Paper M2 — Formal Constraint Dynamics and Emergent Constructors

Version: v0.1
Theory: Cohesion Dynamics (CD)
Level: M (Formal Metaphysics)

Abstract

This paper provides a formal, non-teleological account of how stable, self-maintaining, and replicative structures (constructors) emerge from constraint systems acting on informational states. Building on the conceptual foundations of CD, we define admissibility, mismatch, tolerance, and precedence entirely at the level of relations over state space, without invoking time, computation, or physical execution. We show that constructor-like behaviour arises as a structural consequence of local error tolerance combined with precedence-restricted admissible transitions, and that higher-order constructors follow naturally via lifted constraint structure. This formal layer supplies the minimal mathematical substrate upon which physical instantiations (e.g. spacetime, quantum dynamics) may later be derived.

1. Scope and Intent

This paper does not propose a physical model, substrate, or dynamics in time.
Instead, it formalises:

What it means for informational states to be cohesive
How error tolerance enables scalability
Why local precedence produces persistence, repair, and replication
How hierarchical constructors arise without additional primitives

All physical interpretations (time, locality, energy, fields) are deferred to later papers.

2. Informational State Space

Let $S$ denote the set of informational states.
An element $s \in S$ is a complete informational configuration.

No metric, topology, or temporal ordering is assumed on $S$ .

3. Constraint Systems and Admissibility

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

We write:

s \vdash_{\mathcal{C}}

to mean that state $s$ satisfies $\mathcal{C}$ .

Constraint satisfaction is binary at the ontological level: a state either satisfies the constraint system or it does not.

4. Mismatch and Tolerance

To enable resilience and scalability, constraint satisfaction is mediated through tolerance.

Let:

$M : S \to \mathbb{R}_{\ge 0}$ be a mismatch measure
$W$ be a tolerance bound

We define admissibility as:

s \vdash_{\mathcal{C}} \iff M(s) \le W

Mismatch is not a violation of constraint logic; it is a bounded allowance that enables error correction and prevents brittle collapse.

Without tolerance, any deviation would immediately partition the state space and preclude stable structure.

5. Admissible Transitions

Let $\Delta s$ denote a candidate state difference.

Define the admissible transition set:

\mathcal{A}(s) = \{ \Delta s \mid s + \Delta s \vdash_{\mathcal{C}} \}

Important clarifications:

$\mathcal{A}(s)$ does not imply execution
All elements of $\mathcal{A}(s)$ are equally valid continuations
Transitions outside $\mathcal{A}(s)$ correspond to partitioned domains, not forbidden reality

The existence of $\mathcal{A}(s)$ is structural, not procedural.

6. Precedence and Minimal Mismatch

We now introduce the key mechanism discovered empirically in the constructor emergence programme.

Define precedence-restricted admissibility:

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

This expression does not describe a choice or optimisation performed by the system.

Instead, it characterises the subset of admissible continuations that remain maximally cohesive under repeated constraint application.

All other admissible transitions remain ontologically valid but rapidly partition from the cohesive domain.

7. Persistence as Structural Invariance

A structure is persistent if repeated application of precedence-restricted admissibility yields invariant or cyclic equivalence classes of states.

Formally, a subset $P \subset S$ is persistent if:

\forall s \in P,\quad s + \Delta s^* \in P

Persistence is therefore not imposed, but emerges from constraint geometry.

8. Self-Repair and Error Correction

Let $s \in P$ and let $s'$ be a perturbed state such that:

M(s') > M(s)

If $s'$ remains admissible, then precedence guarantees:

M(s' + \Delta s^*) < M(s')

Thus, repair follows automatically from local precedence.

No additional repair rule is required.

9. Replication via Template Constraints

Consider two admissible states $s_1, s_2$ whose interaction admits joint admissible transitions.

If $s_1$ belongs to a persistent equivalence class $P$ , then precedence-restricted admissibility constrains $s_2$ toward $P$ .

Replication occurs when:

s_2 + \Delta s^* \in P

The original structure remains intact while inducing a new instance.

Replication is therefore a constraint propagation phenomenon, not copying in time.

10. Hierarchical Constructors

Let $[s]$ denote an equivalence class of states under cohesion.

We lift mismatch:

M([s]) := \text{aggregate mismatch over constituents}

Define lifted admissibility:

\mathcal{A}([s]) = \{ \Delta [s] \mid [s] + \Delta [s] \vdash_{\mathcal{C}} \}

Define lifted precedence:

\Delta [s]^* \in \arg\min_{\Delta [s]} M([s] + \Delta [s])

This recursive structure enables constructors of constructors without new axioms.

11. Summary of Formal Results

From the above definitions alone, we obtain:

Persistence (Phase 11)
Self-repair (Phase 12)
Replication (Phase 12)
Hierarchical composition (Phase 14)

All arise from:

Tolerant constraint satisfaction
Admissible transition sets
Precedence toward minimal mismatch

No appeal is made to:

time
computation
energy
forces
global optimisation

12. Relation to Physical Theories

This paper establishes necessary structure, not sufficient physical detail.

Later papers instantiate:

State space as a substrate (Paper A)
Continuum limits (Paper B)
Mode structure and quantum behaviour (future M/A papers)

The present formalism constrains what such instantiations must respect.

13. Conclusion

Constructors are not fundamental objects.
They are fixed points of tolerant constraint systems under precedence-restricted admissibility.

Once tolerance and precedence exist, constructor emergence is unavoidable.

This completes the formal metaphysical layer required for Cohesion Dynamics.