Non-Narrative Café v17A: Go Figure (Carbon Rule Reboot)

Posted: January 4, 2026 | Author: Dr. Ernie | Filed under: Centroids | Tags: ideas, philosophy, systems |2 Comments

Version 1.1 2026-01-04

Sequel 2 to Non-Narrative Café v16: Whitehead’s Reachability

Obsoletes Non-Narrative Café v17: Noticing Causality

Where I literally ignore causality to actually rewrite everything by hand
(with help from Claude)
Ernest Prabhakar

Let’s start over.

Preamble: The Carbon Rule

– Systems just are — they do not begin in our descriptions of them

• The most generative systems scale — the only systems that matter are those that survive long enough to be noticed
• What can survive? — what persists under constraint, as opposed to what can be said or calculated
• Earn, don’t assume — concepts from meaning, mathematics, and narrative must prove they can survive before they are used

Part I: Firstness (Systemic Noticing)

1. Systems

We are looking at something
Anything
Let’s call it a system

2. Perspective

We cannot
See all of the system

We do not
Get to define the system

We can only
Experience the system

We call this reality
Perspective

3. Fragments

We notice something

A collection
A scene
A moment

Call it a fragment

And then
We can notice more of them

4. Figures

Then we notice
Something new

We can distinguish
Elements within fragments

Let’s call them figures
Delightfully ambiguous
Could mean “number” or “person”
Noun or verb
Real or illusion

All that matters
Is that we can notice them

5. Reference

Before we can track figures
We must be able
To point to them

This is reference
Not meaning
Not interpretation

Just addressability
The ability to say
“That one”

6. Indexing

Now we can notice
The same figure
In different fragments

And even
Repeated
Within the same fragment

We call this ability
Indexing

7. Withness

Using indexing
We can identify
Every figure in a fragment

This means
They have something in common
Which we call Withness

8. Sameness

Using indexing
We can identify
A given figure
Across different fragments

This means
They share a property
Which we call Sameness

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Part II: Secondness (Structured Emergence)

9. Recurrence

Within a fragment
We can use indexing
To distinguish
Which figures recur
And which do not

We call that difference level

The ones that recur
We call background

The ones that don’t
We call foreground

Every figure in a fragment
Has a level:
Either background
Or foreground

11. Surprise

Now we can notice
Something else

Some fragments
Have the same background
But different foreground

This feels
Like the first thing
We can call surprise

12. Configuration

We can collect
All the fragments we’ve noticed
That share the same background

And call that
A configuration

We can call
The shared background
Its type

We can call
Any foreground figures
Its values

And add values
From any new fragments
We notice later
With the same type

(This is monotonicity —
configurations can only gain values,
never lose them)

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Part III: Thirdness (Relational Structure)

12. Reachability

Sameness creates a reachability graph
Showing which configurations
Are topologically connected to each other
And which are not

13. Chains

Alternating links
Of Withness and Sameness
Create what we call chains
Between multiple configurations

Note that this means
Chains cannot connect
Across foreground figures
Since those are always
In distinct fragments

14. Configuration Space

The shortest chain
Between configurations
Provides a rough metric

Which we could use
To turn our topology
Into a configuration space

Hold that thought
For later

15. Loops

When a chain
Leads back
To the same configuration
We call it a loop

16. Parity

Then we notice
Something odd
And even

The number of times
A loop crosses
Between background
And foreground

Determines
Whether it will end
On the same level
It started from

If it has an odd
Number of crossings
We say that the loop
Has Parity

17. Tension

Pick two loops X and Y
That both pass through
The configurations A and B

Call this the intersection
Of X and Y at A and B

If X has parity
And Y does not
We say there is tension
Between X and Y

To be continued

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Author’s Note: Why The Rewrite

I originally wrote the narratives (or rather, had ChatGPT write them) then tacked on the formalism later.

While working on “Inescapability” I suddenly realized my existing definitions of parity and causality were inconsistent.

