🔢 The Leonov Sequence: A Recursive Formula for Chaotic Memory and Adaptive Form

#mathematics #chaos #cognitivecomputing #rust

What if memory wasn't fixed in neural networks or static matrices — but emerged from a recursive, chaotic shape?

Welcome to the Leonov sequence, a novel numerical recurrence that generates structured chaos with memory and serves as the core of wave-memory.

📐 The Formula

The sequence is defined recursively:

L_n = |a_n · L_{n-1} + b_n · L_{n-2} + c_n · L_{n-d_n} / (1 + n)|

Where:

a_n, b_n, c_n are random coefficients in range [-2, 2]
d_n is a random delay (1 to 10)
L_0 = 1, L_1 = 1
The |·| ensures non-negative dynamics

This formula creates a dynamic, nonlinear, chaotic sequence with built-in decaying memory — unlike traditional sequences such as Fibonacci.

📊 Why Is It Interesting?

Property	Leonov Sequence	Fibonacci	Logistic Map	Random Walk
Memory	✅ Yes (d_n lag)	❌ No	❌ No	❌ No
Chaos	✅ Tunable	❌ None	✅ Full chaos	✅ Noise
Entropy	🟡 Moderate	🔵 Low	🔴 High	🔴 High
Autocorrelation	✅ Present	✅ High	❌ None	✅ Random
Application	🧠 Geometry, cognition	Math demo	Population models	Noise models

🦀 Example in Rust

use wave_memory::{GeoForm, Wave};

fn main() {
    // Create a geometric form using the Leonov sequence
    let mut form = GeoForm::from_leonov(64, 123);

    // Initialize a wave with an impulse at the center
    let mut wave = Wave::new(64, 32);

    // Run the wave and let the form adapt to its energy
    for _ in 0..50 {
        wave.step(&form);
        let energy = wave.energy();
        form.adapt(&energy, 0.1);
    }

    // Inspect a value from the adapted form
    println!("Form at center: {:.4}", form.data[32]);
}