diff --git a/crates/ailang-codegen/src/lib.rs b/crates/ailang-codegen/src/lib.rs
index dd13021..a7ac544 100644
--- a/crates/ailang-codegen/src/lib.rs
+++ b/crates/ailang-codegen/src/lib.rs
@@ -2320,6 +2320,13 @@ impl<'a> Emitter<'a> {
                     let _ = b;
                     Ok(("true".into(), "i1".into()))
                 }
+                "Float" => {
+                    let dst = self.fresh_ssa();
+                    self.body.push_str(&format!(
+                        "  {dst} = fcmp oeq double {a}, {b}\n"
+                    ));
+                    Ok((dst, "i1".into()))
+                }
                 other => Err(CodegenError::Internal(format!(
                     "`==` not supported for type `{other}` \
                      (ADT and user-defined types lack a structural-equality scheme)"
@@ -3003,4 +3010,70 @@ mod tests {
             "Float arithmetic missing (no `fadd double` in IR): {ir}"
         );
     }
+
+    /// Floats iter 4.3 RED: `(< 1.5 2.5)` lowers as `fcmp olt double`;
+    /// `(!= 1.5 1.5)` lowers as `fcmp UNE double` (NOT `one`); `(== 1.5
+    /// 1.5)` lowers as `fcmp oeq double`. Int regressions still emit
+    /// `icmp slt i64` / `icmp ne i64` / `icmp eq i64`.
+    #[test]
+    fn lowers_float_comparison_dispatched() {
+        use ailang_core::ast::*;
+        fn fn_def(name: &str, body: Term) -> Def {
+            Def::Fn(FnDef {
+                name: name.into(),
+                ty: Type::Fn {
+                    params: vec![],
+                    ret: Box::new(Type::bool_()),
+                    effects: vec![],
+                    param_modes: vec![],
+                    ret_mode: ParamMode::Implicit,
+                },
+                params: vec![],
+                body,
+                suppress: vec![],
+                doc: None,
+            })
+        }
+        fn cmp(op: &str, a: Term, b: Term) -> Term {
+            Term::App {
+                callee: Box::new(Term::Var { name: op.into() }),
+                args: vec![a, b],
+                tail: false,
+            }
+        }
+        let f1 = Term::Lit { lit: Literal::Float { bits: 0x3ff8_0000_0000_0000u64 } };
+        let f2 = Term::Lit { lit: Literal::Float { bits: 0x4004_0000_0000_0000u64 } };
+        let i1 = Term::Lit { lit: Literal::Int { value: 1 } };
+        let i2 = Term::Lit { lit: Literal::Int { value: 2 } };
+        let main_def = Def::Fn(FnDef {
+            name: "main".into(),
+            ty: Type::Fn {
+                params: vec![], ret: Box::new(Type::unit()), effects: vec![],
+                param_modes: vec![], ret_mode: ParamMode::Implicit,
+            },
+            params: vec![], body: Term::Lit { lit: Literal::Unit },
+            suppress: vec![], doc: None,
+        });
+        let m = Module {
+            schema: ailang_core::SCHEMA.to_string(),
+            name: "t".into(),
+            imports: vec![],
+            defs: vec![
+                fn_def("flt_f", cmp("<", f1.clone(), f2.clone())),
+                fn_def("fne_f", cmp("!=", f1.clone(), f1.clone())),
+                fn_def("feq_f", cmp("==", f1.clone(), f1.clone())),
+                fn_def("flt_i", cmp("<", i1.clone(), i2.clone())),
+                fn_def("fne_i", cmp("!=", i1.clone(), i2.clone())),
+                fn_def("feq_i", cmp("==", i1.clone(), i2.clone())),
+                main_def,
+            ],
+        };
+        let ir = emit_ir(&m).unwrap();
+        assert!(ir.contains("fcmp olt double"), "missing `fcmp olt double`: {ir}");
+        assert!(ir.contains("fcmp une double"), "missing `fcmp une double` (note: `une` not `one`): {ir}");
+        assert!(ir.contains("fcmp oeq double"), "missing `fcmp oeq double`: {ir}");
+        assert!(ir.contains("icmp slt i64"), "Int `<` regressed: {ir}");
+        assert!(ir.contains("icmp ne i64"), "Int `!=` regressed: {ir}");
+        assert!(ir.contains("icmp eq i64"), "Int `==` regressed: {ir}");
+    }
 }
diff --git a/crates/ailang-codegen/src/synth.rs b/crates/ailang-codegen/src/synth.rs
index b014b6f..daa8448 100644
--- a/crates/ailang-codegen/src/synth.rs
+++ b/crates/ailang-codegen/src/synth.rs
@@ -281,6 +281,15 @@ pub(crate) fn builtin_binop_typed(
         ("<=", true, _) => Some(("icmp sle", "i64", "i1")),
         (">", true, _) => Some(("icmp sgt", "i64", "i1")),
         (">=", true, _) => Some(("icmp sge", "i64", "i1")),
+        // Float comparison arms (Floats iter 4.3). Note: `!=` Float
+        // uses `fcmp une` ("unordered or not equal") — NOT `one`
+        // ("ordered and not equal"), which would return `false` for
+        // `nan != nan` and violate IEEE / spec A5.
+        ("!=", _, true) => Some(("fcmp une", "double", "i1")),
+        ("<",  _, true) => Some(("fcmp olt", "double", "i1")),
+        ("<=", _, true) => Some(("fcmp ole", "double", "i1")),
+        (">",  _, true) => Some(("fcmp ogt", "double", "i1")),
+        (">=", _, true) => Some(("fcmp oge", "double", "i1")),
         _ => None,
     }
 }