Paper M3 — Modes as Emergent Constraint Eigenstructures

Cohesion Dynamics (CD)
Draft v0.2

1. Purpose and Scope

This paper formalises the concept of modes as emergent, discrete structures arising from constrained informational dynamics. The goal is to explain how stable, repeatable, and finite classes of behaviour arise without postulating quantisation, particles, energy levels, or spacetime as primitives.

This paper operates at the M-level (metaphysical formalisation) within Cohesion Dynamics. It builds only on:

Information as ontologically primitive
Exact, binary constraint satisfaction
Tolerance as an additional admission constraint
Local precedence selection among admissible updates

No assumptions are made about physical instantiation, measurement, probability, or the specific dynamics of our universe.

2. The Problem: Why Discrete Structure Requires Explanation

Unconstrained information admits no structure. Every configuration is equally permissible, and no identity can persist.

Purely local constraints, without further structure, also fail to produce discreteness. They permit continuous drift through state space, yielding:

No finite set of stable outcomes
No reproducibility
No basis for functional identity

However, empirical results from the constructor emergence programme demonstrate convergence toward specific, repeatable, and stable configurations. These configurations are neither arbitrary nor continuously variable.

The purpose of this paper is to explain why such discrete structures must arise, and what they are.

3. Informational State Space and Constraints

Let $s \in \mathcal{S}$ denote an informational state.

Let $\mathcal{C}$ denote a constraint system acting on $\mathcal{S}$ .

Constraints do not select outcomes. They define admissibility.

Constraint satisfaction is exact and binary: a state either satisfies $\mathcal{C}$ or it does not.

For a given admissible state $s$ , define the set of admissible updates:

\mathcal{A}(s) = \{ \Delta s \mid s + \Delta s \text{ satisfies all constraints in } \mathcal{C} \}

Tolerance does not soften constraint satisfaction. Instead, tolerance is itself an additional constraint that defines the boundary of a cohesive domain. States exceeding tolerance are not “approximately valid”; they are inadmissible within that domain and result in partitioning into a different domain.

4. Mismatch and Precedence

Define a mismatch function:

M : \mathcal{S} \rightarrow \mathbb{R}_{\ge 0}

where $M(s)$ measures the degree of internal tension or instability within the space of admissible states. Importantly:

$M(s)$ does not measure constraint violation
$M(s)$ is defined only for admissible states

Lower values correspond to greater coherence under the active constraint system.

When multiple admissible updates exist, Cohesion Dynamics resolves them via precedence selection:

\Delta s^* = \arg\min_{\Delta s \in \mathcal{A}(s)} M(s + \Delta s)

This rule is:

Local
Deterministic
Non-teleological

Precedence introduces error-corrective bias without invoking optimisation over inadmissible states or any global objective.

5. Emergence of Modes

5.1 Definition

A mode is an equivalence class of informational states $\mathcal{M} \subset \mathcal{S}$ such that:

For any $s \in \mathcal{M}$ , repeated application of admissible updates under precedence returns the system to $\mathcal{M}$ .

Formally, define the induced update operator:

\Phi(s) = s + \Delta s^*

Then $\mathcal{M}$ is a mode if:

\Phi(\mathcal{M}) \subseteq \mathcal{M}

Modes are therefore invariant structures of constrained update dynamics, not static configurations.

5.2 Modes as Eigenstructures

Modes are analogous to eigenstructures in the following structural sense:

Eigenvectors are invariant under a linear operator
Modes are invariant under constrained, precedence-governed updates

No linearity, metric, or vector space structure is assumed. The term “eigenstructure” is used descriptively, not spectrally.

6. Why Modes Are Discrete

Discreteness arises from three jointly necessary features.

6.1 Constraint Tolerance

Tolerance groups nearby admissible states into bounded equivalence classes. Small perturbations remain within the same domain of admissibility.

6.2 Precedence Selection

Among admissible updates, those reducing mismatch are preferred. Random walk and unconstrained drift are suppressed.

6.3 Basin Formation and Partitioning

Together, tolerance and precedence produce basins of attraction in informational state space. States outside these basins either:

Collapse into an existing basin under precedence, or
Exceed tolerance and become inadmissible, causing domain partitioning

Only a finite number of such basins exist for a given constraint system.

These basins are modes.

7. Modes Are Not Compositions

Modes are not constructed by assembling components.

The same informational constituents may realise different modes
A mode is defined by constraint satisfaction and update invariance, not aggregation

Composition operates within a mode; modes determine which compositions persist.

8. Modes Are Not Energy Levels

Energy does not appear at this level.

Later physical instantiations may associate scalar quantities with transitions between modes, but modes themselves are:

Prior to energy
Prior to quantisation
Prior to measurement

Energy labels transitions; modes define what can exist.

9. Mode Families and Degeneracy

If a constraint system is symmetric, multiple distinct modes may satisfy it equally.

Such collections form mode families.

Small asymmetries or perturbations select specific members, producing apparent randomness despite fully deterministic dynamics.

This provides a structural basis for later probabilistic descriptions without invoking indeterminism.

10. Relation to Constructors

Constructors are entities that maintain or propagate modes.

A constructor is not a mode, but it depends on modes for:

Identity
Function
Reproducibility

Without modes, there is nothing to preserve or replicate. Modes are therefore a prerequisite for constructors, not a consequence of them.

11. Implications and Forward Path

This paper establishes that:

Discrete stable structures arise inevitably from constrained informational dynamics
No primitive quantisation is required
Modes precede energy, particles, and measurement
Constructor behaviour depends on mode structure

Subsequent work will address:

Measurement as basin selection
Probability as mode degeneracy
Physical instantiation of constraint systems

12. Summary

Modes are emergent, discrete, stable eigenstructures of constrained informational dynamics governed by exact constraint satisfaction, tolerance-based domain admission, and local precedence selection.

They explain how finite, repeatable structures arise prior to physics and provide the necessary substrate for constructor emergence and later physical law.