Mark Tracy

---

Imagine living in a world where patterns repeat in a meaningful sense—but never in exactly the same way.

The Sun rises every day.
The Moon goes through phases.
The seasons return each year.

But if you look closely, these cycles do not line up perfectly. The lunar month does not divide evenly into the year. The year does not divide neatly into lunar cycles. The phases slowly drift relative to one another.

And that drifting turns out to be extremely important.

Two related ideas explore why.

---

Part I: Why Humans Developed Our Sense of Time

Human beings did not invent time from nothing. Instead, our sense of time likely emerged from the structure of the natural world around us.

An observer on Earth experiences four recurring processes:

• the solar day (day and night)
• the lunar cycle (phases of the Moon)
• the solar year (seasons)
• the circadian rhythm (the body’s sleep–wake cycle)

Three of these—day, lunar phase, and year—are astronomical cycles. Crucially, they do not share simple ratios. Their periods are incommensurate: they never line up perfectly.

Because of this, the relationships between them continually drift.

For example:

• The Moon may be full during winter one year,
• but appear during a different phase at the same calendar date the next year.

This drifting creates a world where patterns are stable but never identical.

That environment rewards certain kinds of thinking:

• counting
• memory
• pattern comparison
• estimating ratios between cycles

Over time, societies began representing these relationships explicitly through:

• calendars
• ratios such as 365 days per year
• continuous mathematical models of time

At the same time, the circadian rhythm divides experience into discrete episodes—waking and sleeping.

Together these produce a dual picture of time:

• continuous celestial motion
• discrete lived moments

Human temporal cognition may have emerged from learning to navigate this quasi-periodic structure.

---

Part II: A Toy World That Teaches a Machine the Same Lesson

The second document builds a simple mathematical experiment to study a related question:

Under what conditions can a learning system discover hidden structure in a process that never exactly repeats?

To explore this, the paper constructs a minimal world.

The world

The environment contains three rotating variables—think of them as three independent clocks.

Each rotates steadily, but their speeds are chosen so that their cycles never perfectly align.

Mathematically, this produces a quasi-periodic system. The configuration continually changes but always remains within a structured pattern.

Over time, the system wanders across a geometric surface called a three-dimensional torus.

What the agent can see

The learning agent cannot see the clocks themselves.

Instead it observes only relationships between them—their phase differences.

This mirrors human perception. We never observe absolute positions in space; we only observe relations between objects, such as distances or angles. For example, we perceive where the Moon appears relative to the Sun in the sky, not their absolute locations in the universe.

Modern physics reinforces this relational view. In relativity, there is no absolute position or absolute time. As emphasized by Einstein, the fundamental quantities of the physical world are intervals—spatial distances, time differences, and spacetime separations between events.

What the agent tries to do

The agent’s task is simple:

Predict what the relational observations will be next.

Whenever its prediction is wrong, it updates its internal model.

Through repeated prediction and correction, the agent gradually builds an internal representation of the underlying structure that generates the observations.

The central insight

Even though the world never exactly repeats, it still contains stable structure.

If the learning system

• observes relational patterns,
• predicts future observations,
• and updates its model from prediction error,

then it may eventually discover dynamical invariants—the hidden constants governing the system.

In other words, perhaps prediction alone can drive the emergence of meaningful internal representations.

---

The Shared Idea

Both documents revolve around the same deep principle.

Stable understanding does not require perfect repetition.

Instead, understanding can emerge when:

1. The world has structured dynamics
2. Patterns drift without repeating exactly
3. An observer repeatedly compares and predicts relations

In nature, this may have helped produce human temporal cognition.

In artificial systems, it may allow learning agents to discover latent structure without explicit supervision.

The world does not have to repeat for intelligence to arise.

It only has to have relationships that drift in a structured way.