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Cohesion Dynamics (CD)

A-OPS — Operational Semantics of Mismatch, Closure, and Tolerance

Abstract

Cohesion Dynamics (CD) describes physical behaviour in terms of informational constraints, admissibility, and closure rather than forces or equations of motion. While prior A- and M-series papers defined the formal primitives of the theory, the operational semantics of mismatch, relaxation, reconciliation, and tolerance WW have remained partially implicit. This paper clarifies those semantics without introducing new axioms or mechanisms. In particular, we reinterpret mismatch as unassigned informational degrees of freedom, closure as the sole committing operation, and tolerance WW as a finite admissibility window that sustains mutual cohesion during exploratory relaxation. Partition and closure-stable invariance are shown to arise as necessary consequences when reconciliation is inadmissible, not as independent substrate events. This clarification unifies recent simulator developments and M-series results while preserving full backward compatibility with existing axioms.

1. Purpose and Scope

This paper serves a single purpose: to make explicit how the substrate of Cohesion Dynamics behaves during relaxation and attempted reconciliation.

It does not:

  • introduce new axioms,
  • alter the definition of partition or closure,
  • add new conserved quantities,
  • propose specific physical models.

Instead, it clarifies the operational meaning of existing terms—mismatch, constraint, relaxation, closure, and tolerance WW—as they are already used across the theory and simulator.

2. Substrate Non-Agency

The Cohesion Dynamics substrate does not act, optimise, select, or minimise.

There is:

  • no controller,
  • no objective function,
  • no algorithmic search.

The substrate consists solely of:

  • informational configurations,
  • constraints on admissible configurations,
  • rules governing which configurations may be jointly closed.

All apparent dynamics arise from changes in admissibility, not from directed action.

3. Constraints and Admissible Configurations

A constraint specifies a set of configurations that may participate in closure. Constraints do not act on information; they delimit which representations are admissible.

A configuration is admissible if it satisfies all constraints relevant to a candidate closure.

No configuration is privileged prior to closure.

4. Mismatch as Unassigned Information

Mismatch is not excess information and not error. It is information that cannot be stably assigned to any constraint slot under the current admissibility conditions.

Formally, mismatch exists when:

  • no closure-stable representation exists,
  • but multiple provisional representations remain admissible during relaxation.

Mismatch therefore corresponds to under-determination, not violation.

A configuration with mismatch is not unstable; it is non-fixed.

5. Relaxation as Exploratory Admissibility

Relaxation is the process by which provisional configurations are explored within the admissible configuration space.

Key properties:

  • Relaxation does not commit state.
  • Multiple provisional representations may coexist.
  • Mismatch persists as long as no closure requires assignment.

Relaxation is inherently relational: exploration of admissibility depends on which configurations overlap and for how long.

6. Overlap and Mutual Cohesion

For two CIUs to attempt reconciliation, their relaxation windows must overlap sufficiently to allow mutual constraint evaluation.

This overlap is neither instantaneous nor global:

  • there is no global time,
  • no universal synchronisation,
  • only local, relational availability.

Exploration cannot occur in isolation; it presupposes temporary mutual cohesion.

7. Closure as the Sole Committing Act

Closure is the only operation that commits configuration.

At closure:

  • a specific admissible configuration is selected,
  • all under-determination is resolved,
  • mismatch either vanishes or becomes invariant.

There are no committed states prior to closure.

8. Reconciliation and Inadmissibility

When multiple provisional configurations attempt to share closure, two outcomes are possible:

  1. Reconciliation admissible
    There exists at least one closure that preserves mutual cohesion.

  2. Reconciliation inadmissible
    No admissible closure preserves cohesion.

In the second case, partition occurs per AX-PAR.

Partition is not a process; it is a categorical fact about closure admissibility.

9. Closure-Stable Invariance

When reconciliation is inadmissible, any admissible closure must preserve the distinction between configurations.

Consequently:

  • mismatch becomes invariant under all admissible closures,
  • structural distinctions persist categorically.

This is the phenomenon described in M-8 as closure-stable invariance.

No additional substrate event is required.

9a. CIU Boundary Principle

The identity of a Cohesive Informational Unit (CIU) is determined by closure, not by arbitrary spatial or informational boundaries.

CIU Boundary Principle
A CIU is the minimal constraint subgraph that admits repeated closure at a given scale.
When internal closure requires satisfaction of an external constraint, the CIU ceases to exist at that scale and is replaced by a larger CIU that includes the binding constraint.

This principle clarifies:

  • CIU identity is closure-defined: A CIU boundary is where independent closure ends.
  • Scale dependence: CIU identity shifts when closure dependencies change.
  • Structural emergence: When constraints bind (chemistry, fusion, gravity), independent closures merge, and CIU identity is redefined at the composite scale.
  • Non-arbitrariness: CIU boundaries are not imposed externally but arise from constraint-pressure dynamics and closure requirements.

Implications:

  1. Chemistry and bonding: When two previously independent CIUs cannot close independently due to a shared binding constraint, they cease to exist as separate CIUs and form a composite CIU.

  2. Fusion: When constraint binding removes the possibility of independent closure for constituent structures, CIU identity shifts to the fused system scale.

  3. Gravitational collapse: When closure-gradient pressure binds constraints across previously independent regions, CIU identity is redefined at the larger cohesive scale.

This principle grounds structural emergence in closure mechanics and aligns with the original constraint-pressure motivation of Cohesion Dynamics, without introducing new axioms.

10. The Role of Tolerance WW

The tolerance vector WW does not act after closure. It constrains admissibility during attempted reconciliation.

Operationally, WW defines:

  • how much provisional divergence may exist,
  • for how long mutual exploration can persist,
  • whether reconciliation can even be attempted.

WW therefore functions as a finite admissibility window sustaining exploratory relaxation.

If exploratory relaxation cannot remain mutually admissible within WW, reconciliation is inadmissible and partition follows.

11. Rigidity and Zero Mismatch

A configuration with zero mismatch is fully determined.

Such a configuration:

  • admits no internal reallocation,
  • cannot absorb further information without partition,
  • is maximally rigid.

Zero mismatch is therefore not equilibrium but structural brittleness.

12. Summary

This paper clarifies that:

  • Mismatch is unassigned information, not violation.
  • Relaxation explores admissible configurations without commitment.
  • Closure is the only committing operation.
  • Reconciliation is an admissibility question, not a process.
  • Partition is categorical, not dynamical.
  • Tolerance WW is an exploratory admissibility window, not a static threshold.

These clarifications unify existing axioms, simulator behaviour, and recent M-series results without introducing new primitives.

13. Relationship to Other Work

  • A-series: Clarifies but does not modify axioms
  • M-8: Provides the operational basis for closure-stable invariance.
  • M-9: Aligns with symmetry-structured reconciliation without assuming provenance tracking.
  • W-series: Establishes the semantic foundation required for meaningful parameter narrowing.

14. Outlook

A companion paper may introduce minimal toy models to illustrate these semantics concretely. Such models will be illustrative only and will not alter the substrate definitions established here.