Filter<F> {
     Filter::new(
         vec![(
-            Column::single(COL_MAP.general.push().push_prover_input_not_kernel),
+            Column::single(COL_MAP.general.push().is_not_kernel),
             Column::single(COL_MAP.op.push_prover_input),
         )],
         vec![],
diff --git a/evm_arithmetization/src/witness/operation.rs b/evm_arithmetization/src/witness/operation.rs
index 233e47ff6..07d48819b 100644
--- a/evm_arithmetization/src/witness/operation.rs
+++ b/evm_arithmetization/src/witness/operation.rs
@@ -388,7 +388,7 @@ pub(crate) fn generate_push<F: Field, T: Transition<F>>(
 
     // This is necessary to filter out PUSH instructions from the BytePackingStark
     // CTl when happening in the KERNEL context.
-    row.general.push_mut().push_prover_input_not_kernel = F::ONE - row.is_kernel_mode;
+    row.general.push_mut().is_not_kernel = F::ONE - row.is_kernel_mode;
 
     if code_context != KERNEL_CONTEXT {
         byte_packing_log(state, base_address, bytes);