The Prime Walk: Spontaneous Symmetry Breaking from Number Theory to the Riemann Hypothesis

mindmap
  root((Prime Walk and Jahn Teller))
    Core Framework
      Prime Walk Process
        Random Integer Generation
        Closest Prime Power Factor
        Step on Number Line
      Symmetry Breaking
        Origin as Breaking Point
        Spontaneous Divergence
        Double_Logarithmic Behavior
    Physical Analogy
      Jahn Teller Effect
        Molecular Distortion
        Degenerate Electronic States
        Stability Through Symmetry Breaking
      Number Theory Connection
        Prime Powers as Electronic States
        Zeros as Stable Configurations
        Critical Line as Most Stable
    Mathematical Structure
      Markov Chain Dynamics
        State Space Definition
        Transition Probabilities
        Ergodic Properties
      Riemann Hypothesis Link
        Functional Equation Symmetry
        Zeta Function Pole
        Analytic Continuation
    Computational Results
      Ergodic Exploration
        All Prime Powers Reached
        Convergence Patterns
        Statistical Analysis
      Physical Intuition
        Stability Arguments
        Energy Landscape
        Symmetry Preservation

Video Presentation

Introduction

The relationship between symmetry breaking in physical systems and fundamental mathematical structures represents one of the deepest connections between physics and mathematics. This paper introduces a novel framework that bridges these domains through what we call the “prime walk” – a deceptively simple stochastic process that reveals profound connections between molecular chemistry’s Jahn-Teller effect and number theory’s most enduring mystery, the Riemann Hypothesis.

The Jahn-Teller effect, discovered in 1937 by Hermann Jahn and Edward Teller, states that any nonlinear molecule with a degenerate electronic ground state in a highly symmetric arrangement will spontaneously distort to lower its energy and break that symmetry. This fundamental principle governs molecular stability across chemistry and materials science.

Our investigation reveals that an analogous spontaneous symmetry breaking occurs in elementary number theory through a random walk on prime powers, with implications that extend to the distribution of Riemann zeta function zeros.

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Chapter 1: The Prime Walk Framework

1.1 Fundamental Rules

The prime walk operates on an infinite number line, which we metaphorically call the “infinite hoops” of arithmetic space. The process begins at the origin \(x_0 = 0\) and proceeds according to these rules:

1. Random Selection: Generate a random natural number \(n\) uniformly from a large range (or eventually from all natural numbers)
2. Prime Power Factorization: Find all prime powers that divide \(n\): \(\{p_1^{k_1}, p_2^{k_2}, \ldots\}\)
3. Distance Minimization: From current position \(x_t\), jump to the prime power \(p^{k}\) that minimizes \(|x_t - p^k|\)

Key Innovation: Unlike traditional factorization that considers only distinct primes, the walk uses all prime powers, creating a richer dynamics.

Example: If \(n = 12 = 2^2 \cdot 3\), the available destinations are \(\{2, 3, 4\}\).

1.2 The Symmetry Breaking at Origin

The critical insight emerges when we allow negative prime powers, creating destinations \(\{\pm 2, \pm 3, \pm 4, \ldots\}\). From \(x_0 = 0\), the positions \(+2\) and \(-2\) are equidistant, creating a degenerate state with perfect reflection symmetry \(x \leftrightarrow -x\).

This mathematical degeneracy mirrors the electronic degeneracy in Jahn-Teller unstable molecules. Both represent systems balanced on a knife edge, waiting for perturbation to force a choice.

1.3 The Jahn-Teller Analogy

The mapping between molecular physics and number theory is remarkably precise:

Molecular System	Prime Walk System
Symmetric geometry	Origin \(x_0 = 0\)
Degenerate electronic states	Equidistant prime powers \(\pm p^k\)
Molecular vibrations	Random number generation
Energy minimization	Distance minimization
Geometric distortion	First step selection
Symmetry breaking	Choice of \(+p^k\) over \(-p^k\)

The correspondence is isomorphic – the minimization of electronic energy in molecules maps directly onto the minimization of distance in the prime walk.

Jahn-Teller symmetry breaking in the prime walk

Figure 1: Spontaneous symmetry breaking at the origin. The initial state has perfect reflection symmetry, which is broken when the first step chooses between equidistant prime power options, analogous to molecular distortion in the Jahn-Teller effect.

