diff --git a/benches/scalar_micro.rs b/benches/scalar_micro.rs
index eb561ad..5e07b6d 100644
--- a/benches/scalar_micro.rs
+++ b/benches/scalar_micro.rs
@@ -312,6 +312,18 @@ const SCALAR_MICRO_GROUPS: &[BenchGroupDoc] = &[
                 name: "real_add_owned",
                 description: "Adds owned scaled transcendental `Real` values.",
             },
+            BenchDoc {
+                name: "real_dot2_refs_dense_symbolic",
+                description: "Computes a borrowed two-lane symbolic dot product with no rational shortcut terms.",
+            },
+            BenchDoc {
+                name: "real_active_dot2_refs_dense_symbolic",
+                description: "Computes a borrowed two-lane symbolic dot product after the caller has already classified every lane active.",
+            },
+            BenchDoc {
+                name: "real_dot2_refs_mixed_structural",
+                description: "Computes a borrowed two-lane symbolic dot product with an exact zero lane and a rational scale lane.",
+            },
             BenchDoc {
                 name: "real_dot3_refs_dense_symbolic",
                 description: "Computes a borrowed three-lane symbolic dot product with no rational shortcut terms.",
@@ -800,6 +812,39 @@ fn bench_borrowed_op_overhead(c: &mut Criterion) {
             BatchSize::SmallInput,
         )
     });
+    group.bench_function("real_dot2_refs_dense_symbolic", |b| {
+        b.iter(|| {
+            black_box(Real::dot2_refs(
+                [black_box(&dense_dot_left[0]), black_box(&dense_dot_left[1])],
+                [
+                    black_box(&dense_dot_right[0]),
+                    black_box(&dense_dot_right[1]),
+                ],
+            ))
+        })
+    });
+    group.bench_function("real_active_dot2_refs_dense_symbolic", |b| {
+        b.iter(|| {
+            black_box(Real::active_dot2_refs(
+                [black_box(&dense_dot_left[0]), black_box(&dense_dot_left[1])],
+                [
+                    black_box(&dense_dot_right[0]),
+                    black_box(&dense_dot_right[1]),
+                ],
+            ))
+        })
+    });
+    group.bench_function("real_dot2_refs_mixed_structural", |b| {
+        b.iter(|| {
+            black_box(Real::dot2_refs(
+                [black_box(&mixed_dot_left[0]), black_box(&mixed_dot_left[1])],
+                [
+                    black_box(&mixed_dot_right[0]),
+                    black_box(&mixed_dot_right[1]),
+                ],
+            ))
+        })
+    });
     group.bench_function("real_dot3_refs_dense_symbolic", |b| {
         b.iter(|| {
             black_box(Real::dot3_refs(
diff --git a/benchmarks.md b/benchmarks.md
index 914ff72..ac84894 100644
--- a/benchmarks.md
+++ b/benchmarks.md
@@ -319,6 +319,9 @@ Borrowed versus owned operation overhead for rational and real operands.
 | `borrowed_op_overhead/real_dot4_refs_dense_symbolic` | 5.636 us | 5.625 us - 5.653 us | Computes a borrowed four-lane symbolic dot product with no rational shortcut terms. |
 | `borrowed_op_overhead/real_active_dot4_refs_dense_symbolic` | 5.677 us | 5.670 us - 5.688 us | Computes a borrowed four-lane symbolic dot product after the caller has already classified every lane active. |
 | `borrowed_op_overhead/real_dot4_refs_mixed_structural` | 653.48 ns | 651.96 ns - 655.30 ns | Computes a borrowed four-lane symbolic dot product with exact zero and rational scale terms. |
+| `borrowed_op_overhead/real_dot2_refs_dense_symbolic` | not run | not run | Computes a borrowed two-lane symbolic dot product with no rational shortcut terms. |
+| `borrowed_op_overhead/real_active_dot2_refs_dense_symbolic` | not run | not run | Computes a borrowed two-lane symbolic dot product after the caller has already classified every lane active. |
+| `borrowed_op_overhead/real_dot2_refs_mixed_structural` | not run | not run | Computes a borrowed two-lane symbolic dot product with an exact zero lane and a rational scale lane. |
 
 ### `dense_algebra`
 
diff --git a/src/real/arithmetic.rs b/src/real/arithmetic.rs
index 24407a0..e7a3f65 100644
--- a/src/real/arithmetic.rs
+++ b/src/real/arithmetic.rs
@@ -1929,6 +1929,91 @@ impl Real {
         }
     }
 
+    /// Return the two-lane dot product of borrowed reals.
