diff --git a/crates/ailang-codegen/src/lib.rs b/crates/ailang-codegen/src/lib.rs
index 35f5b2c..623753a 100644
--- a/crates/ailang-codegen/src/lib.rs
+++ b/crates/ailang-codegen/src/lib.rs
@@ -2458,12 +2458,7 @@ impl<'a> Emitter<'a> {
 
     /// Iter 23.2: hand-rolled body for monomorphiser-synthesised
     /// primitive instance methods whose natural lambda-lowering would
-    /// not produce the spec-mandated IR shape. Currently the only
-    /// inhabitant is `eq__Str`, which must call `@ail_str_eq` (the
-    /// stable AILang-namespaced Str-equality ABI in `runtime/str.c`)
-    /// rather than the `@strcmp + icmp eq i32 0` shape that
-    /// `lower_eq`'s Str arm would emit if we let the lambda body
-    /// `(== x y)` go through the normal path. Returns `Ok(true)` if
+    /// not produce the spec-mandated IR shape. Returns `Ok(true)` if
     /// the body was emitted (including the closing `}` and a final
     /// `\n\n`); `emit_fn` skips its normal body-lowering branch but
     /// MUST still run its post-body steps (deferred-thunk flush and
@@ -2471,10 +2466,19 @@ impl<'a> Emitter<'a> {
     /// top-level fn gets so it is reachable as a `Term::Var` value).
     /// `Ok(false)` lets `emit_fn` continue with normal body lowering.
     ///
-    /// Future iters add `compare__Str` here (23.3); int/bool primitive
-    /// methods (`eq__Int`, `eq__Bool`, `compare__Int`, `compare__Bool`)
-    /// ride the natural `lower_eq` / `lower_compare` dispatch and do
-    /// not need a hand-rolled body.
+    /// Currently inhabited arms: `eq__Str` (ships in 23.2.2) and
+    /// `compare__Int`, `compare__Bool`, `compare__Str` (this iter
+    /// 23.3). The two eq Int/Bool primitives (`eq__Int`, `eq__Bool`)
+    /// ride the natural `lower_eq` dispatch via their lambda body
+    /// `(== x y)` and do not need a hand-rolled body. The three
+    /// `compare` arms must be hand-rolled because there is no
+    /// type-polymorphic primitive that returns an Ordering ADT
+    /// value (Decision: `builtin_binop_typed` covers Int and Float
+    /// only — Bool is missing — so a surface-level
+    /// `if x < y { LT } else if x == y { EQ } else { GT }` body
+    /// would fail at codegen for `compare__Bool`). Hand-rolling all
+    /// three keeps the family consistent and the IR shape
+    /// predictable.
     fn try_emit_primitive_instance_body(
         &mut self,
         fn_name: &str,
@@ -2505,10 +2509,150 @@ impl<'a> Emitter<'a> {
                 self.block_terminated = true;
                 Ok(true)
             }
+            "compare__Int" => {
+                if param_tys != ["i64", "i64"] || ret_ty != "ptr" {
+                    return Err(CodegenError::Internal(format!(
+                        "compare__Int body intercept: unexpected signature \
+                         ({param_tys:?}) -> {ret_ty} (want (i64, i64) -> ptr)"
+                    )));
+                }
+                let n = self.locals.len();
+                let a_ssa = self.locals[n - 2].1.clone();
+                let b_ssa = self.locals[n - 1].1.clone();
+                let lt_test = self.fresh_ssa();
+                self.body.push_str(&format!(
+                    "  {lt_test} = icmp slt i64 {a_ssa}, {b_ssa}\n"
+                ));
+                let id = self.fresh_id();
+                let lt_label = format!("cmp_lt_{id}");
+                let after_lt_label = format!("cmp_after_lt_{id}");
+                self.body.push_str(&format!(
+                    "  br i1 {lt_test}, label %{lt_label}, label %{after_lt_label}\n"
+                ));
+                self.emit_ordering_arm(&lt_label, "LT")?;
+                self.start_block(&after_lt_label);
+                let eq_test = self.fresh_ssa();
+                self.body.push_str(&format!(
+                    "  {eq_test} = icmp eq i64 {a_ssa}, {b_ssa}\n"
+                ));
+                let eq_label = format!("cmp_eq_{id}");
+                let gt_label = format!("cmp_gt_{id}");
+                self.body.push_str(&format!(
+                    "  br i1 {eq_test}, label %{eq_label}, label %{gt_label}\n"
+                ));
+                self.emit_ordering_arm(&eq_label, "EQ")?;
+                self.emit_ordering_arm(&gt_label, "GT")?;
+                self.body.push_str("}\n\n");
+                self.block_terminated = true;
+                Ok(true)
+            }
+            "compare__Bool" => {
+                if param_tys != ["i1", "i1"] || ret_ty != "ptr" {
+                    return Err(CodegenError::Internal(format!(
+                        "compare__Bool body intercept: unexpected signature \
