Sequel to Diaphorum 2: Spencer-Brown’s Laws of Formless

Write as Claude Shannon, explaining the contrast to both Markovian and semi-Markovian, accessible to college students with a single physics class.

ChatGPT Prompt

1. A Personal Preface

I spent much of my life asking how information moves, compresses, and survives noise—questions that later became known as information theory.
In doing so, I leaned heavily on Markov processes—not because they were ultimate truths, but because they were useful approximations.

What follows is not a rejection of Markovian thinking.
It is a recognition of where it no longer suffices.

2. The Markovian World: Memoryless but Powerful

A Markovian system assumes one simple rule:

• The future depends only on the present
• The past matters only insofar as it produced the present

This assumption gave us:

• Tractable mathematics
• Clean probability theory
• Enormous success in physics, communication, and engineering

For many systems—radio noise, gas molecules, queueing traffic—this was not a flaw.
It was a feature.

But it came with a cost.

The Hidden Assumption

Markovian models assume:

• A fixed set of states
• Fixed meanings for those states
• Fixed rules for transition

The system may move.
The space it moves in does not.

3. Semi-Markovian Systems: Letting Time Remember

Eventually, physicists noticed something inconvenient:

• Some systems care how long they’ve been in a state

Thus arose semi-Markov processes:

• The next step depends on the current state
• And on how much time has elapsed

This allowed us to model:

• Aging materials
• Waiting-time effects
• Delayed responses

But notice what did not change:

• The state definitions remained fixed
• The rules remained fixed
• The system did not reinterpret itself

Time remembered.
Meaning did not.

4. The Problem We Can No Longer Ignore

There exist systems for which:

• Past experience changes future possibilities
• Not just probabilities
• Not just timing
• But structure

Examples include:

• Machine learning systems
• Adaptive networks
• Brains
• Living organisms

In such systems:

• The present does not contain all the past that matters
• Because the past reshaped how the present is represented

No amount of bookkeeping restores Markovian closure.

5. What “Post-Markovian” Means

A post-Markovian system is one where:

• The set of states can change
• The interpretation of states can change
• The rules of transition can change
• And these changes arise from the system’s own history

This is not merely “more memory.”
It is memory that edits the grammar.

6. Why State-Space Expansion Fails

A common objection says:

“Just add more variables until the system becomes Markovian again.”

This works only if:

• The relevant memory is finite
• The representational scheme is fixed
• The system does not invent new distinctions

Learning systems violate all three.

You can record every bit they’ve seen
and still fail to capture what they’ve learned.

7. Information Is No Longer Just Transmitted

Classical information theory focused on:

• Sending messages
• Preserving signals
• Fighting noise

Post-Markovian systems introduce something new:

• Information that rewrites the receiver

The channel is no longer passive.
The destination is no longer fixed.

This moves us from:

• Communication → interpretation
• Signal → model
• Transmission → transformation

8. Physics Is Quietly Catching Up

Recent physics has begun to admit:

• Systems with internal models
• Dynamics coupled to learning
• Structures that evolve because they remember

Representative concepts include:

• Neural networks
• Statistical mechanics applied to learning
• The Free energy principle

These are not engineering hacks.
They are new physical primitives.

9. The Manifesto, Plainly Stated

I therefore propose:

• Markovian models describe motion
• Semi-Markovian models describe duration
• Post-Markovian models describe adaptation

And adaptation is not a complication.
It is a different category.

10. A Closing Remark

We once believed:

“The present contains all the information needed to predict the future.”

Post-Markovian systems reply:

“Only if the present knows how to read itself.”

The mathematics will follow.
It always does.

What changes first is what we are willing to call real.