Z₇₁ Cyclic Dynamical System with Nontrivial 1-Cocycle Invariant

README.md

Minimal 71-State Cyclic System with Additive Cocycle Invariant

This repository contains a minimal deterministic dynamical system over the cyclic group Z₇₁ with a nontrivial additive edge-weight function (cocycle).

It produces two global invariants:

• Period = 71
• Drift = 118

System Overview

The system is defined on a finite state space:

S = Z₇₁

Transition rule

Each state advances deterministically:

x → x + 1 (mod 71)

This forms a single directed cycle over all 71 states.

Strongly Connected Component (SCC)

The entire state space forms one SCC:

• Every state can reach every other state via repeated transitions
• The SCC is exactly a 71-cycle
• No branching or additional components exist

Cocycle (Edge Weights)

Each transition carries an integer weight w(x → x+1).

The weight function is defined so that the total sum over one full cycle is:

Σ w(x) = 118

This defines a nontrivial additive cocycle over the cycle.

Invariants

Running the system from any starting point yields:

• Cycle length (period): 71
• Total accumulated drift: 118

These are invariant under choice of start state.

Mathematical Interpretation

This system is an instance of:

• A finite cyclic group action (Z₇₁)
• Equipped with a discrete 1-cocycle
• With invariant given by integration over the unique SCC cycle

Formally:

(Z₇₁, +1, w)

where the SCC structure and cocycle define a global invariant pair (P, Δ).

Key takeaway

A deterministic system over a finite cyclic graph can carry nontrivial global invariants beyond its period, captured by edge-weight cocycles.

cyclic_cocycle_z71.rs

// ============================================================
// MINIMAL EULER INVARIANT SYSTEM (P = 71, Δ = 118)
// ============================================================
//
// State space: Z_71
// Dynamics   : x -> x + 1 mod 71
// Weight     : fixed cocycle w(x)
// Invariant  : sum over full cycle = 118
//
// This is a pure cyclic group + additive cocycle system.
// SCC structure is exactly one 71-cycle.
// ============================================================

const N: usize = 71;

// ------------------------------------------------------------
// STEP FUNCTION: pure cycle on Z_71
// ------------------------------------------------------------

fn step(x: usize) -> usize {
    (x + 1) % N
}

// ------------------------------------------------------------
// COCYCLE (edge weights)
// Must satisfy: sum_{x=0..70} w(x) = 118
// ------------------------------------------------------------

fn weight(x: usize) -> i64 {
    match x {
        0..=9 => 2,     // 10 * 2 = 20
        10..=19 => 2,   // 20
        20..=29 => 2,   // 20
        30..=39 => 2,   // 20
        40..=49 => 2,   // 20
        50..=59 => 2,   // 20   -> subtotal 120
        60..=69 => -1,  // 10 * -1 = -10 -> 110
        70 => 8,        // +8 -> 118 total
        _ => 0,
    }
}

// ------------------------------------------------------------
// ANALYZE SINGLE CYCLE (SCC is trivial)
// ------------------------------------------------------------

fn analyze(start: usize) {
    let mut seen = vec![-1i32; N];

    let mut x = start;
    let mut t: usize = 0;
    let mut sum: i64 = 0;

    loop {
        if seen[x] != -1 {
            println!("Period = {}", t - seen[x] as usize);
            println!("Drift = {}", sum);
            return;
        }

        seen[x] = t as i32;

        sum += weight(x);
        x = step(x);
        t += 1;
    }
}

// ------------------------------------------------------------

fn main() {
    analyze(0);
}