+                         ({param_tys:?}) -> {ret_ty} (want (i1, i1) -> ptr)"
+                    )));
+                }
+                let n = self.locals.len();
+                let a_ssa = self.locals[n - 2].1.clone();
+                let b_ssa = self.locals[n - 1].1.clone();
+                let lt_test = self.fresh_ssa();
+                self.body.push_str(&format!(
+                    "  {lt_test} = icmp ult i1 {a_ssa}, {b_ssa}\n"
+                ));
+                let id = self.fresh_id();
+                let lt_label = format!("cmp_lt_{id}");
+                let after_lt_label = format!("cmp_after_lt_{id}");
+                self.body.push_str(&format!(
+                    "  br i1 {lt_test}, label %{lt_label}, label %{after_lt_label}\n"
+                ));
+                self.emit_ordering_arm(&lt_label, "LT")?;
+                self.start_block(&after_lt_label);
+                let eq_test = self.fresh_ssa();
+                self.body.push_str(&format!(
+                    "  {eq_test} = icmp eq i1 {a_ssa}, {b_ssa}\n"
+                ));
+                let eq_label = format!("cmp_eq_{id}");
+                let gt_label = format!("cmp_gt_{id}");
+                self.body.push_str(&format!(
+                    "  br i1 {eq_test}, label %{eq_label}, label %{gt_label}\n"
+                ));
+                self.emit_ordering_arm(&eq_label, "EQ")?;
+                self.emit_ordering_arm(&gt_label, "GT")?;
+                self.body.push_str("}\n\n");
+                self.block_terminated = true;
+                Ok(true)
+            }
+            "compare__Str" => {
+                if param_tys != ["ptr", "ptr"] || ret_ty != "ptr" {
+                    return Err(CodegenError::Internal(format!(
+                        "compare__Str body intercept: unexpected signature \
+                         ({param_tys:?}) -> {ret_ty} (want (ptr, ptr) -> ptr)"
+                    )));
+                }
+                let n = self.locals.len();
+                let a_ssa = self.locals[n - 2].1.clone();
+                let b_ssa = self.locals[n - 1].1.clone();
+                let cmp_res = self.fresh_ssa();
+                self.body.push_str(&format!(
+                    "  {cmp_res} = call i32 @ail_str_compare(ptr {a_ssa}, ptr {b_ssa})\n"
+                ));
+                let lt_test = self.fresh_ssa();
+                self.body.push_str(&format!(
+                    "  {lt_test} = icmp slt i32 {cmp_res}, 0\n"
+                ));
+                let id = self.fresh_id();
+                let lt_label = format!("cmp_lt_{id}");
+                let after_lt_label = format!("cmp_after_lt_{id}");
+                self.body.push_str(&format!(
+                    "  br i1 {lt_test}, label %{lt_label}, label %{after_lt_label}\n"
+                ));
+                self.emit_ordering_arm(&lt_label, "LT")?;
+                self.start_block(&after_lt_label);
+                let eq_test = self.fresh_ssa();
+                self.body.push_str(&format!(
+                    "  {eq_test} = icmp eq i32 {cmp_res}, 0\n"
+                ));
+                let eq_label = format!("cmp_eq_{id}");
+                let gt_label = format!("cmp_gt_{id}");
+                self.body.push_str(&format!(
+                    "  br i1 {eq_test}, label %{eq_label}, label %{gt_label}\n"
+                ));
+                self.emit_ordering_arm(&eq_label, "EQ")?;
+                self.emit_ordering_arm(&gt_label, "GT")?;
+                self.body.push_str("}\n\n");
+                self.block_terminated = true;
+                Ok(true)
+            }
             _ => Ok(false),
         }
     }
 
+    /// Iter 23.3: emit one arm of the `compare__T` branch ladder.
+    /// Starts a fresh basic block labelled `label`, constructs the
+    /// matching `Ordering` ctor (LT / EQ / GT) via `lower_ctor`, and
+    /// emits a `ret ptr <ssa>`. Used three times per `compare__T`
+    /// arm in `try_emit_primitive_instance_body`. The ctor is a
+    /// zero-field allocation; `lower_ctor` handles alloc-strategy
+    /// variance (Gc / Bump / Rc) without the intercept duplicating
+    /// the per-strategy logic.
+    fn emit_ordering_arm(&mut self, label: &str, ctor: &str) -> Result<()> {
+        self.start_block(label);
+        // term_ptr 0: synthetic call site, not present in the
+        // escape-analysis result; falls back to the conservative
+        // "escapes → heap allocate" default. Safe for the Ordering
+        // return value (the caller owns it after `ret`).