I also realized that:

• Fragments should be made from figures, rather than vice versa
• Withness could emerge from figures and/or fragments
• Parity might emerge from foreground/background rather than positing a global orientation

So, took a day to rewrite the foundations, to avoid unnecessary complexity and clarify my own thinking.

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Carbon Rule Formalism (CRF)

Introduction

CRF is a formal system for describing structure that emerges from noticing patterns. There is no time, no process, no construction – only what persists under observation.

This document defines the language precisely enough to implement.

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I. Basic Types

Figure

A figure is a primitive element. Written as lowercase letters:

f, g, h, k.

Figures have no properties except identity – two figures are either the same or different.

Fragment

A fragment is a bag (multiset) of figures. Order doesn’t matter, but multiplicity does.

{f, f, g, g, h}

This fragment contains two fs, two gs, and one h.

Configuration

A configuration is a set of fragments that share the same background figures.

[{f, f, g, g, h}, {f, f, g, g, k}]

Both fragments have background {f, g}, so they belong to the same configuration.

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II. Properties

Fragment Properties

Background

The background of a fragment is the set of figures that appear more than once.

Background({f, f, g, g, h}) = {f, g}

Foreground (Distinctives)

The foreground of a fragment are the figures that appear exactly once.

Foreground({f, f, g, g, h}) = {h}

Also called “distinctives.”

Level

Every figure in a fragment has a level: either Background or Foreground.

Level(f, {f, f, g, g, h}) = Background

Level(h, {f, f, g, g, h}) = Foreground

The level distinction emerges from recurrence within a single fragment.

Configuration Properties

Type

The type of a configuration is its shared background.

Type([{f, f, g, g, h}, {f, f, g, g, k}]) = {f, g}

All fragments in a configuration must have identical backgrounds. If two fragments have different backgrounds, they belong to different configurations.

Values

The values of a configuration are all the foreground figures from all its fragments.

Values([{f, f, g, g, h}, {f, f, g, g, k}]) = {h, k}

Configurations exhibit monotonicity: they can only gain values, never lose them. Once a foreground figure appears in a configuration, it remains.

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III. Relations

Reference

Reference is the primitive ability to address or point to a specific figure. It is not indexing (tracking identity) or meaning (interpretation). It is simply addressability – being able to say “that one.”

In the formalism, this is implicit in our ability to write figure identifiers like f, g, h.

Sameness

Two figures are related by Sameness if they have the same identifier and appear in different fragments.

Given fragments:

F1 = {f, f, g} and F2 = {f, f, h}:

Sameness(f in F1, f in F2) = true

Background figures automatically have Sameness relations because they appear in multiple fragments by definition.

Withness

Two figures are related by Withness if they appear in the same fragment.

Given fragment F = {f, f, g, h}:

• Withness(f, g) = true
• Withness(f, h) = true
• Withness(g, h) = true

Important: Withness only connects figures within a single fragment instance. It cannot connect across fragments, even within the same configuration.

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IV. Higher-Order Structure

Reachability and Chains

Reachability

Sameness creates a reachability graph showing which configurations are topologically connected (share at least one figure identifier) and which are isolated.

Two configurations are reachable from each other if there exists at least one figure identifier that appears in both.

Chains

A chain is a path through figures, alternating between Withness (within fragments) and Sameness (between fragments).

f₁ --Same--> f₂ --With--> h₂ --Same--> h₃ --With--> g₃

Where:

• f₁ and g₃ are in the same fragment
• f₂ is the same figure identifier as one of the previous figures, but in a different fragment
• h₂ is in the same fragment as f₂

Constraint: Since foreground figures appear only once per fragment, Withness cannot traverse from a foreground figure to another foreground figure (there are no other figures in that “foreground space”). You can arrive at a foreground figure via Withness, but you must leave via Sameness or through background figures.

Loops and Configuration Space

Loops

A loop is a chain that returns to its starting fragment.

