diff --git a/code/verify_math.py b/code/verify_math.py
index 7d854df..aec8737 100644
--- a/code/verify_math.py
+++ b/code/verify_math.py
@@ -289,6 +289,30 @@ def v8_cross_period_terms():
     return ok
 
 
+def v9_score_variance_ratio():
+    """Per-digit score-variance ratio of Proposition 1's proof: the mean
+    of sum_j e_j^4 / sum_j e_j^2 e'_j^2 over ordered codeword pairs of
+    the trained unit codebook, quoted as 2.8 in the manuscript."""
+    import torch
+    from exp_full import main_model
+    m = main_model()
+    B = m.unit_codebook().detach().cpu().double()
+    B = B / B.norm(dim=1, keepdim=True)
+    n = B.shape[0]
+    num = (B ** 4).sum(1)
+    ratios = []
+    for i in range(n):
+        for j in range(n):
+            if i != j:
+                ratios.append(float(num[i] / (B[i] ** 2 * B[j] ** 2).sum()))
+    mean = sum(ratios) / len(ratios)
+    ok = abs(mean - 2.8) < 0.05
+    print("V9 per-digit score-variance ratio: %.3f (quoted 2.8)" % mean)
+    ROWS.append(("V9 score-variance ratio", "2.8", "%.4f" % mean,
+                 "%.4f" % abs(mean - 2.8), "0.05", "PASS" if ok else "FAIL"))
+    return ok
+
+
 def main():
     print(f"config d={D} U={U} V={V}\n")
     results = {
@@ -300,6 +324,7 @@ def main():
         "V6": v6_coded_oma_outage(),
         "V7": v7_symbolic_identities(),
         "V8": v8_cross_period_terms(),
+        "V9": v9_score_variance_ratio(),
     }
     print("\nsummary:", {k: ("PASS" if v else "FAIL") for k, v in results.items()})
     print("ALL PASS" if all(results.values()) else "SOME FAILED")
diff --git a/data/verify_math.csv b/data/verify_math.csv
index a7267e8..e300001 100644
--- a/data/verify_math.csv
+++ b/data/verify_math.csv
@@ -5,10 +5,11 @@ V2b eve SER @ 0dB,0.99609375,0.995,0.0010937500000000044,0.015,PASS
 V2b eve SER @ 10dB,0.99609375,0.9888333333333333,0.007260416666666658,0.015,PASS
 V2b eve SER @ 20dB,0.99609375,0.9881666666666666,0.007927083333333362,0.015,PASS
 V2b eve SER @ 80dB,0.99609375,0.9896666666666667,0.0064270833333333055,0.015,PASS
-V3b random mask E|corr|,0.09973557010035818,0.10187042771408686,0.002134857613728683,0.009973557010035819,PASS
+V3b random mask E|corr|,0.09973557010035818,0.10187042771408686,0.002134857613728683,0.002992067103010745,PASS
 V4a blind jammer projection mean,0.0,0.00026396107284673695,0.00026396107284673695,0.003,PASS
-V4b blind jammer projection variance,0.01558576233568703,0.015476787953278994,0.00010897438240803532,0.0007792881167843516,PASS
+V4b blind jammer projection variance,0.01558576233568703,0.015476787953278994,0.00010897438240803532,0.0001558576233568703,PASS
 V5 matched bias over blind RMS,8.0,8.0,0.04,1.0,PASS
 V6 coded-OMA outage @ 10 dB,0.0406,0.040575,0,0,REFERENCE
 V7 symbolic identities,exact,exact,0,0,PASS
-V8 cross-period remainder,0.0,0.000337,0.000337,0.01,PASS
+V8 cross-period remainder,0.0,0.000337,0.000337,0.0005,PASS
+V9 score-variance ratio,2.8,2.8252,0.0252,0.05,PASS