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Cohesion Dynamics (CD)

Paper A-NET — Relational Closure and Network Semantics in Cohesion Dynamics

Abstract

Cohesion Dynamics operates on Cohesive Informational Units (CIUs) and admissible closure relations between them. While prior foundational papers define closure, mismatch, admissibility, and tolerance, the relational structure implied by these mechanisms has remained implicit.

This paper formalises the network semantics of Cohesion Dynamics: the existence and properties of reconciliation relations, their availability, and the structure of reconciliation chains. No spacetime substrate, vacuum adjacency, or background graph is assumed. Instead, all relational structure arises solely from admissible interaction between CIUs.

These definitions establish a minimal relational substrate upon which representational constructs such as distance, geometry, and gravity may later be summarised, without introducing new ontology.

1. Scope and Intent

1.1 Purpose

This paper exists to make explicit the relational primitives already required by Cohesion Dynamics, but previously treated informally.

Specifically, it defines:

  • Reconciliation relations between CIUs
  • Admissibility and availability of such relations
  • Reconciliation chains
  • Minimal network invariants

These concepts are introduced without invoking space, time, geometry, or propagation as ontological entities.

1.2 Non-Scope

This paper does not:

  • Introduce new axioms
  • Propose gravitational, geometric, or cosmological models
  • Perform coarse-graining or empirical matching
  • Define distance, curvature, or force

Those belong to downstream representational series.

2. Ontological Commitments

This section enumerates what exists at the relational level.

2.1 Cohesive Informational Units (CIUs)

CIUs are as defined in Paper A:

  • Finite informational structures
  • Governed by internal constraints
  • Capable of closure under admissibility
  • Carrying mismatch during relaxation

No modification to the CIU definition is made here.

2.2 Reconciliation Relations

A reconciliation relation exists between CIUs (A) and (B) iff:

  • Mutual constraint interaction is admissible under tolerance (W)
  • Closure between (A) and (B) is possible in principle

Reconciliation relations are:

  • Not spatial
  • Not persistent unless maintained
  • Not edges in a pre-existing graph

They exist only while admissibility holds.

3. Admissibility and Availability

3.1 Availability as a Relational Primitive

Each admissible reconciliation relation carries an availability:

A(A↔B)∈(0,1]A(A \leftrightarrow B) \in (0,1]

Availability characterises:

  • Ease of reconciliation
  • Typical closure delay
  • Robustness under perturbation

Availability is:

  • Constrained by tolerance (W)
  • Influenced by CIU constraint saturation and symmetry
  • Not a force, field, or flow

3.2 Absence of Vacuum Relations

Where no CIU exists, no reconciliation relation exists.

There are:

  • No vacuum nodes
  • No background adjacency
  • No implicit relations through empty space

All relational structure is CIU-mediated.

Lemma 3.1 — Absence of Availability Implies Absence of Interaction

If no admissible reconciliation relation exists between a CIU (A) and any CIU (B), then no reconciliation, closure, mismatch transfer, or provenance extension involving (A) and (B) occurs.

In particular:

  • No influence propagates partially or unsuccessfully
  • No state accumulates downstream
  • No closure delay, distance, or geometric bookkeeping is defined

Absence of availability is not obstruction or absorption; it is the absence of relational structure.

Proof sketch.
Reconciliation chains exist only as sequences of admissible relations. If no admissible relation exists, no chain may be formed. Since closure, mismatch redistribution, and provenance extension are defined only along such chains, none may occur. ∎

4. Reconciliation Chains

4.1 Definition

A reconciliation chain (\chi) is an ordered sequence of admissible relations:

C0↔C1↔⋯↔CnC_0 \leftrightarrow C_1 \leftrightarrow \dots \leftrightarrow C_n

Such chains:

  • Are contextual and contingent
  • May appear or vanish as admissibility changes
  • Are not trajectories through space

4.2 Chain Cost and Delay

Define a chain cost functional:

C(χ)=∑(i,j)∈χf(A(i,j))C(\chi) = \sum_{(i,j)\in\chi} f(A(i,j))

where (f) is a monotonic decreasing function (e.g. (f(x) = -\log x)).

This cost measures reconciliation delay or difficulty.
It is not distance, which is a representational construct introduced later.

5. Saturation and Local Network Effects

5.1 CIU Constraint Saturation

Each CIU carries a scalar constraint saturation:

σ(CIU)≥0\sigma(\text{CIU}) \ge 0

This measures proximity to closure overload.

5.2 Effect on Relations

Increasing saturation:

  • Reduces availability of incident reconciliation relations
  • Biases admissible chains
  • Does not propagate through vacuum

All downstream “influence” effects arise solely from this local relational impact.

6. Minimal Network Invariants

The following quantities may be derived from reconciliation structure:

  • Minimal chain cost between CIUs
  • Number of admissible chains
  • Saturation neighbourhoods
  • Availability asymmetry

These are network invariants, not spatial fields.

They serve as inputs to later representational summaries.

7. Negative Ontology

For clarity and stability, we explicitly deny the existence of:

  • Vacuum CIUs
  • Background graphs or lattices
  • Spatial embedding of relations
  • Fields defined over empty space
  • Propagation through regions without CIUs
  • Intrinsic metric or geometric structure

Any appearance of such constructs downstream must be explicitly representational.

8. Relationship to Other Series

8.1 Relation to Paper A and A-OPS

This paper:

  • Extends operational clarity
  • Introduces no new axioms
  • Makes implicit relational assumptions explicit

8.2 Relation to G-Series

The G-series:

  • Treats distance, geometry, and gravity as representational summaries
  • Must not introduce new relational primitives
  • Operates entirely on structures defined here

8.3 Relation to W-Series

The W-series:

  • Narrows admissibility parameters
  • Does not alter relational structure
  • Applies tolerance to relations defined in this paper

9. Summary

Cohesion Dynamics operates on CIUs and their admissible relations alone.
Networks are not embedded in space; space is a summary of networks.

By fixing relational semantics at the substrate level, this paper enables gravity, geometry, and cosmology to be developed as effective descriptions without ontological ambiguity.

End of Paper A-NET