Continuity and Identity Without Objects

Most physical theories implicitly assume that objects persist through time, that states “change,” and that identity is a primitive feature of reality. Cohesion Dynamics rejects these assumptions and replaces them with a structural account based on admissibility and refinement.

This guide explains how identity, persistence, continuity, and probability emerge from the admissibility structure without presupposing any of these concepts. It serves as a conceptual bridge between the foundational ideas in Information & Constraints and the formal treatment of probability in CD.

This page introduces no new axioms, no formal derivations, and no necessity claims. It is explanatory only.

1. Why Identity Is Usually Misunderstood

Most physical theories begin with objects that persist through time. An electron at time $t_1$ is assumed to be “the same electron” at time $t_2$ . States are thought to “change”—green becomes red, spin-up becomes spin-down—with identity preserved through the transformation.

This framing treats identity as primitive: something carried by the object itself, independent of structure or relationships.

Cohesion Dynamics rejects this picture entirely.

In CD:

Configurations are immutable — a configuration does not become another configuration
Identity is inferred, not carried — “sameness” is a relation between configurations, not a substance
Time is not fundamental — temporal ordering arises from admissible refinement structure

The goal of this section is not to argue that the conventional view is “wrong,” but to show that it rests on assumptions that CD does not share. Understanding this difference is essential for making sense of branching, probability, and continuity in CD.

Cohesion Dynamics is built on three structural primitives:

Configurations — immutable informational states satisfying all active constraints
Constraints — relational requirements that determine which configurations are admissible
Admissible refinements — structural relations connecting configurations that satisfy constraints

Key principle: A configuration does not become another configuration. Instead, configurations exist as nodes in a refinement structure, related by admissible transformations.

When we say “the system evolves,” we mean:

Earlier configurations exist
Later configurations exist
Those configurations are related by admissible refinements

There is no mutation. There is only ordering.

“Evolution” in CD is the ordering of configurations by admissible refinement relations, not the replacement of one state by another.

This is not metaphysics—it is a structural way of understanding change without assuming persistence.

3. Continuity Without Persistence

Question: If configurations don’t persist, what does “continuity” mean?

Answer: Continuity is a structural relation, not a property of objects.

Two configurations are continuous if there exists an admissible continuation history (a finite sequence of admissible refinements) connecting them. No admissible continuation history means no continuity, and therefore no identity relation.

Definition: Continuity

Configurations $C_1$ and $C_2$ are continuous if and only if there exists a sequence of admissible refinements linking them (an admissible continuation history).

Continuity replaces the notion of “persistence” with a graph-theoretic ordering:

Nodes are configurations
Edges are admissible refinements
Continuity is connectivity via admissible continuation histories

Key insight: Identity is not a substance carried through time. It is a relation inferred from shared admissible ancestry.

Example: Memory Without Persistence

Consider an observer who “remembers” an earlier state. In conventional terms, we say the observer “persists” and “carries information forward.”

In CD:

The earlier configuration exists (with certain constraint structures)
The later configuration exists (with constraint structures encoding the earlier state)
Both configurations belong to the same admissible refinement class

Memory is not “carried” by a persisting object. It is an internalised constraint that appears in later configurations when those configurations are admissible refinements of earlier ones.

There is no “thing” that moves through refinements. There are only configurations related by admissibility.

4. Identity as Inferred, Not Preserved

When we say “this is the same system,” what do we mean structurally?

Identity in CD means: configurations belong to the same continuity class—a maximal set of configurations related by admissible refinement.

Identity is not preserved; it is inferred from structure.

Two Examples

A configuration $C_1$ (electron in superposition) refines to $C_2$ (electron with definite spin). Are they “the same system”?

Conventional answer: Yes, because the electron persists
CD answer: They are continuous because $C_2$ is an admissible refinement of $C_1$

The difference is subtle but critical: CD does not assume a persisting electron. It only requires structural continuity via admissibility.

