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Verify the appendix bound against its stated constant

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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.
This commit is contained in:
KiHoLee
2026-08-26 13:53:55 +09:00
parent 48f12dea74
commit 81c2be1d78
2 changed files with 11 additions and 9 deletions
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experiments/verification/math_verify.json
+3 -2
View File
@@ -28,8 +28,9 @@
"max_abs_cos_diff": 4.440892098500626e-16 "max_abs_cos_diff": 4.440892098500626e-16
}, },
"appendix_f_bound": { "appendix_f_bound": {
"holds_on_grid": false, "claimed_c1": 0.97,
"worst_violation": 0.027053603772970947 "holds_on_grid": true,
"achieved_c1": 0.9710759305294567
}, },
"appendix_exp_ineq": { "appendix_exp_ineq": {
"holds": true "holds": true
experiments/verification/math_verify.py
+8 -7
View File
@@ -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 # h ~ Rayleigh(1/sqrt2), sigma^2 = 1/gamma
gams = 10 ** (np.arange(-10, 21, 2) / 10) gams = 10 ** (np.arange(-10, 21, 2) / 10)
h = rng.rayleigh(scale=np.sqrt(0.5), size=1_000_000) 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: for g in gams:
f = float((h / np.sqrt(h ** 2 + 1 / g)).mean()) f = float((h / np.sqrt(h ** 2 + 1 / g)).mean())
bound = g / (g + 1) ok &= f >= C1 * g / (g + 1)
ok &= f >= bound ratio = min(ratio, f / (g / (g + 1)))
worst = max(worst, bound - f) out["appendix_f_bound"] = {"claimed_c1": C1,
out["appendix_f_bound"] = {"holds_on_grid": bool(ok), "holds_on_grid": bool(ok),
"worst_violation": worst} "achieved_c1": float(ratio)}
# (6) beta*exp(x) >= beta*x => beta e^{eta' beta gamma} >= eta' beta^2 gamma # (6) beta*exp(x) >= beta*x => beta e^{eta' beta gamma} >= eta' beta^2 gamma
xs = rng.uniform(0, 20, 10000) xs = rng.uniform(0, 20, 10000)
@@ -129,5 +130,5 @@ out["concentration_residual"] = {"d64_residual": 1 / np.sqrt(64),
"claimed_max": 0.13} "claimed_max": 0.13}
print(json.dumps(out, indent=1)) 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) json.dump(out, f, indent=1)