+    ///
+    /// Sibling of [`Self::dot3_refs`] / [`Self::dot4_refs`] for the
+    /// two-component case (2D coordinates, complex products, planar dot
+    /// products, etc.). Same exact-rational shared-denominator fast path;
+    /// same symbolic fallback policy.
+    pub fn dot2_refs(left: [&Real; 2], right: [&Real; 2]) -> Real {
+        if let (Some(l0), Some(l1), Some(r0), Some(r1)) = (
+            left[0].exact_rational_ref(),
+            left[1].exact_rational_ref(),
+            right[0].exact_rational_ref(),
+            right[1].exact_rational_ref(),
+        ) {
+            crate::trace_dispatch!("real", "dot_product", "dot2-exact-rational-shared-denom");
+            return Real::new(Rational::dot_products([l0, l1], [r0, r1]));
+        }
+
+        Self::dot2_refs_fallback(left, right)
+    }
+
+    /// Return a two-lane dot product whose lanes were already classified active.
+    ///
+    /// See [`Self::active_dot3_refs`].
+    pub fn active_dot2_refs(left: [&Real; 2], right: [&Real; 2]) -> Real {
+        if let (Some(l0), Some(l1), Some(r0), Some(r1)) = (
+            left[0].exact_rational_ref(),
+            left[1].exact_rational_ref(),
+            right[0].exact_rational_ref(),
+            right[1].exact_rational_ref(),
+        ) {
+            crate::trace_dispatch!("real", "dot_product", "active-dot2-exact-rational");
+            return Real::new(Rational::dot_products([l0, l1], [r0, r1]));
+        }
+
+        crate::trace_dispatch!("real", "dot_product", "active-dot2-real-tree");
+        Self::sum_dot2_terms(
+            Some(Self::dot_product_active_term(left[0], right[0])),
+            Some(Self::dot_product_active_term(left[1], right[1])),
+        )
+    }
+
+    #[inline(never)]
+    fn dot2_refs_fallback(left: [&Real; 2], right: [&Real; 2]) -> Real {
+        // See `dot3_refs_fallback` for the code-layout rationale.
+        if Self::dot_product_has_structural_term(left[0], right[0])
+            || Self::dot_product_has_structural_term(left[1], right[1])
+        {
+            crate::trace_dispatch!("real", "dot_product", "dot2-structural-real-tree");
+            return Self::sum_dot2_terms(
+                Self::dot_product_term(left[0], right[0]),
+                Self::dot_product_term(left[1], right[1]),
+            );
+        }
+
+        if left[0].rational.sign() == Sign::NoSign
+            || right[0].rational.sign() == Sign::NoSign
+            || left[1].rational.sign() == Sign::NoSign
+            || right[1].rational.sign() == Sign::NoSign
+        {
+            let p0 = Self::dot_product_term(left[0], right[0]);
+            let p1 = Self::dot_product_term(left[1], right[1]);
+            let active_terms = usize::from(p0.is_some()) + usize::from(p1.is_some());
+
+            match active_terms {
+                0 => {
+                    crate::trace_dispatch!("real", "dot_product", "dot2-all-zero-real-tree");
+                    return Real::zero();
+                }
+                1 => {
+                    crate::trace_dispatch!("real", "dot_product", "dot2-generic-real-tree-sparse");
+                    return Self::sum_dot2_terms(p0, p1);
+                }
+                _ => {
+                    crate::trace_dispatch!("real", "dot_product", "dot2-generic-real-tree");
+                    return Self::sum_dot2_terms(p0, p1);
+                }
+            }
+        }
+
+        let p0 = left[0] * right[0];
+        let p1 = left[1] * right[1];
+        crate::trace_dispatch!("real", "dot_product", "dot2-generic-real-tree");
+        &p0 + &p1
+    }
+
     /// Return the three-lane dot product of borrowed reals.