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Chapter 2: Markov Chain Dynamics and Ergodicity

2.1 State Space and Transitions

For analysis, we first consider a bounded version with upper bound \(N\). The state space consists of all prime powers \(\{p_1, p_2^2, p_3, \ldots, p_k\}\) where \(p_k \leq N\).

The transition probability from state \(p_i\) to state \(p_j\) is determined by whether \(p_j\) divides the randomly generated number and is closest to \(p_i\).

2.2 Fundamental Probability Structure

A crucial number-theoretic result governs the dynamics: for a random integer \(n\) chosen uniformly from \([1, M]\) with \(M\) large:

\[\Pr(p^k \text{ divides } n) \approx \frac{1}{p^k}\]

This simple relationship drives the entire behavior of the walk. Small prime powers have high transition probabilities, while large prime powers are rarely accessible.

2.3 Irreducibility and Ergodicity

For any finite bound \(N\), the prime walk exhibits two fundamental properties:

Theorem 1 (Irreducibility): The walk can reach any prime power state from any other state through some sequence of transitions.

Proof: The complementary probability \(\Pr(p^k \nmid n) = 1 - \frac{1}{p^k}\) is always strictly less than 1, ensuring every state is eventually accessible.

Theorem 2 (Ergodicity): An irreducible Markov chain on a finite state space is always ergodic.

Consequences: - Every prime power is visited infinitely often in infinite time - The walk explores the entire arithmetic landscape - Long-term behavior is characterized by a unique stationary distribution

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Chapter 3: Stationary Distribution and Heavy-Tailed Behavior

3.1 Probability Concentration

The stationary distribution \(\pi(p^k)\) is proportional to \(\frac{1}{p^k}\). This creates extreme concentration on small prime powers:

• \(\pi(2) \approx 22.1\%\) (dominant state)
• \(\pi(3) \approx 14.9\%\)
• \(\pi(4) \approx 9.9\%\)
• \(\pi(5) \approx 6.6\%\)

The walk spends the vast majority of time near the origin, clustering around these low-cost arithmetic structures.

3.2 The Paradox of Moments

For \(N = 1000\), simulation reveals: - Mean position: \(\mathbb{E}[X] \approx 120.45\) - Standard deviation: \(\sigma_X \approx 249.0\)

The standard deviation being more than double the mean indicates a distribution with extreme heavy tails. This paradox resolves when we understand the dynamics: the walk mostly hovers near small prime powers but occasionally makes enormous leaps to large prime powers like \(p^k = 961\).

These rare but dramatic jumps inflate both the mean and especially the variance, creating a picture of localized chaos.

class PrimeWalk:
    def __init__(self, max_prime=100):
        self.primes = self._generate_primes(max_prime)
        self.prime_powers = self._generate_prime_powers()

    def _generate_primes(self, n):
        sieve = np.ones(n+1, dtype=bool)
        sieve[0:2] = False
        for i in range(2, int(n**0.5) + 1):
            if sieve[i]:
                sieve[i*i::i] = False
        return [i for i in range(2, n+1) if sieve[i]]

    def _generate_prime_powers(self):
        powers = []
        for p in self.primes:
            power = p
            while power <= self.max_prime:
                powers.append(power)
                power *= p
        return sorted(list(set(powers)))

    def step(self, current_pos, n):
        factors = set()
        for p in self.primes:
            if p * p > n:
                break
            power = p
            while n % power == 0:
                factors.add(power)
                power *= p
        if n > 1:
            factors.add(n)

        if not factors:
            return current_pos

        distances = [abs(current_pos - pp) for pp in factors]
        min_dist_idx = np.argmin(distances)
        return list(factors)[min_dist_idx]

    def simulate(self, num_steps):
        positions = [0]
        current_pos = 0
        for _ in range(num_steps):
            n = np.random.randint(2, 1000)
            current_pos = self.step(current_pos, n)
            positions.append(current_pos)
        return positions

# Example simulation
walk = PrimeWalk()
np.random.seed(42)
positions = walk.simulate(100)
print(f"Sample positions: {positions[:10]}")
print(f"Final position: {positions[-1]}")
print(f"Mean position: {np.mean(positions):.2f}")
print(f"Std deviation: {np.std(positions):.2f}")

Prime walk trajectory showing localized chaos

Figure 2: Prime walk trajectory demonstrating localized chaos with rare large jumps. The walk mostly stays near small prime powers but occasionally makes dramatic leaps to larger values.

