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Explicit Formula Engine (from v3, enhanced for v_mix)
Uses the von Mangoldt explicit formula with 100,000 zeros.
"""
import numpy as np
from typing import List, Dict
from dataclasses import dataclass
@dataclass
class PrimePrediction:
predicted_value: int
confidence: float
actual_is_prime: bool = False
error: int = 0
method: str = ""
class ExplicitFormulaEngine:
def __init__(self, zeros: List[float]):
self.zeros = np.array(zeros, dtype=np.float64)
self._log = []
def log(self, msg: str):
self._log.append(msg)
print(f" [ExplicitFormula] {msg}")
def compute_psi(self, x: float, n_zeros: int = None) -> float:
if n_zeros is None:
n_zeros = len(self.zeros)
n_zeros = min(n_zeros, len(self.zeros))
if x <= 1:
return 0.0
log_x = np.log(x)
sqrt_x = np.sqrt(x)
psi = x
gamma_subset = self.zeros[:n_zeros]
denom = 0.25 + gamma_subset ** 2
cos_term = np.cos(gamma_subset * log_x) * 0.5
sin_term = np.sin(gamma_subset * log_x) * gamma_subset
psi -= np.sum(2 * sqrt_x * (cos_term + sin_term) / denom)
psi -= np.log(2 * np.pi)
if x > 1.01:
psi -= 0.5 * np.log(1 - 1 / (x * x))
return psi
def compute_psi_vectorized(self, x_values: np.ndarray, n_zeros: int = None) -> np.ndarray:
if n_zeros is None:
n_zeros = min(10000, len(self.zeros))
n_zeros = min(n_zeros, len(self.zeros))
x_values = np.asarray(x_values, dtype=np.float64)
log_x = np.log(np.maximum(x_values, 1.01))
sqrt_x = np.sqrt(np.maximum(x_values, 0))
psi = x_values.copy()
gamma_arr = self.zeros[:n_zeros]
for gamma in gamma_arr:
denom = 0.25 + gamma * gamma
cos_terms = np.cos(gamma * log_x) * 0.5
sin_terms = np.sin(gamma * log_x) * gamma
psi -= 2 * sqrt_x * (cos_terms + sin_terms) / denom
psi -= np.log(2 * np.pi)
mask = x_values > 1.01
psi[mask] -= 0.5 * np.log(1 - 1 / (x_values[mask] ** 2))
return psi
def count_predicted_primes(self, x_max: int, n_zeros: int = None) -> int:
if n_zeros is None:
n_zeros = min(5000, len(self.zeros))
x_range = np.arange(2, x_max + 1, dtype=np.float64)
psi_prev = self.compute_psi_vectorized(x_range - 0.5, n_zeros=n_zeros)
psi_next = self.compute_psi_vectorized(x_range + 0.5, n_zeros=n_zeros)
jumps = psi_next - psi_prev
expected = np.log(x_range)
predicted = np.sum(jumps > expected * 0.5)
return int(predicted)
def minimum_zeros_for_primes(self, x_max: int, max_zeros: int = 100000) -> Dict:
sieve = np.ones(x_max + 1, dtype=bool)
sieve[:2] = False
for i in range(2, int(x_max ** 0.5) + 1):
if sieve[i]:
sieve[i * i::i] = False
actual_primes = np.sum(sieve)
N_tests = [10, 50, 100, 200, 500, 1000, 2000, 5000,
10000, 20000, 50000, 100000]
N_tests = [n for n in N_tests if n <= max_zeros]
results = []
for N in N_tests:
predicted = self.count_predicted_primes(x_max, n_zeros=N)
results.append({
'N': N,
'predicted_primes': predicted,
'actual_primes': int(actual_primes),
'accuracy': predicted / actual_primes if actual_primes > 0 else 0,
'correct': predicted == actual_primes,
})
if predicted == actual_primes:
break
return {
'x_max': x_max,
'results': results,
'minimum_N': next((r['N'] for r in results if r['correct']), None),
}
def prime_oscillation_contributions(self, x: float, n_zeros: int = 1000) -> Dict:
n_zeros = min(n_zeros, len(self.zeros))
log_x = np.log(x)
sqrt_x = np.sqrt(x)
contributions = []
cumulative = 0.0
for i in range(n_zeros):
gamma = self.zeros[i]
denom = 0.25 + gamma * gamma
cos_term = np.cos(gamma * log_x) * 0.5
sin_term = np.sin(gamma * log_x) * gamma
actual = -2 * sqrt_x * (cos_term + sin_term) / denom
cumulative += actual
contributions.append({
'index': i + 1,
'gamma': gamma,
'contribution': float(actual),
'cumulative': float(cumulative),
})
return {
'x': x,
'total_contribution': float(cumulative),
'contributions': contributions,
}
def get_log(self):
return self._log
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