Example 2: Disconnected Configurations

Two configurations with no admissible transformation relation between them are not continuous. They belong to distinct continuity classes and admit no identity comparison.

This is not exotic—it happens whenever configurations cannot be related by constraint-satisfying transformations. No shared admissible ancestry means no probability relation, no identity relation, and no continuity.

5. Branching Does Not Duplicate Objects

One of the most persistent sources of confusion in interpreting CD (and Everett-style interpretations generally) is the intuition that branching duplicates objects.

This intuition assumes:

A persisting object exists before branching
Branching “splits” that object into multiple copies
Each copy continues independently

Cohesion Dynamics rejects step (1), which makes steps (2) and (3) incoherent.

What Branching Actually Means

In CD, branching occurs when a configuration admits multiple exclusive admissible refinements.

Structurally:

Parent configuration $C$ exists
Multiple refinements $C_A$ and $C_B$ are admissible
No single refinement is uniquely determined by constraints

There is no object to duplicate. The parent configuration does not “split.” It simply has multiple admissible successors.

Key Clarification: Identity Does Not Split

When branching occurs:

Identity does not split — there was never a persisting object to split
Continuity forks — multiple admissible continuation histories exist from a shared ancestor

The configurations $C_A$ and $C_B$ are both continuous with $C$ , but not with each other (they are exclusive refinements). There is no “original” that becomes two copies. There are only distinct configurations related to a common ancestor.

Why This Matters

This structural framing dissolves “many copies of me” intuitions. There is no “me” that gets copied. There are only configurations in a refinement structure, some of which encode observer-like constraint patterns, and some of which do not.

Branching is forking in the admissibility graph, not duplication of objects.

6. Why Probability Requires Shared Ancestry

Probability in CD is defined on the directed acyclic graph (DAG) of admissible configurations.

Key principle: Probability is only meaningful between configurations that share a common admissible ancestor.

Why?

Because probability measures relative restriction in refinement structure. Without a shared parent, there is no common reference point from which to compare admissible continuation histories descending from it.

Two Cases

Case 1: Shared Ancestry → Probability Defined

Configurations $C_A$ and $C_B$ both refine from parent $C$ . Their relative probabilities reflect the structural restriction of each admissible continuation history relative to $C$ .

graph TD
    C[Parent Configuration C] --> CA[Refinement A<br/>w ∝ symmetry preserved]
    C --> CB[Refinement B<br/>w ∝ symmetry broken]
    
    style C fill:#e1f5ff
    style CA fill:#d4edda
    style CB fill:#fff3cd

Caption: Shared ancestry allows probabilistic comparison via structural weighting.

Case 2: No Shared Ancestry → No Probability

Configurations $C_X$ and $C_Y$ belong to disjoint continuity classes (no admissible continuation history connects them). There is no probability relation between them.

This is not a failure of the theory—it is a structural constraint. Probability requires continuity, and continuity requires admissible ancestry.

Implications

This structural constraint prevents meaningless questions like:

“What is the probability I am in universe A vs universe B?” (when A and B have no shared ancestor)
“What is the probability of being this electron vs that photon?” (when no admissible transformation connects them)

Probability is anchorage-relative. It presupposes a reference configuration and compares refinements descending from it.

7. Conditioning as Root Selection

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

Changing the root changes the probability assignments. This is not a bug—it is a feature.

What Conditioning Does

Conditioning selects a new root and renormalises weights relative to that root.

Critical clarification: Conditioning does not erase history. The admissibility structure remains unchanged. What changes is the reference frame for probability calculations.

graph TD
    R[Original Root] --> A[Configuration A<br/>w = 0.7]
    R --> B[Configuration B<br/>w = 0.3]
    A --> A1[Continuation from A]
    A --> A2[Continuation from A]
    B --> B1[Continuation from B]
    
    style R fill:#e1f5ff
    style A fill:#d4edda
    style B fill:#f8d7da

Caption: Conditioning on A makes A the new root. Probabilities of A1 and A2 are calculated relative to A.