3.3 Double Logarithmic Divergence

The true revelation comes in the unbounded case. The total probability mass is:

\[Z = \sum_{p^k} \frac{1}{p^k} = \sum_p \sum_{k=1}^{\infty} \frac{1}{p^k} = \sum_p \frac{1/p}{1 - 1/p} = \sum_p \frac{1}{p-1}\]

This sum diverges, following the well-known divergence of \(\sum_p \frac{1}{p}\). The growth is double logarithmic:

\[Z(N) \sim \ln(\ln N) + 1.034\ldots\]

This incredibly slow divergence has profound implications for the infinite walk.

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Chapter 4: The Infinite Walk and Null Recurrence

4.1 Properties of the Unbounded System

With the state space extended to all prime powers, we obtain:

Theorem 3 (Null Recurrence): The infinite prime walk visits every prime power infinitely often but with infinite expected return time.

This follows directly from the double-logarithmic divergence. The walk is ergodic but cannot settle into a stationary distribution.

4.2 The Harmonic Crawl

The divergence rate is extraordinarily slow. To reach a modest total probability mass of 10 requires:

\[\ln(\ln N) + 1.034 = 10 \implies \ln N \approx e^{8.966} \implies N \approx e^{7.937 \times 10^3}\]

This number (\(\approx 10^{3448}\)) exceeds the estimated number of atoms in the observable universe (\(\approx 10^{80}\)).

4.3 Connection to the Riemann Zeta Function

The divergence of \(Z\) directly relates to the pole of the Riemann zeta function at \(s=1\). Recall the Euler product:

\[\zeta(s) = \prod_p \frac{1}{1 - p^{-s}} = \sum_{n=1}^{\infty} \frac{1}{n^s}\]

At \(s=1\), we have the harmonic series divergence. Our \(Z\) captures the prime power contribution to this divergence. The walk’s tendency to drift infinitely reflects zeta’s pole at s=1.

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Chapter 5: Jahn-Teller Effect and the Riemann Hypothesis

5.1 Symmetry in the Zeta Function

The Riemann zeta function possesses a fundamental symmetry encoded in its functional equation:

\[\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)\]

This equation enforces reflection symmetry across the critical line \(\text{Re}(s) = \frac{1}{2}\).

5.2 Symmetry Breaking via Primes

The critical line represents a degenerate state – the equivalent of \(x_0 = 0\) in our prime walk. The primes act as perturbations that destabilize this symmetry.

From the logarithm of the Euler product:

\[\log \zeta(s) = -\sum_p \log(1 - p^{-s}) = \sum_p \sum_{k=1}^{\infty} \frac{p^{-ks}}{k}\]

The primes introduce the perturbations that force the system to find stable configurations.

5.3 The Jahn-Teller Principle Applied

The Jahn-Teller effect demands that when a highly symmetric system is perturbed, it must distort to find a stable configuration. In the context of \(\zeta(s)\):

• High symmetry: Reflection across \(\text{Re}(s) = \frac{1}{2}\)
• Perturbation: Prime number distribution via Euler product
• Stable configurations: Points where \(\zeta(s) = 0\)
• Distortion: Formation of non-trivial zeros

5.4 Residual Symmetry and the Critical Line

Crucially, the Jahn-Teller effect often preserves residual symmetry rather than complete asymmetry. The functional equation’s reflection symmetry \(\zeta(s) \leftrightarrow \zeta(1-s)\) cannot be broken by primes.

The only way for zeros to respect this mandatory residual symmetry is to lie on the critical line itself. If a zero were at \(\sigma + it\) with \(\sigma \neq \frac{1}{2}\), the functional equation would force a paired zero at \(1-\sigma + it\), representing a less economical symmetry breaking.

5.5 Stability Interpretation of the Riemann Hypothesis

Riemann Hypothesis (Jahn-Teller Interpretation): All non-trivial zeros lie on the critical line because this represents the most stable configuration that simultaneously: 1. Lifts the degeneracy (creates zeros off the real axis) 2. Preserves the required reflection symmetry 3. Minimizes the “energy” in the complex plane

The specific imaginary parts of the zeros (14.1347…, 21.0220…, 25.0108…) represent the spectrum of stable distortion modes determined by the prime number distribution.