To close a loop:

1. Use Sameness to return to any figure in the starting fragment
2. If needed, use Withness to reach the starting figure

Example loop:

Loop Example

• f₀ in fragment F₁
• --Same--> f₂ (same figure, different fragment F₂)
• --With--> g₂ (same fragment as f₂)
• --Same--> g₃ (different fragment F₃)
• --With--> h₃ (same fragment as g₃)
• --Same--> f₀ (back to starting fragment F₁)

Configuration Space

The shortest chain between two configurations provides a rough metric – a measure of their topological distance. This metric can be used to understand the structure of the space of all configurations.

Configuration space is not assumed a priori but emerges from the pattern of chains and their lengths. It is a shadow cast by the reachability structure.

Parity

Definition

Parity measures whether a loop twists through levels.

Count the number of times the loop crosses between Background and Foreground:

• Background → Foreground: counts as one crossing
• Foreground → Background: counts as one crossing
• Background → Background: not a crossing
• Foreground → Foreground: not a crossing
• Parity = P if crossings mod 2 = 1 (odd)
• Parity = ≠P if crossings mod 2 = 0 (even)

Properties:

• Parity is the same no matter where you start counting in the loop
• Loops with Parity = P have a twist (like a Möbius strip)
• Loops with Parity = ≠P have no twist (like a simple circle)

Intersection and Tension

Intersection

Two loops have an intersection if they both pass through at least one common configuration.

`

• Loop X passes through configurations {A, B, C}
• Loop Y passes through configurations {B, C, D}
• Intersection(X, Y) = {B, C}

Tension

Two loops have tension if:

1. They intersect (share at least one configuration), AND
2. One has Parity = P and the other has Parity = ≠P

• Tension(X, Y) = true iff:
• Intersection(X, Y) ≠ ∅ AND
• Parity(X) ≠ Parity(Y)
  “`

Tension is a property of the pair of loops, not of their intersection. The intersection just tells us where they overlap, but tension exists between the complete loops.

Properties:

• Tension is symmetric: if X has tension with Y, then Y has tension with X
• Tension requires exactly one loop with parity
• Two loops with the same parity (both P or both ≠P) have no tension

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V. Examples

Example 1: Basic Fragment

Fragment: {f, f, g, g, h}

Background: {f, g}

Foreground: {h}

Levels:

• Level(f) = Background
• Level(g) = Background
• Level(h) = Foreground

Example 2: Configuration

Fragments:

• F₁ = {f, f, g, g, h}
• F₂ = {f, f, g, g, k}

Both have Background = {f, g}

Configuration: C = [F₁, F₂]

• Type(C) = {f, g}
• Values(C) = {h, k}

Example 3: Parity Calculation

This example uses the notation defined in section IV (Higher-Order Structure) for tracking level crossings in loops.

Loop:

f₁ v=v f₂ v~^ j₂ ^=v h₃ v~v g₃ v=^ k₄ ^~v f₄ v=v f₁

Crossings:

• v~^ (1),
• ^=v (2),
• v=^ (3),
• ^~v (4)

Total: 4 (even) → Parity = ≠P

Notation key:

• v=v = Sameness at Background level
• v~v = Withness at Background level
• v~^ = Withness CROSSING Background to Foreground
• ^=^ = Sameness at Foreground level
• ^=v = Sameness CROSSING Foreground to Background
• ^~v = Withness CROSSING Foreground to Background
• v=^ = Sameness CROSSING Background to Foreground

DISALLOWED:

• ^~^ = Withness at Foreground level (foreground figures are in different fragments)

Example 4: Tension

• Loop X: 3 crossings → Parity = P
• Loop Y: 2 crossings → Parity = ≠P

Both pass through configurations A and B.

Analysis:

• Intersection(X, Y) = {A, B} ≠ ∅
• Parity(X) ≠ Parity(Y)

Therefore: Tension(X, Y) = true

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