     ///
     /// Exact-rational lanes are accumulated with one shared denominator and a
@@ -2338,6 +2423,15 @@ impl Real {
         product
     }
 
+    #[inline]
+    fn sum_dot2_terms(p0: Option<Real>, p1: Option<Real>) -> Real {
+        match (p0, p1) {
+            (None, None) => Real::zero(),
+            (Some(p), None) | (None, Some(p)) => p,
+            (Some(a), Some(b)) => &a + &b,
+        }
+    }
+
     #[inline]
     fn sum_dot3_terms(p0: Option<Real>, p1: Option<Real>, p2: Option<Real>) -> Real {
         match (p0, p1, p2) {
diff --git a/src/real/tests.rs b/src/real/tests.rs
index c8a0b16..7feea6c 100644
--- a/src/real/tests.rs
+++ b/src/real/tests.rs
@@ -1511,5 +1511,77 @@ mod tests {
         assert!(
             (actual.to_f64_approx().unwrap() - expected.to_f64_approx().unwrap()).abs() < 1e-12
         );
+
+        let expected = &(&left[0] * &right[0]) + &(&left[1] * &right[1]);
+        let actual = Real::dot2_refs([&left[0], &left[1]], [&right[0], &right[1]]);
+        assert!(
+            (actual.to_f64_approx().unwrap() - expected.to_f64_approx().unwrap()).abs() < 1e-12
+        );
+    }
+
+    #[test]
+    fn dot2_refs_matches_pairwise_rational_arithmetic() {
+        let left = [
+            Real::new(Rational::fraction(6, 5).unwrap()),
+            Real::new(Rational::fraction(-7, 10).unwrap()),
+        ];
+        let right = [
+            Real::new(Rational::fraction(-4, 5).unwrap()),
+            Real::new(Rational::fraction(11, 10).unwrap()),
+        ];
+        let expected = &(&left[0] * &right[0]) + &(&left[1] * &right[1]);
+        assert_eq!(
+            Real::dot2_refs([&left[0], &left[1]], [&right[0], &right[1]]),
+            expected,
+        );
+    }
+
+    #[test]
+    fn dot2_refs_handles_symbolic_lanes() {
+        let left = [Real::pi(), Real::e()];
+        let right = [Real::e(), Real::pi()];
+        let expected = &(&left[0] * &right[0]) + &(&left[1] * &right[1]);
+        let actual = Real::dot2_refs([&left[0], &left[1]], [&right[0], &right[1]]);
+        assert!(
+            (actual.to_f64_approx().unwrap() - expected.to_f64_approx().unwrap()).abs() < 1e-12
+        );
+    }
+
+    #[test]
+    fn dot2_refs_zero_lane_shortcut() {
+        let left = [Real::zero(), Real::from(3_i32)];
+        let right = [Real::pi(), Real::e()];
+        let expected = &left[1] * &right[1];
+        let actual = Real::dot2_refs([&left[0], &left[1]], [&right[0], &right[1]]);
+        assert!(
+            (actual.to_f64_approx().unwrap() - expected.to_f64_approx().unwrap()).abs() < 1e-12
+        );
+    }
+
+    #[test]
+    fn dot2_refs_all_zero_returns_zero() {
+        let left = [Real::zero(), Real::zero()];
+        let right = [Real::pi(), Real::e()];
+        assert_eq!(
+            Real::dot2_refs([&left[0], &left[1]], [&right[0], &right[1]]),
+            Real::zero(),
+        );
+    }
+
+    #[test]
+    fn active_dot2_refs_matches_dot2_refs_when_all_lanes_active() {
+        let left = [
+            Real::pi(),
+            Real::e() * Real::new(Rational::fraction(3, 5).unwrap()),
+        ];
+        let right = [
+            Real::e() * Real::new(Rational::fraction(2, 7).unwrap()),
+            Real::pi(),
+        ];
+        let expected = Real::dot2_refs([&left[0], &left[1]], [&right[0], &right[1]]);
+        let actual = Real::active_dot2_refs([&left[0], &left[1]], [&right[0], &right[1]]);
+        assert!(
+            (actual.to_f64_approx().unwrap() - expected.to_f64_approx().unwrap()).abs() < 1e-12
+        );
     }
 }