diff --git a/bend2/comp.ts b/bend2/comp.ts
index 5a24c7b1b3..f73b4ce4a6 100644
--- a/bend2/comp.ts
+++ b/bend2/comp.ts
@@ -180,6 +180,7 @@ const OPERATIONS: Record<string, Intr> = Object.setPrototypeOf({
   u32_mul: { C: "U32_BIN($0, *, $1)", JS: "(Math.imul($0, $1) >>> 0)" },
   // C only: a JS file runs Base's U32.pow, as intr_of skips an op with no JS
   u32_pow: { C: "u32_pow($0, $1)" },
+  u32_log2: { C: "(31 - CLZ((u32)$0 | 1))" },
   u32_div: {
     C:  "((u32)($1) == 0 ? 0 : (u64)U32_QUO((u32)($0), (u32)($1)))",
     JS: "($1 === 0 ? 0 : ($0 / $1) >>> 0)",
@@ -233,6 +234,8 @@ const OPERATIONS: Record<string, Intr> = Object.setPrototypeOf({
   ...tpl_ops("nat_", "add:$1 double:$0", "nat_chk(e, $0 + $o)",
     "nat_chk($0 + $o)"),
   nat_mul: { C: "nat_mul(e, $0, $1)", JS: "nat_chk($0 * $1)" },
+  nat_pow: { C: "nat_pow(e, $0, $1)" },
+  map_msb_u: { C: "map_msb_u($0, $1)" },
   nat_cmp: { C: "(($0 > $1) + ($0 >= $1))", JS: "cmp_new($0, $1)" },
   ...tpl_ops("nat_", "sub", "($0 < $1 ? 0 : $0 - $1)"),
   ...tpl_ops("nat_", "is_lt:<", "($0 $o $1)"),
@@ -384,6 +387,22 @@ INLINE u64 nat_mul(Env e, u64 a, u64 b) {
   return nat_chk(e, b != 0 && a > NAT_IMM / b ? NAT_IMM + 1 : a * b);
 }
 
+INLINE u64 nat_pow(Env e, u64 a, u64 n) {
+  u64 r = 1;
+  while (n > 0) {
+    if (n & 1) r = nat_mul(e, r, a);
+    n >>= 1;
+    if (n) a = nat_mul(e, a, a);
+  }
+  return r;
+}
+
+INLINE u64 map_msb_u(u64 n, u32 x) {
+  if (n == 0 || x == 0) return 0;
+  u64 width = 32 - CLZ(x);
+  return n < width ? n : width;
+}
+
 #if DEVICE
 
 #define f32_show(e, x) (err_post(e.mem, ERR_FIDS), 0)
diff --git a/guide/GUIDE.md b/guide/GUIDE.md
index 56f91fcb3d..87d48692c3 100644
--- a/guide/GUIDE.md
+++ b/guide/GUIDE.md
@@ -638,10 +638,10 @@ A `Nat` literal past `256n` is `U32.to_nat(n)` underneath, up to `4294967295n`.
 
 ## Under the Hood
 
-Bend's compiler emits one C file, and that file is both the CPU program and the
-GPU kernel. clang compiles it for the CPU. Metal (on Apple) or CUDA (on NVIDIA)
-compiles the same file for the GPU. So a `!` call runs the same code on
-whichever chip it lands on.
+Bend emits one C file for CPU and GPU: clang builds the CPU program,
+Metal the Apple kernel, CUDA the NVIDIA kernel. Native scalar paths keep
+Base's definitions: checked repeated squaring for `Nat.pow`, count-leading-zero
+queries for `U32.log2` and fuel-bounded `Map.msb.u`.
 
 A term is one 64-bit word: small values are stored inline, everything else is a
 pointer into a single heap shared by every core and by the GPU. There is no
diff --git a/tests/run/nat_power_overflow.bend b/tests/run/nat_power_overflow.bend
new file mode 100644
index 0000000000..0bbf53f088
--- /dev/null
+++ b/tests/run/nat_power_overflow.bend
@@ -0,0 +1,8 @@
+# Repeated squaring must check the 48-bit Nat wall, not just u64 overflow.
+import Base
+
+def main() -> IO(Unit):
+  IO.print(Nat.show(Nat.pow(2n, 48n)))
+
+#|bend: a Nat past the largest immediate 2^48-1
+#|exit 1
\ No newline at end of file
diff --git a/tests/run/nat_power_wide_overflow.bend b/tests/run/nat_power_wide_overflow.bend
new file mode 100644
index 0000000000..11d2d82c3e
--- /dev/null
+++ b/tests/run/nat_power_wide_overflow.bend
@@ -0,0 +1,9 @@
+# Squaring the largest immediate must fail before the product wraps u64.
+import Base
+
+def main() -> IO(Unit):
+  largest = Nat.add(Nat.mul(16777215n, 16777216n), 16777215n)
+  IO.print(Nat.show(Nat.pow(largest, 2n)))
+
+#|bend: a Nat past the largest immediate 2^48-1
+#|exit 1
\ No newline at end of file
diff --git a/tests/run/native_numeric_boundaries.bend b/tests/run/native_numeric_boundaries.bend
new file mode 100644
index 0000000000..18ef86afdc
--- /dev/null
+++ b/tests/run/native_numeric_boundaries.bend
@@ -0,0 +1,53 @@
+# Native scalar paths agree with Base at zero, bit-width and Nat boundaries.
+import Base
+
+def main() -> IO(Unit):
+  +largest = Nat.add(Nat.mul(16777215n, 16777216n), 16777215n)
+  do IO<Unit>:
+    IO.print(Nat.show(Nat.pow(0n, 0n)))
+    IO.print(Nat.show(Nat.pow(0n, 7n)))
+    IO.print(Nat.show(Nat.pow(3n, 7n)))
+    IO.print(Nat.show(Nat.pow(1n, 257n)))
+    IO.print(Nat.show(Nat.pow(2n, 47n)))
+    IO.print(Nat.show(Nat.pow(largest, 0n)))
+    IO.print(Nat.show(Nat.pow(largest, 1n)))
+    IO.print(Nat.show(U32.log2(0)))
+    IO.print(Nat.show(U32.log2(1)))
+    IO.print(Nat.show(U32.log2(2)))
+    IO.print(Nat.show(U32.log2(3)))
+    IO.print(Nat.show(U32.log2(4)))
+    IO.print(Nat.show(U32.log2(2147483647)))
+    IO.print(Nat.show(U32.log2(2147483648)))
+    IO.print(Nat.show(U32.log2(4294967295)))
+    IO.print(Nat.show(Map.msb.u(0n, 4294967295)))
+    IO.print(Nat.show(Map.msb.u(1n, 0)))
+    IO.print(Nat.show(Map.msb.u(1n, 4294967295)))
+    IO.print(Nat.show(Map.msb.u(31n, 4294967295)))
+    IO.print(Nat.show(Map.msb.u(32n, 4294967295)))
+    IO.print(Nat.show(Map.msb.u(99n, 4294967295)))
+    IO.print(Nat.show(Map.msb.u(32n, 1)))
+    IO.print(Nat.show(Map.msb.u(99n, 2147483648)))
+
+#|1
+#|0
+#|2187
+#|1
+#|140737488355328
+#|1
+#|281474976710655
+#|0
+#|0
+#|1
+#|1
+#|2
+#|30
+#|31
+#|31
+#|0
+#|0
+#|1
+#|31
+#|32
+#|32
+#|1
+#|32
\ No newline at end of file