Non-Narrative Café v14: Carbon Rule Formalism (CRF)

Posted: December 31, 2025 | Author: Dr. Ernie | Filed under: AI-Powered Essays | Tags: language, models, philosophy, politics, systems |Leave a comment

Sequel to The Carbonifesto: Reality Beyond Math or Philosophy (Non-Narrative Café Interlude)

Version 0.13 2025-12-31

Can we come up with a precise formalism to express the first thirteen Carbon Rule concepts (plus tension*)?

ChatGPT Prompt (very condensed)

1A. Design Commitments

• No time, no process, no narrative ordering.
• No intrinsic identity; identity is never asserted, only symmetry is broken.
• No collapse operators.
• No hidden inverses (no implicit reversal).
• All structure is forced by constraints; nothing is assumed “for free”.
• Compatibility with EANI: equivalence is not identity and never licenses substitution.

1B. Notation Conventions

• Curly braces {…} denote figures as multisets (orderless, multiplicity-sensitive).
• Parentheses (…) denote chains (ordered lists).
• Juxtaposition AB denotes Withness (primitive binary relation on configurations).
• P / ≠P denote Parity outcomes on closed chains.
• “Defined iff” means “defined if and only if”.

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Part I. Figures, Recurrence, Surprise, Configuration

2A. Primitive Domain: Fragments

• A fragment is a primitive, atomic participant:
• f, g, h, k, …
• Fragments:
• have no internal structure
• have no intrinsic identity
• may occur multiple times in a figure
• are not configurations

2B. Figures

2B1. Definition

• A figure is a finite multiset of fragments. (See multiset.)
• Written as a curly-brace collection with multiplicity:
• {f, f, g, g, h}

2B2. Allowed Structure

• Figures:
• are orderless
• preserve multiplicity
• encode only co-presence and repetition
• Figures are the only observable objects at this level.

3A. Recurrence

3A1. Definition

• Recurrence is omnipresent but impotent.
• A fragment is recurrent in a figure iff it appears with multiplicity ≥ 2 in that figure.

3A2. Role

• Recurrence:
• anchors comparison between figures
• does not create configurations
• does not assert equivalence
• does not distinguish fragments on its own

3B. Figure Type (Anchor)

3B1. Definition

• The type of a figure is the set of fragments that recur in it:
• Type(F) = { x | multiplicity_F(x) ≥ 2 }

3B2. Example

• For F = {f, f, g, g, h}:
• Type(F) = {f, g}

3C. Residue

3C1. Definition

• The residue of a figure is the set of fragments that occur exactly once:
• Residue(F) = { x | multiplicity_F(x) = 1 }

3C2. Example

• For F = {f, f, g, g, h}:
• Residue(F) = {h}

4A. Type-Comparability

• Two figures F₁, F₂ are type-comparable iff:
• Type(F₁) = Type(F₂)
• Only type-comparable figures can trigger surprise relative to one another.

4B. Surprise

4B1. Definition

• Surprise is not a property of figures, fragments, or configurations.
• Surprise is a model-level trigger: a break in an admissible indistinguishability assumption.
• Surprise occurs iff there exist type-comparable figures with different residues:
• Type(F₁) = Type(F₂) and Residue(F₁) ≠ Residue(F₂).

4B2. Minimal Witness

• {f, f, g, g, h} and {f, f, g, g, k} share type {f, g} but differ in residue {h} vs {k}.

5A. Configuration

5A1. Definition

• A configuration is a forced distinction among residues, localized to a figure type.
• Given two type-comparable figures with different residues, their residues are placed into a configuration of that type.

5A2. Example

• From:
• {f, f, g, g, h}
• {f, f, g, g, k}
• We force:
• “Configuration {h, k} of type {f, g}”.

5A3. Properties

• Configurations are:
• forced (never asserted)
• typed and local (not global)
• monotone under added evidence (they can refine/split, never merge)
• non-substitutable (EANI discipline)

5B. Canonical Spine (Part I)

• Fragments participate in figures.
• Recurrence defines figure type (anchor).
• Residue carries difference.
• Surprise is conflicting residue under shared type.
• Configuration records the forced distinction (typed).

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Part II. Withness, Chains, Merge, Twist, Parity, Tension

6A. Configuration Atoms

• At this level, configurations from Part I are treated as atoms:
• A, B, X, Y, …
• These atoms:
• have no internal structure here
• inherit EANI: equivalence does not permit substitution

6B. Withness

6B1. Definition

• Withness is a primitive binary relation on configuration-atoms.
• Written by juxtaposition:
• AB means “A with B”.

