Non-Narrative Café v14: OLD Carbon Rule Formalism

Posted: December 31, 2025 | Author: Dr. Ernie | Filed under: AI-Powered Essays | Tags: language, models, philosophy, politics, systems |4 Comments

Obsoleted by Non-Narrative Café v17A: Go Figure (Carbon Rule Reboot) on 2026-01-03

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

Version 0.17 2026-01-01

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

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”.

---

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).

---

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

9B1. Definition

Tension is a property of mixed closure in parity-bearing systems, expressed as non-commutation of parity-licensed substitutions under merge.

Equivalently:

Tension occurs when equivalence authorized by parity fails to be preserved under enforced closure.

---

9B1.1 Minimal Pattern

Let X, Y be parity-licensed substitutions and P a parity class.

• (A X B X A) = P
• (A Y B Y A) ≠ P
• (A X B Y A) = P

Tension is present iff:

• substitutions are individually parity-valid,
• closure is enforced,
• and no re-ordering removes the non-equivalence.

---

9B2. What Tension Is

Tension is:

• incompatibility of substitutions under merge,
• non-commutation revealed by mixed closure,
• persistence of non-equivalence after admissible reduction.

---

9B3. What Tension Is Not

Tension is not:

• parity itself,
• contradiction,
• narrative or dynamical conflict,
• generic non-commutativity,
• difference absent parity.

---

9B4. Selection

Not all mixed loops exhibit tension.

A mixed loop exhibits tension iff:

1. parity authorizes substitution,
2. closure is enforced,
3. substitution order affects equivalence.

Absent parity, mixed loops yield complexity only.

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.

Part III: Individuation

11. Reachability Formalism

11A. Motivation

In the CRF setting, Withness (AB) is a primitive directed binary relation among configuration-atoms that need not be symmetric (AB ≠ BA) and admits castability into chains. Reachability arises from repeated Merge of chains (ordered lists of atoms) under endpoint intersection. The reachability graph is not an added structure but is what is already present once the primitives of CRF are taken seriously.

11B. Reachability Graph Definition

Let:

• Nodes be configuration-atoms (CRF Part II, §6A).
• Edges be directed continuations corresponding to Withness (AB) cast to chains (A B).
• Paths be finite sequences of Merge-composed chains such that if (A … B) and (B … C) are chains, then Merge((A … B), (B … C)) = (A … … C) is defined.

The reachability graph is the directed graph (G = (V, E)):

• V = set of all configuration-atoms.
• E = set of all directed edges A → B instantiated by Withness AB.

A node (v) is reachable from (u) iff there is a finite sequence of merges of chain representations that yields a path from u to v.

11C. Connectedness in CRF

A reachability graph is connected if for every pair of atoms (u, v ∈ V), either:

• (v) is reachable from (u), or
• (u) is reachable from (v),

under admissible chain merges.
Connectedness is necessary for inescapability, since disconnected subgraphs witness disjoint Withness loops.

11D. Avoidance and Inescapability

Given a set of atoms (S ⊆ V) (e.g., a constraint or asymmetry), define:

• An avoidance path as a path in (G) that does not include any atom in (S).
• Inescapability relative to (S) holds iff no avoidance path exists; every admissible path encounters some member of (S).

Formally:

Inescapability of (S) in (G) holds ⇔
∀(u, v ∈ V), every path from (u) to (v) intersects (S).

This notion does not presuppose closure, law, or identity — only reachability and admissible continuation.

11E. Strongly Connected Components

A subset (C ⊆ V) is a strongly connected component (SCC) iff for all (u, v ∈ C):

• (v) is reachable from (u), and
• (u) is reachable from (v).

SCCs in the reachability graph represent closed Withness loops under CRF primitives.

12. Causal Graph

In the Carbon Rule Formalism (CRF), a causal graph is the directed structure that emerges from the primitives of configuration and relation under the only new constraint of unidirectionality.

Building on CRF’s earlier definitions of figures, configurations, Withness, and chains, the causal graph is defined as follows:

• Let nodes be configuration-atoms (the irreducible outcomes of forced distinctions among residues in figures) as introduced in Part II of the CRF framework.
• Let edges be directed continuations arising from Withness relations (AB cast to a 2-chain (A B)).
• A path is a finite sequence of merges of chains whose endpoints intersect, so that if (A … B) and (B … C) are chains, merging them yields (A … C).
• The causal graph (G = (V, E)) is then the directed graph with vertex set (V) equal to all configuration-atoms and edge set (E) equal to all directed edges instantiated by Withness.

In this view, reachability—whether one atom can be reached from another by successive merges of admissible chains—is not added by fiat but noticed once the primitives of CRF are taken seriously. The directed edges reflect the structural non-equivalence of reversal under the Equivalence Assumption of Non-Identity (EANI) and parity, which ensures that formally symmetric paths can differ extensionally once constraints accrue. The causal graph thus records the local asymmetries of constraint and continuation that cannot be avoided under CRF’s universality-free, identity-disciplined ontology.

---

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

---

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

Continued in Non-Narrative Café v15: Resolving Tension

---