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AlphaEvolve-style Evolutionary Formula Discovery for Zeros
=============================================================
Inspired by arXiv:2511.02864 (AlphaEvolve, Terence Tao et al.)
Key idea: Use an evolutionary coding agent to discover empirical formulas
for zeta zero statistics. Instead of searching in proof space, we search
in CODE space — Python expressions that approximate observed data.
Fitness function: agreement with empirical zero statistics (pair correlation,
spacing distribution, etc.).
This is the "AlphaFold for math" approach: end-to-end search for formulas
that match data, then verify if they generalize.
"""
import numpy as np
from typing import Dict, List, Callable
import random
import math
class FormulaGene:
"""A candidate formula as a Python expression string."""
def __init__(self, expression: str, fitness: float = None):
self.expression = expression
self.fitness = fitness
def __repr__(self):
return f"Gene({self.expression}, fitness={self.fitness})"
class AlphaFoldMathEvolver:
"""
Evolutionary search for formulas matching zero statistics.
Target: Find a closed-form formula f(n) that approximates the n-th
zero spacing s_n = γ_{n+1} - γ_n.
Expression grammar:
- Variables: n, log(n), sqrt(n)
- Constants: pi, e, numbers 1-10
- Operations: +, -, *, /, exp, log, sqrt, sin, cos
Fitness: L2 distance between f(n) and actual normalized spacings.
"""
def __init__(self, zeros: List[float]):
self.zeros = np.array(zeros)
self.spacings = np.diff(self.zeros)
self.normalized_spacings = self.spacings / np.mean(self.spacings)
self.results = {}
def _random_expression(self, depth: int = 0, max_depth: int = 4) -> str:
"""Generate a random mathematical expression."""
if depth >= max_depth or random.random() < 0.3:
# Terminal
terminals = ['n', 'math.log(n+1)', 'math.sqrt(n)',
'math.pi', 'math.e', '1', '2', '0.5']
return random.choice(terminals)
# Non-terminal
ops = ['+', '-', '*', '/']
funcs = ['math.exp', 'math.log', 'math.sqrt', 'math.sin', 'math.cos']
if random.random() < 0.5:
op = random.choice(ops)
left = self._random_expression(depth + 1, max_depth)
right = self._random_expression(depth + 1, max_depth)
# Protect against division by zero
if op == '/':
return f"({left}) / ({right} + 0.001)"
return f"({left}) {op} ({right})"
else:
func = random.choice(funcs)
arg = self._random_expression(depth + 1, max_depth)
# Protect log domain
if func == 'math.log':
return f"{func}(abs({arg}) + 0.001)"
return f"{func}({arg})"
def _evaluate_expression(self, expr: str, n_vals: np.ndarray) -> np.ndarray:
"""Safely evaluate expression for array of n values."""
try:
# Create safe namespace
namespace = {'n': n_vals, 'math': math, 'np': np}
result = eval(expr, {"__builtins__": {}}, namespace)
result = np.array(result, dtype=float)
# Ensure 1D array of correct length
if result.ndim == 0:
result = np.full(len(n_vals), float(result))
# Filter invalid values
result = np.where(np.isfinite(result), result, 0)
return result
except Exception:
return np.zeros(len(n_vals))
def _fitness(self, expr: str, sample_size: int = 1000) -> float:
"""Compute fitness: negative MSE between formula and actual spacings."""
n_vals = np.arange(1, min(sample_size + 1, len(self.normalized_spacings)))
predicted = self._evaluate_expression(expr, n_vals)
actual = self.normalized_spacings[:len(n_vals)]
if len(predicted) != len(actual):
return -1e10
# Normalize predicted to match actual scale
if np.std(predicted) > 0:
predicted = (predicted - np.mean(predicted)) / np.std(predicted)
predicted = predicted * np.std(actual) + np.mean(actual)
mse = np.mean((predicted - actual) ** 2)
# Also reward simplicity (shorter expressions)
complexity_penalty = len(expr) * 0.0001
return -mse - complexity_penalty
def _mutate(self, expr: str) -> str:
"""Randomly mutate an expression."""
mutations = [
lambda e: e + f" + {random.choice(['1', '0.1', 'n*0.01'])}",
lambda e: f"math.sin({e})" if 'sin' not in e else e,
lambda e: f"math.log(abs({e}) + 0.001)" if 'log' not in e else e,
lambda e: f"({e}) * {random.choice(['0.9', '1.1', '2'])}",
lambda e: self._random_expression(depth=1, max_depth=3),
]
return random.choice(mutations)(expr)
def _crossover(self, expr1: str, expr2: str) -> str:
"""Combine two expressions."""
if random.random() < 0.5:
return f"({expr1}) * 0.5 + ({expr2}) * 0.5"
return f"({expr1}) / (abs({expr2}) + 0.001)"
def evolve(self, population_size: int = 50, generations: int = 30,
sample_size: int = 500) -> Dict:
"""Run evolutionary search."""
print(f" [AlphaFoldMath] Evolving formulas for {sample_size} spacings...")
# Initialize population
population = [FormulaGene(self._random_expression()) for _ in range(population_size)]
best_fitness_history = []
for gen in range(generations):
# Evaluate fitness
for gene in population:
if gene.fitness is None:
gene.fitness = self._fitness(gene.expression, sample_size)
# Sort by fitness
population.sort(key=lambda g: g.fitness, reverse=True)
best_fitness_history.append(population[0].fitness)
if gen % 10 == 0:
print(f" Gen {gen}: best fitness = {population[0].fitness:.6f}")
# Selection: keep top 20%
elite_count = max(1, population_size // 5)
new_population = population[:elite_count]
# Generate offspring
while len(new_population) < population_size:
if random.random() < 0.7 and len(population) >= 2:
# Crossover
p1, p2 = random.sample(population[:elite_count*2], 2)
child_expr = self._crossover(p1.expression, p2.expression)
else:
# Mutation
parent = random.choice(population[:elite_count*2])
child_expr = self._mutate(parent.expression)
new_population.append(FormulaGene(child_expr))
population = new_population
# Final evaluation
for gene in population:
if gene.fitness is None:
gene.fitness = self._fitness(gene.expression, sample_size)
population.sort(key=lambda g: g.fitness, reverse=True)
best = population[0]
n_vals = np.arange(1, min(sample_size + 1, len(self.normalized_spacings)))
predicted = self._evaluate_expression(best.expression, n_vals)
self.results = {
'strategy': 'alphafold_math_evolutionary_formula',
'population_size': population_size,
'generations': generations,
'best_expression': best.expression,
'best_fitness': float(best.fitness),
'fitness_history': [float(f) for f in best_fitness_history],
'predicted_vs_actual': {
'predicted_mean': float(np.mean(predicted)),
'actual_mean': float(np.mean(self.normalized_spacings[:sample_size])),
'predicted_std': float(np.std(predicted)),
'actual_std': float(np.std(self.normalized_spacings[:sample_size])),
},
'interpretation': "Negative fitness = -MSE. Higher = better agreement.",
}
return self.results
def summary(self) -> str:
r = self.results
s = f"AlphaFold-Math Formula Evolver\n{'='*50}\n"
s += f"Best formula: {r['best_expression']}\n"
s += f"Best fitness: {r['best_fitness']:.6f}\n"
pv = r['predicted_vs_actual']
s += f"Predicted mean={pv['predicted_mean']:.4f} vs actual={pv['actual_mean']:.4f}\n"
s += f"Predicted std={pv['predicted_std']:.4f} vs actual={pv['actual_std']:.4f}\n"
s += f"Note: This is an empirical formula discovered by evolution, not proven.\n"
return s
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