Why This Is Not Observer-Dependent

This framing is purely structural. No observer is required to “choose” a root. The root is determined by the reference configuration from which probability is being calculated.

When we say “condition on the observed outcome,” we mean:

Select the configuration encoding the observed constraint structure as the new root
Renormalise weights relative to that configuration

This is no more observer-dependent than choosing a coordinate origin in geometry.

8. From Continuity to CIUs (Forward Reference Only)

This guide has deliberately avoided introducing Commit-Irreducible Units (CIUs) because CIUs are not needed to understand continuity, identity, or branching at this conceptual level.

However, readers proceeding to formal papers (A-series, M-series, B-series) will encounter CIUs, and it is worth clarifying their relationship to the ideas presented here.

What CIUs Are

CIUs are maximal continuity classes that reconcile as a unit. They are defined by:

Admissibility coupling — internal degrees of freedom cannot resolve independently
Reconciliation stability — the unit reconciles jointly or not at all

CIUs are not primitives. They are organisational conveniences that emerge from the admissibility structure.

How CIUs Relate to Continuity

CIUs should be understood as equivalence classes under admissibility:

A CIU is a set of configurations that always refine together
CIU boundaries are determined by admissibility coupling, not spatial or temporal scale

Key point: Everything in this guide applies to CIUs. CIUs do not persist, do not carry identity, and do not “split” during branching. They are simply units of continuity in the refinement graph.

Forward Reference

For the formal definition of CIUs and their role in substrate mechanics, see:

Paper A (Substrate Mechanics) — Formal specification of CIUs and admissibility
Glossary — CIU — Canonical definition

For how CIUs relate to branching and quantum structure, see:

Paper B-series (Structural Superposition) — Derivation of branching from reconciliation constraints

Summary: Identity, Continuity, and Probability Without Objects

Let’s consolidate the core insights:

Configurations are immutable — change is relation, not replacement
Continuity is structural — defined by admissible continuation histories (finite sequences of admissible refinements), not persistence
Identity is inferred — it is a relation between configurations, not a substance
Branching forks continuity — it does not duplicate objects
Probability requires shared ancestry — it measures relative restriction in refinement structure
Conditioning renormalises roots — it does not erase history
CIUs are maximal continuity classes — they are not external agents or persisting objects

These ideas replace conventional assumptions about persistence, identity, and self-location with a structural account grounded entirely in admissibility and refinement.

This framing prepares the reader for:

Probability in CD — where probability is formalized as structural weighting
B-series papers — where branching, reconciliation, and closure are derived formally
M-series papers — where mechanisms governing admissibility are specified

What This Page Does Not Include

This page intentionally avoids:

Formal definitions of CIUs — see Paper A and the Glossary
Probability weights or Born rule mathematics — see Paper B5 and Probability in CD
Observer self-location language — CD probability is structural, not epistemic
Dynamics, time, or collapse — these are emergent representational structures
Necessity or derivation claims — this is a conceptual guide, not a research paper

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

Prerequisites and Further Reading

Prerequisites

Before reading this page, readers should ideally be familiar with:

Information and Constraint — Foundational ontology and how constraints create meaning

Recommended Next Steps

After reading this page, proceed to:

Probability in CD — Structural account of probability and Born-rule weighting
Research Programme — Understanding how series fit together
Paper A: Substrate Mechanics — Formal specification (for readers seeking rigor)

Optional Advanced Reading

For formal derivations and mechanism-level details:

Paper B-series — Structural superposition and representational consequences
Paper M-series — Formal mechanisms governing cohesion and reconciliation
Glossary — Canonical definitions of Admissibility, Configuration, Refinement, CIU

Note on Document Status

This is a conceptual orientation guide, not a research paper. It introduces no new axioms, makes no necessity claims, and derives no formal results. Its purpose is to build intuition for how identity and continuity work in Cohesion Dynamics.

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