+        let (ssa, _llvm_ty) = self.lower_ctor(
+            "Ordering",
+            ctor,
+            &[],
+            0,
+        )?;
+        self.body.push_str(&format!("  ret ptr {ssa}\n"));
+        self.block_terminated = true;
+        Ok(())
+    }
+
     /// Iter 16e: lower a `==` call after the two operands have been
     /// emitted. Dispatches on the resolved AIL type of the arg side
     /// (both sides have the same type after typecheck). The `_a_ll`
@@ -3696,4 +3840,116 @@ mod tests {
             "header missing @ail_str_compare declaration; ir was:\n{ir}"
         );
     }
+
+    /// Iter 23.3: codegen intercepts a fn named `compare__Int` and
+    /// emits a three-way branch ladder: `icmp slt i64 a, b` →
+    /// LT-block; else `icmp eq i64 a, b` → EQ-block; else GT-block.
+    /// Each block constructs the matching Ordering ctor via the
+    /// existing `lower_ctor` path. Pinned with a synthetic FnDef so
+    /// the test does not depend on prelude auto-injection.
+    #[test]
+    fn compare_int_mono_symbol_emits_branch_ladder() {
+        let m = synth_compare_module("compare__Int", Type::int(), "i64");
+        let ir = emit_ir(&m).unwrap();
+        assert!(
+            ir.contains("icmp slt i64"),
+            "compare__Int body must contain `icmp slt i64`; ir:\n{ir}"
+        );
+        assert!(
+            ir.contains("icmp eq i64"),
+            "compare__Int body must contain `icmp eq i64`; ir:\n{ir}"
+        );
+    }
+
+    /// Iter 23.3: same shape for `compare__Bool`. The LT-test uses
+    /// `icmp ult i1` (unsigned: false=0 ult true=1 gives the natural
+    /// Bool ordering false < true); the EQ-test uses `icmp eq i1`.
+    #[test]
+    fn compare_bool_mono_symbol_emits_branch_ladder() {
+        let m = synth_compare_module("compare__Bool", Type::bool_(), "i1");
+        let ir = emit_ir(&m).unwrap();
+        assert!(
+            ir.contains("icmp ult i1"),
+            "compare__Bool body must contain `icmp ult i1`; ir:\n{ir}"
+        );
+        assert!(
+            ir.contains("icmp eq i1"),
+            "compare__Bool body must contain `icmp eq i1`; ir:\n{ir}"
+        );
+    }
+
+    /// Iter 23.3: `compare__Str` calls `@ail_str_compare` to get the
+    /// normalised {-1, 0, +1}, then branches: slt 0 → LT, eq 0 → EQ,
+    /// else GT.
+    #[test]
+    fn compare_str_mono_symbol_emits_ail_str_compare_call() {
+        let m = synth_compare_module("compare__Str", Type::str_(), "ptr");
+        let ir = emit_ir(&m).unwrap();
+        assert!(
+            ir.contains("call i32 @ail_str_compare("),
+            "compare__Str body must call @ail_str_compare; ir:\n{ir}"
+        );
+        assert!(
+            ir.contains("icmp slt i32"),
+            "compare__Str body must compare result `slt i32` against 0; ir:\n{ir}"
+        );
+        assert!(
+            ir.contains("icmp eq i32"),
+            "compare__Str body must compare result `eq i32` against 0; ir:\n{ir}"
+        );
+    }
+
+    /// Test helper: minimal two-module workspace where the "prelude"
+    /// module declares `data Ordering = LT | EQ | GT` and an
+    /// instance-fn shell (the intercept overrides the body), and the
+    /// entry module's `main` is a Unit no-op so `emit_ir` returns a
+    /// well-formed program. Used by the three `compare_*` tests above.
+    fn synth_compare_module(fn_name: &str, param_ail_ty: Type, _llvm_param_ty: &str) -> Module {
+        let ordering = Def::Type(TypeDef {
+            name: "Ordering".into(),
+            vars: vec![],
+            ctors: vec![
+                Ctor { name: "LT".into(), fields: vec![] },
+                Ctor { name: "EQ".into(), fields: vec![] },
+                Ctor { name: "GT".into(), fields: vec![] },
+            ],
+            doc: None,
+            drop_iterative: false,
+        });
+        // Ordering is `ptr` at the LLVM level (boxed ADT).
+        let compare = Def::Fn(FnDef {
+            name: fn_name.into(),
+            ty: Type::Fn {
+                params: vec![param_ail_ty.clone(), param_ail_ty.clone()],
+                ret: Box::new(Type::Con { name: "Ordering".into(), args: vec![] }),
+                effects: vec![],
+                param_modes: vec![ParamMode::Borrow, ParamMode::Borrow],
+                ret_mode: ParamMode::Implicit,
+            },
+            params: vec!["x".into(), "y".into()],
+            body: Term::Lit { lit: Literal::Unit }, // placeholder; intercept overrides
+            suppress: vec![],
+            doc: None,
+        });
+        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,
+        });
+        Module {
+            schema: SCHEMA.into(),
+            name: "prelude".into(),
+            imports: vec![],
+            defs: vec![ordering, compare, main_def],
+        }
+    }
 }