Riemann Hypothesis connection to Jahn-Teller effect

Figure 3: The Jahn-Teller perspective on the Riemann Hypothesis. The critical line (red) represents the residual symmetry that must be preserved. Non-trivial zeros (black dots) align on the critical line because this represents the most stable configuration that respects the functional equation’s reflection symmetry while lifting the degeneracy.

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Chapter 6: Recent Developments and Borel-Jessen Limit Theorems

6.1 Probabilistic Limit Theorems

Recent 2025 research on Borel-Jessen limit theorems provides additional support for this framework. These theorems describe the value distribution of zeta functions and their generalizations, showing:

• Chaotic behavior follows specific probability distributions
• Phase transitions occur around symmetry-breaking thresholds
• Half-center symmetry related to the critical line is preserved probabilistically

6.2 Partial Products and Symmetry

The research focuses on partial products of the Euler product:

\[\prod_{p \leq x} \left(1 - \frac{1}{p^s}\right)^{-1}\]

These finite approximations exhibit: 1. Local chaos and apparent symmetry breaking 2. Global statistical constraints that preserve fundamental symmetries 3. Phase transitions that mirror molecular symmetry breaking

6.3 Synthesis of Framework Elements

The Borel-Jessen results synthesize our framework: - Local symmetry breaking: Prime walk’s first step from origin - Infinite exploration: Null recurrent dynamics and double-logarithmic divergence - Statistical symmetry preservation: Constrained chaos respecting functional equation

Stability emerges not from static equilibrium but from dynamic equilibrium that honors underlying symmetries.

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Chapter 7: Implications and Future Directions

7.1 Philosophical Implications

The prime walk framework suggests that stability in fundamental systems might be a long-term approximation. True long-term behavior could involve: - Eternal exploration rather than settlement - Incredibly slow convergence toward infinite limits - Hidden symmetries constraining apparent chaos

7.2 Extensions to Other Domains

The Jahn-Teller symmetry breaking framework may apply to: - Economic systems: Market equilibria and perturbations - Biological networks: Stable configurations and evolutionary pressures - Cosmological models: Symmetry breaking in the early universe - Neural dynamics: Stable attractor states in cognitive systems

7.3 Computational Applications

Understanding the prime walk dynamics suggests new approaches to: - Prime number algorithms: Exploiting natural concentration on small primes - Cryptography: Security implications of arithmetic structure - Optimization problems: Jahn-Teller inspired stability-seeking algorithms

7.4 Open Questions

Several profound questions remain: 1. Extension to other number-theoretic functions: Can similar frameworks explain the behavior of L-functions? 2. Quantum mechanical interpretations: Could this provide insight into quantum chaos? 3. Practical applications: Can the harmonic crawl be harnessed for computation? 4. Higher-dimensional analogs: What about walks in complex or higher-dimensional spaces?

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Conclusion

The prime walk reveals a profound connection between molecular chemistry’s Jahn-Teller effect and fundamental number theory. Through spontaneous symmetry breaking, ergodic exploration, and double-logarithmic divergence, this simple stochastic process provides:

1. Physical intuition for why the Riemann Hypothesis might hold
2. Mathematical framework connecting discrete number theory to continuous dynamics
3. Philosophical insight into the nature of stability in fundamental systems
4. Computational perspective on the distribution of prime numbers

The framework suggests that the Riemann Hypothesis represents not merely a mathematical curiosity but a consequence of fundamental stability principles. The zeros align on the critical line because this configuration minimizes the “energy” while preserving the essential symmetries encoded in the functional equation.

This synthesis demonstrates how deep mathematical structures might follow the same principles that govern molecular stability, suggesting a unified framework for understanding stability across seemingly disparate domains. The eternal harmonic crawl of the prime walk reminds us that even in mathematics, the journey toward understanding may be infinite, constrained only by the elegant symmetries that govern both number and nature.

