Verify the appendix bound against its stated constant

The check tested f(gamma) >= gamma/(gamma+1), but the manuscript claims
f(gamma) >= c1*gamma/(gamma+1) with c1 >= 0.97. It now tests the stated
constant and records the achieved one, 0.9711.

experiments/verification/math_verify.py

@@ -109,14 +109,15 @@ out["prop3_blind_eq_ofdma"] = {"max_abs_cos_diff": float(np.abs(cb - co).max())}
 #     h ~ Rayleigh(1/sqrt2), sigma^2 = 1/gamma
 gams = 10 ** (np.arange(-10, 21, 2) / 10)
 h = rng.rayleigh(scale=np.sqrt(0.5), size=1_000_000)
-ok, worst = True, 0.0
+C1 = 0.97
+ok, ratio = True, np.inf
 for g in gams:
     f = float((h / np.sqrt(h ** 2 + 1 / g)).mean())
-    bound = g / (g + 1)
-    ok &= f >= bound
-    worst = max(worst, bound - f)
-out["appendix_f_bound"] = {"holds_on_grid": bool(ok),
-                           "worst_violation": worst}
+    ok &= f >= C1 * g / (g + 1)
+    ratio = min(ratio, f / (g / (g + 1)))
+out["appendix_f_bound"] = {"claimed_c1": C1,
+                           "holds_on_grid": bool(ok),
+                           "achieved_c1": float(ratio)}
 
 # (6) beta*exp(x) >= beta*x  =>  beta e^{eta' beta gamma} >= eta' beta^2 gamma
 xs = rng.uniform(0, 20, 10000)
@@ -129,5 +130,5 @@ out["concentration_residual"] = {"d64_residual": 1 / np.sqrt(64),
                                  "claimed_max": 0.13}
 
 print(json.dumps(out, indent=1))
-with open("math_verify.json", "w") as f:
+with open("data/math_verify.json", "w") as f:
     json.dump(out, f, indent=1)