feat(factor): C1 因子相关性分析（横截面 Spearman 矩阵 + API）

- evaluation.factor_correlation_report：多因子共同日期 ∩ 后逐日横截面 Spearman 相关
  取均值 → FactorCorrelationReport（冗余剔除前置，v3 §12 Correlation→Redundancy）
- ResearchService.run_factor_correlation + POST /api/factor-correlations
  （universe/factors/period；与其它研究同装配口径）
- tests/test_factor_correlation.py：矩阵对角=1/近线性±相关符号/对称/无共同日期补零、
  API 冒烟；全量 pytest 通过

backend/app/api/research.py

@@ -6,13 +6,20 @@ Phase 3 为同步执行（样本有限）；Phase 4 将改为 Job + SSE 异步
 
 from __future__ import annotations
 
+from datetime import date
+
 from fastapi import APIRouter, HTTPException
+from pydantic import BaseModel, Field
 
 from app.api.deps import ResearchServiceDep
 from app.domain.entities.research import (
     BacktestResult,
+    FactorCorrelationReport,
+    FactorSpec,
     FactorTestReport,
     ResearchSpec,
+    SelectionSpec,
+    UniverseSpec,
 )
 from app.quant.factors import FactorError
 
@@ -54,3 +61,38 @@ def last_backtest() -> BacktestResult:
     if "default" not in _LAST_BACKTEST:
         raise HTTPException(status_code=404, detail="尚无回测结果，请先 POST /api/backtests")
     return _LAST_BACKTEST["default"]
+
+
+class FactorCorrelationRequest(BaseModel):
+    """因子相关性分析请求（v3 §12）。"""
+
+    universe: UniverseSpec = UniverseSpec()
+    factors: list[FactorSpec] = Field(min_length=2)
+    period: tuple[date, date]
+    price_adjustment: str = Field(default="none", pattern="^(none|qfq)$")
+
+
+@router.post(
+    "/factor-correlations",
+    response_model=FactorCorrelationReport,
+    summary="多因子两两相关（横截面 Spearman）",
+)
+def run_factor_correlation(
+    req: FactorCorrelationRequest,
+    service: ResearchServiceDep,
+) -> FactorCorrelationReport:
+    if req.period[0] >= req.period[1]:
+        raise HTTPException(status_code=400, detail="period 必须满足 start < end")
+    spec = ResearchSpec(
+        type="factor_test",
+        universe=req.universe,
+        price_adjustment=req.price_adjustment,
+        factors=req.factors,
+        selection=SelectionSpec(top_n=10),
+        rebalance="monthly",
+        period=req.period,
+    )
+    try:
+        return service.run_factor_correlation(spec)
+    except (ValueError, FactorError) as exc:
+        raise HTTPException(status_code=400, detail=str(exc)) from exc