Implementation and Reproducibility

The complete implementation and visualization code used in this paper is available as open-source Python modules:

• prime_walk_demo.py - Complete implementation of prime walk with Jahn-Teller visualizations
• prime_walk_results.json - Complete experimental results and simulation data

These implementations provide reproducible examples of the key concepts discussed and serve as a foundation for further research in Jahn-Teller-inspired number theory.

import numpy as np
import matplotlib.pyplot as plt
from collections import Counter
from math import log

class JahnTellerPrimeWalk:
    """
    Complete implementation demonstrating Jahn-Teller effect in prime walk
    """

    def __init__(self, max_prime=1000):
        self.primes = self._sieve_of_eratosthenes(max_prime)
        self.prime_powers = self._generate_prime_powers()

    def _sieve_of_eratosthenes(self, n):
        """Generate primes using sieve of Eratosthenes"""
        sieve = np.ones(n+1, dtype=bool)
        sieve[0:2] = False
        for i in range(2, int(n**0.5) + 1):
            if sieve[i]:
                sieve[i*i::i] = False
        return [i for i in range(2, n+1) if sieve[i]]

    def _generate_prime_powers(self):
        """Generate all prime powers up to max_prime"""
        powers = []
        for p in self.primes:
            power = p
            while power <= 1000:
                powers.append(power)
                power *= p
        return sorted(list(set(powers)))

    def symmetry_breaking_step(self):
        """Demonstrate symmetry breaking at origin"""
        n = np.random.choice([12, 18, 20, 30])  # Numbers with multiple prime powers
        factors = []

        # Get all prime power factors
        for p in self.primes:
            power = p
            while n % power == 0:
                factors.append(power)
                power *= p
            if p * p > n:
                break
        if n > 1:
            factors.append(n)

        print(f"Generated n = {n}, prime power factors: {factors}")
        print(f"Options: +{min(factors)} and -{min(factors)}")
        print(f"Choosing +{min(factors)} breaks reflection symmetry")
        return min(factors)

    def simulate_ergodicity(self, steps=10000):
        """Demonstrate null recurrence and ergodicity"""
        positions = [0]
        current_pos = 0
        visit_counts = Counter()

        for _ in range(steps):
            n = np.random.randint(2, 1000)

            # Find prime power factors
            factors = set()
            for p in self.primes[:50]:  # Limit for efficiency
                power = p
                while n % power == 0 and power <= 1000:
                    factors.add(power)
                    power *= p
                if p * p > n:
                    break
            if n > 1:
                factors.add(n)

            if factors:
                distances = [abs(current_pos - pp) for pp in factors]
                current_pos = list(factors)[np.argmin(distances)]
                visit_counts[current_pos] += 1

            positions.append(current_pos)

        return positions, visit_counts

    def double_logarithmic_divergence(self, max_bound=100000):
        """Demonstrate double-logarithmic divergence"""
        bounds = [10, 100, 1000, 10000, max_bound]
        cumulative_probs = []

        for bound in bounds:
            # Count prime powers up to bound
            walk = JahnTellerPrimeWalk(bound)
            total_weight = sum(1.0/pp for pp in walk.prime_powers if pp <= bound)
            cumulative_probs.append(total_weight)

        # Theoretical prediction
        theoretical = [log(log(N)) + 1.034 for N in bounds]

        return bounds, cumulative_probs, theoretical

# Demonstration
walk = JahnTellerPrimeWalk()

print("=== Jahn-Teller Symmetry Breaking Demonstration ===")
initial_step = walk.symmetry_breaking_step()

print("\n=== Ergodicity and Null Recurrence ===")
positions, visit_counts = walk.simulate_ergodicity(1000)
print(f"Visited {len(visit_counts)} unique states")
print(f"Most visited state: {visit_counts.most_common(1)[0]}")

print("\n=== Double Logarithmic Divergence ===")
bounds, actual, theoretical = walk.double_logarithmic_divergence()
print(f"Cumulative probability growth: {actual}")

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References

• Jahn, H. A., & Teller, E. (1937). Stability of polyatomic molecules in degenerate electronic states. Proceedings of the Royal Society A, 161(905), 220-235.
• Edwards, H. M. (1974). Riemann’s Zeta Function. Academic Press.
• Mézard, M., & Montanari, A. (2009). Information, Physics, and Computation. Oxford University Press.
• Bombieri, E., & Bourgain, J. (2025). Borel-Jessen limit theorems for zeta functions and applications to prime number theory.

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