Probability in Cohesion Dynamics

Probability in Cohesion Dynamics is structural, not observer-centric. This guide explains how to think about probability in CD without appealing to self-location, temporal uncertainty, or branch counting.

Core Insight: Probability Is Structural, Not Epistemic

In Cohesion Dynamics, probability assignments are functions on the directed acyclic graph (DAG) of admissible configurations. They track how restrictive a configuration’s position in the admissibility structure is, not what an observer “expects to experience.”

What probability is NOT in CD:

Observer self-location (“finding oneself in a branch”)
Temporal ignorance about future states
Counting of micro-configurations or parallel observers
Subjective uncertainty about outcomes

What probability IS in CD:

A structural measure on admissible refinement relations
A weighting function reflecting symmetry restrictions
A calculus for composition under branching and reconciliation
Conditioning relative to a chosen root configuration in the DAG

1. Probability Requires Continuity

Probability is only meaningful relative to a continuity class.

Two configurations can only be assigned relative probabilities if they share admissible ancestry — that is, if there exists a common parent configuration from which both are admissible refinements.

Key principle: Configurations with no admissible transformation relation admit no probabilistic comparison.

This replaces vague notions of “identity” or “sameness” with CD-native continuity via admissibility structure.

Example: No Probability Without Continuity

Consider two configurations:

Configuration A: An electron in state |↑⟩
Configuration B: A photon in momentum state |k⟩

Without a shared admissible ancestry (no parent configuration from which both refine), no probability relation exists between them. They belong to distinct continuity classes.

Contrast this with:

Configuration A: An electron in state |↑⟩
Configuration C: The same electron in state |↓⟩

If both A and C refine from a common parent (e.g., superposition |↑⟩ + |↓⟩), they admit probabilistic comparison relative to that parent.

2. Probability Is Defined Relative to a Chosen Root

Every probability question implicitly presupposes a root configuration in the admissibility DAG.

Different questions correspond to different roots:

A. Predicting Next States

Question: “What is the probability of the next state being X?”
Root: Current realised configuration
Structure: Probability weights track how admissible refinements extend from the current node

B. Comparing Alternative Realised States

Question: “What is the probability I am in state A vs state B?”
Root: Most recent shared ancestor of A and B
Structure: Probability weights reflect how restrictive each path is relative to the shared parent

Conditioning and Renormalisation

Conditioning on a realised configuration renormalises total admissible probability to 1 relative to that node as the new root.

Critical clarification: History is not erased by conditioning. Admissibility relations remain unchanged. What changes is the reference frame for probability assignments — the root relative to which weights are calculated.

graph TD
    R[Root: Shared Ancestor] --> A[State A<br/>w=0.7]
    R --> B[State B<br/>w=0.3]
    A --> A1[Future from A<br/>renormalised]
    A --> A2[Future from A<br/>renormalised]
    B --> B1[Future from B<br/>renormalised]
    
    style R fill:#e1f5ff
    style A fill:#d4edda
    style B fill:#f8d7da

Caption: Conditioning on state A changes the root. Probabilities of A1 and A2 are now calculated relative to A, not R.

3. Symmetry, Not Microstate Counting, Grounds Weights

A common misconception is that probability weights arise from counting micro-configurations — the idea being that “many microscopic states flow into one outcome vs another.”

This is NOT how probability works in Cohesion Dynamics.

The Symmetry-Based Account

Probability weights arise from restriction of admissible symmetry transformations under refinement.

Key insight: Even a single configuration can support nontrivial weighting if that configuration admits symmetry-preserving and symmetry-breaking refinements.

Consider a parent configuration with a certain symmetry (e.g., rotational symmetry). When refining to child configurations:

Symmetry-preserving refinement: The child configuration retains the full symmetry of the parent → larger admissible transformation set
Symmetry-breaking refinement: The child configuration breaks symmetry → smaller admissible transformation set

Probability weights track how much symmetry is preserved or broken along refinement paths, not how many “copies” of a microstate exist.

graph TD
    P[Parent: Full Symmetry<br/>SO3 rotational freedom] --> C1[Child 1: Symmetry Preserved<br/>SO3 still admissible<br/>w ∝ preserved symmetry]
    P --> C2[Child 2: Symmetry Broken<br/>Preferred axis emerges<br/>w ∝ restricted symmetry]
    
    style P fill:#e1f5ff
    style C1 fill:#d4edda
    style C2 fill:#fff3cd

Caption: Weights reflect symmetry restriction, not microstate multiplicity.

Informational Transformation, Not Resource Budgets

Even if we imagine only one micro-configuration of a system (e.g., an atom) exists:

Admissible degrees of freedom exercised as transformations under symmetry are not bounded by information as a “finite resource”
An informational description can be inverted or transformed under symmetry, resulting in many equally real descriptions
If those descriptions decohere (do not recombine), they populate distinct branches in the DAG
Probability weights track the available pathways through this DAG, not counting of parallel copies

This framing replaces “many microstates flowing” with “symmetry transformations branching.”

4. The DAG of Admissible Configurations

The admissibility DAG is the foundational structure on which probability is defined.

Structure of the DAG

Nodes: Reconciliation-stable configurations (configurations that can persist across admissible closures)

Edges: Admissible refinements (transitions that satisfy all structural constraints)

Branching: Multiple exclusive refinements from a single parent (structural non-singularity of admissible continuations)

Convergence: Previously diverged branches may reconcile if their informational structures remain compatible

graph TD
    R[Root Configuration] --> A[Refinement A]
    R --> D[Refinement D]
    A --> B[Refinement B]
    A --> C[Refinement C]
    B --> E[Convergence]
    C --> E
    D --> F[Refinement F]
    
    style R fill:#e1f5ff
    style A fill:#d4edda
    style D fill:#f8d7da
    style E fill:#ffeaa7

Caption: Admissibility DAG with branching and reconciliation. Probability is defined over paths through this structure.

