feat(joinir): Phase 170-C-2b LoopUpdateSummary wiring into shape detection

Wire LoopUpdateSummary box into real code paths:

- Add `update_summary: Option<LoopUpdateSummary>` field to LoopFeatures
- Add `detect_with_updates()` method to CaseALoweringShape
  - Uses UpdateKind (CounterLike/AccumulationLike) for classification
  - Single CounterLike carrier → StringExamination
  - AccumulationLike present → ArrayAccumulation or IterationWithAccumulation
- Update loop_to_join.rs to use detect_with_updates()
- Update deprecated detect() to use detect_with_updates() internally
- Set update_summary: None in AST-based feature extraction

Carrier name heuristics now encapsulated in LoopUpdateSummary box.
No regression: 15/16 loop smoke tests pass (1 failure is pre-existing).

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

src/mir/join_ir/lowering/loop_scope_shape/case_a_lowering_shape.rs

@@ -202,6 +202,68 @@ impl CaseALoweringShape {
         }
     }
 
+    /// Phase 170-C-2b: LoopUpdateSummary を使った精度向上版
+    ///
+    /// `features.update_summary` から UpdateKind を取得して判定。
+    /// これにより、carrier 名を直接見るコードを CaseALoweringShape から排除できる。
+    ///
+    /// # Arguments
+    /// * `features` - LoopFeatures (update_summary 含む)
+    /// * `carrier_count` - Number of carrier variables
+    /// * `has_progress_carrier` - Whether progress carrier exists
+    ///
+    /// # Returns
+    /// CaseALoweringShape classification based on UpdateSummary
+    pub fn detect_with_updates(
+        features: &crate::mir::loop_pattern_detection::LoopFeatures,
+        carrier_count: usize,
+        has_progress_carrier: bool,
+    ) -> Self {
+        use crate::mir::join_ir::lowering::loop_update_summary::UpdateKind;
+
+        // Case-A requirement: must have a progress carrier
+        if !has_progress_carrier {
+            return CaseALoweringShape::NotCaseA;
+        }
+
+        // Case-A requirement: no complex control flow (continue)
+        if features.has_continue {
+            return CaseALoweringShape::NotCaseA;
+        }
+
+        // Phase 170-C-2b: Use UpdateSummary if available
+        if let Some(ref summary) = features.update_summary {
+            // Single carrier with CounterLike update → StringExamination
+            if summary.carriers.len() == 1
+                && summary.carriers[0].kind == UpdateKind::CounterLike
+            {
+                return CaseALoweringShape::StringExamination;
+            }
+
+            // Any AccumulationLike carrier → ArrayAccumulation (for single carrier)
+            // or IterationWithAccumulation (for multiple carriers)
+            if summary.carriers.iter().any(|c| c.kind == UpdateKind::AccumulationLike) {
+                if carrier_count == 1 {
+                    return CaseALoweringShape::ArrayAccumulation;
+                } else {
+                    return CaseALoweringShape::IterationWithAccumulation;
+                }
+            }
+
+            // Multiple carriers without clear accumulation → IterationWithAccumulation
+            if carrier_count >= 2 {
+                return CaseALoweringShape::IterationWithAccumulation;
+            }
+        }
+
+        // Fallback to carrier count heuristic
+        match carrier_count {
+            0 => CaseALoweringShape::NotCaseA,
+            1 => CaseALoweringShape::Generic,
+            2.. => CaseALoweringShape::IterationWithAccumulation,
+        }
+    }
+
     /// Typical index variable name detection
     ///
     /// StringExamination パターン（skip/trim）で使われる典型的なインデックス名：
@@ -229,6 +291,12 @@ impl CaseALoweringShape {
         let has_progress_carrier = scope.progress_carrier.is_some();
         let carrier_count = scope.carriers.len();
 
+        // Phase 170-C-2b: Build update_summary from carrier names
+        // Note: carriers is BTreeSet<String>, so each item is already a String
+        let carrier_names: Vec<String> = scope.carriers.iter().cloned().collect();
+        let update_summary =
+            crate::mir::join_ir::lowering::loop_update_summary::analyze_loop_updates(&carrier_names);
+
         // Create stub features (Phase 170-B will use real LoopFeatures)
         let stub_features = crate::mir::loop_pattern_detection::LoopFeatures {
             has_break: false,       // Unknown from LoopScopeShape alone
@@ -238,9 +306,10 @@ impl CaseALoweringShape {
             carrier_count,
             break_count: 0,
             continue_count: 0,
+            update_summary: Some(update_summary),
         };
 
-        Self::detect_from_features(&stub_features, carrier_count, has_progress_carrier)
+        Self::detect_with_updates(&stub_features, carrier_count, has_progress_carrier)
     }
 
     /// Is this a recognized lowering shape?