Upload problem_solvers/lehmer_phenomena.py

problem_solvers/lehmer_phenomena.py

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+"""
+PROBLEM 8: Zero Spacing Records — Lehmer Phenomena
+====================================================
+Unusually close zeros are called "Lehmer phenomena".
+OPEN: characterize the distribution of closest pairs among large zeros.
+"""
+
+import numpy as np
+from typing import Dict
+
+
+class LehmerPhenomenaAnalyzer:
+    def __init__(self, zeros: list):
+        self.zeros = np.array(zeros)
+        self.results = {}
+
+    def analyze(self) -> Dict:
+        spacings = np.diff(self.zeros)
+        # Normalize by local mean spacing
+        window = 50
+        normalized = np.zeros(len(spacings))
+        for i in range(len(spacings)):
+            start = max(0, i - window // 2)
+            end = min(len(spacings), i + window // 2)
+            local_mean = np.mean(spacings[start:end])
+            normalized[i] = spacings[i] / local_mean if local_mean > 0 else 1.0
+
+        # Find minimum spacings
+        min_spacing = float(np.min(normalized))
+        min_idx = int(np.argmin(normalized))
+
+        # Small spacing distribution
+        small_threshold = 0.3
+        n_small = int(np.sum(normalized < small_threshold))
+
+        # GUE prediction for small spacings: P(s) ~ (πs)²/2 for small s
+        # Wigner surmise: p(s) = (π/2)s exp(-πs²/4)
+        s_bins = np.linspace(0, 3, 60)
+        hist, _ = np.histogram(normalized, bins=s_bins, density=True)
+        wigner = (np.pi / 2) * s_bins[:-1] * np.exp(-np.pi * s_bins[:-1] ** 2 / 4)
+
+        self.results = {
+            'n_zeros': len(self.zeros),
+            'min_normalized_spacing': min_spacing,
+            'min_spacing_index': min_idx,
+            'min_spacing_gamma': float(self.zeros[min_idx]),
+            'mean_spacing': float(np.mean(spacings)),
+            'std_spacing': float(np.std(spacings)),
+            'n_small_spacings': n_small,
+            'fraction_small': float(n_small / len(normalized)),
+            'spacing_histogram': hist.tolist(),
+            'wigner_prediction': wigner.tolist(),
+            'bin_centers': ((s_bins[:-1] + s_bins[1:]) / 2).tolist(),
+            'top_10_smallest': [float(v) for v in np.sort(normalized)[:10]],
+        }
+        return self.results
+
+    def summary(self) -> str:
+        r = self.results
+        s = f"Lehmer Phenomena Analysis\n{'='*50}\n"
+        s += f"Zeros: {r['n_zeros']:,}\n"
+        s += f"Min normalized spacing: {r['min_normalized_spacing']:.6f}\n"
+        s += f"At γ ≈ {r['min_spacing_gamma']:.2f} (index {r['min_spacing_index']})\n"
+        s += f"Spacings < 0.3: {r['n_small_spacings']} ({r['fraction_small']:.4%})\n"
+        s += f"Top 10 smallest: {r['top_10_smallest'][:5]}\n"
+        return s