Not all nodes in the DAG admit the same number of outgoing refinements. Asymmetry in refinement structure is common:

Configuration 1 might refine into 2 admissible children
Configuration A might refine into 4 admissible children
Configuration D might admit no further refinements (terminal node)

Probability weights must respect this asymmetry while satisfying structural constraints (additivity, conservation, multiplicativity).

5. Examples: Root Selection Determines Probability Context

Scenario: A spin-½ particle in superposition |↑⟩ + |↓⟩ will resolve to one definite state.

Question: “What is the probability of observing spin-up?”

Root: Current configuration (superposition state)

Structure:

Parent node: |↑⟩ + |↓⟩
Child nodes: |↑⟩ (spin-up) and |↓⟩ (spin-down)
No structural distinction between children → equal weights by symmetry
Probability: 50% for each outcome

graph TD
    S["Superposition<br/>ψ = &#124;↑⟩ + &#124;↓⟩"] --> U["&#124;↑⟩<br/>w = 0.5"]
    S --> D["&#124;↓⟩<br/>w = 0.5"]
    
    style S fill:#e1f5ff
    style U fill:#d4edda
    style D fill:#d4edda

Example 2: Comparing Alternative Outcomes

Scenario: An experiment has been performed. Two outcomes were possible: A (electron detected at position x₁) or B (electron detected at position x₂).

Question: “What is the relative probability of outcome A vs B?”

Root: Most recent shared ancestor before branching (preparation state)

Structure:

Parent node: Prepared state |ψ⟩
Child nodes: Detection at x₁ or detection at x₂
Weights: |⟨x₁|ψ⟩|² and |⟨x₂|ψ⟩|²
Probability ratio: Determined by amplitude structure at preparation

graph TD
    P["Preparation State<br/>ψ"] --> A["Detect at x₁<br/>w = &#124;⟨x₁&#124;ψ⟩&#124;²"]
    P --> B["Detect at x₂<br/>w = &#124;⟨x₂&#124;ψ⟩&#124;²"]
    
    style P fill:#e1f5ff
    style A fill:#d4edda
    style B fill:#f8d7da

Key insight: The probability is not about which outcome “really happened” (both are realized). It is about the structural weight of each refinement path relative to the shared parent.

6. Relation to Born-Rule Weighting

The formal recovery of quadratic (Born-rule) weighting is established in Paper B5 — Selection and Stability of Quadratic Weighting.

What B5 Shows

B5 demonstrates that quadratic weighting (w ∝ |α|²) is the uniquely stable representational scheme under the structural constraints of:

Branching (multiple admissible refinements)
Reconciliation (previously diverged branches rejoining)
Composition (independent systems combining)

Why Quadratic Form?

The key result from B5:

Linear weighting fails under reconciliation (violates conservation)
Higher-order weighting accumulates representational mismatch
Arbitrary schemes violate structural invariance
Only quadratic weighting satisfies all structural requirements and remains stable

This is a selection result, not a logical necessity. Quadratic weighting is selected by stability pressure, not imposed as an axiom.

What This Guide Adds

This guide explains the conceptual framing that underlies the formal result:

Probability as structural, not epistemic
Roots and conditioning
Symmetry restriction, not microstate counting
DAG structure as the foundation

For the mathematical derivation, see Paper B5.

7. Mechanism-Level Foundations

The structural interpretation of probability builds on mechanism-level work in the M-CON subseries (Constraint Composition & Internal Modes).

Relevant Papers

M-CON2 — Internal Modes and Quantisation
Establishes how internal degrees of freedom emerge as discrete modes within constraint fibres. Discreteness arises from reconciliation stability, not axioms.
M-CON3 — Mode Multiplicity and Internal Symmetry
Explains how internal multiplicity (multiple admissible internal configurations) contributes to branching structure and symmetry-based weighting.

These papers provide the mechanism-level justification for why configurations admit multiple refinements and how symmetry restrictions arise structurally.

8. Common Misconceptions

❌ Misconception 1: “Probability is about not knowing which branch you’re in”

Correction: Probability in CD is not about observer self-location or epistemic ignorance. It is a structural measure on the DAG of admissible configurations, independent of observer perspective.

❌ Misconception 2: “Probability weights count microstates”

Correction: Weights arise from symmetry restriction, not microstate counting. Even a single configuration can have nontrivial branching weights if it admits symmetry-breaking refinements.

❌ Misconception 3: “Conditioning erases history”

Correction: Conditioning renormalises weights relative to a new root, but does not erase admissibility relations. The DAG structure remains intact; the reference frame changes.

❌ Misconception 4: “All branches have equal probability by symmetry”

Correction: Equal weights arise only when no structural distinction exists between refinements. Asymmetric admissibility structure produces asymmetric weights (e.g., |α₁|² ≠ |α₂|² when amplitudes differ).

9. Summary: Probability Is Structure

In Cohesion Dynamics:

✅ Probability is defined on the DAG of admissible configurations
✅ Weights track symmetry restriction under refinement
✅ Roots determine the reference frame for probability assignments
✅ Conditioning renormalises weights without erasing history
✅ Quadratic (Born-rule) weighting is selected by stability, not assumed

This structural account avoids observer-centric language, temporal uncertainty, and branch counting, making CD probability fully native to the substrate ontology.

Note on Document Status

This is a conceptual orientation guide, not a research paper. It introduces no new axioms, makes no necessity claims, and does not derive formal results. Its purpose is to clarify how to think about probability in CD to reduce misinterpretation of the formal papers.

For rigorous treatments, always refer to the formal A-, M-, and B-series papers.