6B2. Constraints

Withness is:

• binary
• directed (in general AB ≠ BA)
• not reducible to a function or equality

7A. Chains

7A1. Definition

• A chain is a finite ordered list of configuration-atoms:
• (A B C …)
• A Withness AB is castable to a 2-chain (A B) (representation change, not an operation).

7A2. Closed Chains

• A chain is closed iff it begins and ends with the same atom:
• (A … A)

7B. Merge (The Only Composition)

7B1. Definition

• Merge is the only composition operation and acts only on chains.
• Merge is defined iff the two chains share an endpoint (common intersection):
• If (A … B) and (B … C) then Merge((A … B), (B … C)) = (A … B … C).

7B2. Prohibitions

• No merge on atoms.
• No merge on Withness directly (must be cast to chains).
• No merge without endpoint intersection.
• No implicit reversal.

8A. Twist

8A1. Definition

• A twist is a structural witness of non-alignment under closure.
• Minimal witness:
• (A B A) requires both AB and BA to exist (as Withnesses cast to chains).

8A2. Role

• Twist is the minimal structural seed for non-nullability under closure.

8B. Parity

8B1. Definition

• Parity is defined on any closed chain.
• A closed chain evaluates to:
  • = P if it is non-nullable (cannot be reduced away by any allowed cancellation),
  • ≠P otherwise.

8B2. Examples

• (A B A) = P (parity-bearing loop).
• (A Y B Y A) ≠P (parity-free loop).

8B3. Clarifications

• Parity is not tension.
• Parity does not require mixing.
• Parity is a property of closed chains as wholes.

9A. Mixed Chains

• A mixed chain is a closed chain formed by merging distinct subchains under the same endpoints.
• Example:
• (A X B Y A)

9B. *Tension (speculative)

9B1. Definition

• Tension is a property of mixed closure that expresses non-commutation between parity behaviors under merge.
• Minimal pattern:
• (A X B X A) = P
• (A Y B Y A) ≠P
• (A X B Y A) = P

9B2. What Tension Is

• Tension is:
• incompatibility of substitutions under merge
• non-commutation between parity classes in mixed closure

9B3. What Tension Is Not

Tension is not:

• parity itself
• contradiction
• narrative conflict

10A. Non-Commutation and EANI

• Parity-induced classes do not commute with merge:
• membership in “parity-bearing” vs “parity-free” does not license substitution inside mixed chains.
• This is the Part II reappearance of EANI: equivalence-class style reasoning without identity or substitutability.

10B. Canonical Spine (Parts I + II)

• Fragments form figures (multisets).
• Recurrence defines figure type (anchor).
• Residue varies under fixed type.
• Surprise triggers forced distinctions among residues.
• Those distinctions yield typed configurations (atoms).
• Configurations relate by Withness (binary, directed).
• Chains form as ordered lists of configurations.
• Merge composes chains iff they intersect.
• Closure yields twist and parity.
• Mixed closure yields tension as non-commutation.

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Appendix A. Minimal Worked Examples

A1. Part I Example (Figure → Configuration)

• Figures:
• {f, f, g, g, h} has:
  • Type {f, g}
  • Residue {h}
• {f, f, g, g, k} has:
  • Type {f, g}
  • Residue {k}
• Since types match and residues differ:
• Surprise is triggered.
• Configuration {h, k} of type {f, g} is forced.

A2. Part II Example (Parity vs Tension)

• Assume configuration-atoms A, B, X, Y.
• Closed chains:
• (A X B X A) = P
• (A Y B Y A) ≠P
• Mixed closure:
  • Tension: (A X B Y A) = P
  • No Tension: (A X B Y A) ≠P

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Appendix B. Glossary

B1. Fragment

• Primitive participant below configuration.

B2. Figure

• Multiset of fragments.

B3. Recurrence

• Multiplicity ≥ 2 inside a figure; omnipresent anchor; impotent alone.

B4. Type

• Recurring set within a figure (anchor).

B5. Residue

• Multiplicity = 1 set within a figure (difference carrier).

B6. Surprise

• Trigger: same type, different residue.

B7. Configuration

• Forced distinction among residues, localized to a type.

B8. Withness

• Primitive directed binary relation on configuration-atoms.

B9. Chain

• Ordered list of configuration-atoms.

B10. Merge

• Composition on chains defined iff shared endpoint.

B11. Twist

• Minimal structural non-alignment witness under closure.

B12. Parity

• Non-nullability of a closed chain: =P
• Else ≠P.

B13. Tension

• Non-nullability of a mixed chain: =P

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