"""Experiment 2 -- how big is an AUC of 0.5105, really?

Paper 002 reports walk-forward AUC 0.5105 over 11,497 windows and calls it no
skill. That claim is only honest if 0.5105 is inside the range pure noise
produces at that sample size. This measures the null distribution directly
rather than asserting it: labels are a fair coin, scores are independent noise,
so the true AUC is exactly 0.5 by construction.
"""
import numpy as np

N, TRIALS = 11_497, 20_000
OBSERVED = 0.5105

def auc(scores, labels):
    # Mann-Whitney U via ranks, ties averaged. Written out so no library
    # convention is assumed -- the same reason the paper implements its own.
    order = np.argsort(scores, kind="mergesort")
    s = scores[order]
    ranks = np.empty(len(s), float)
    i = 0
    while i < len(s):
        j = i
        while j + 1 < len(s) and s[j + 1] == s[i]:
            j += 1
        ranks[i:j + 1] = 0.5 * (i + j) + 1.0
        i = j + 1
    lab = labels[order]
    n1 = lab.sum(); n0 = len(lab) - n1
    return (ranks[lab == 1].sum() - n1 * (n1 + 1) / 2) / (n1 * n0)

rng = np.random.default_rng(0)
vals = np.empty(TRIALS)
for t in range(TRIALS):
    lab = rng.integers(0, 2, N)
    vals[t] = auc(rng.normal(size=N), lab)

sd = vals.std(ddof=1)
p = float((np.abs(vals - 0.5) >= abs(OBSERVED - 0.5)).mean())
print(f"trials                     : {TRIALS}")
print(f"samples per trial          : {N}")
print(f"null AUC mean              : {vals.mean():.5f}")
print(f"null AUC std dev           : {sd:.5f}")
print(f"null 95% range             : {np.quantile(vals,0.025):.4f} - {np.quantile(vals,0.975):.4f}")
print(f"largest AUC pure noise gave: {vals.max():.4f}")
print(f"observed 0.5105 is         : {abs(OBSERVED-0.5)/sd:.2f} sd from chance")
print(f"two-sided p                : {p:.4f}")