Trajectory-Primitive Systems:

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Trajectory-Primitive Systems:

A Formal Framework for Existence Without State Ontology

Author

James B. Chapman

Xhe-CarpenXer

Abstract

We present a formal framework in which trajectory, rather than state, object, or symbol, is the sole ontological primitive. A trajectory is defined as sustained transformation under constraint. States are shown to be observer-induced projections with no independent ontological status. We introduce baseline symmetry, constraint operators, and termination conditions, and provide an executable semantics that demonstrates existence as continued iteration. The framework applies uniformly across physical, biological, and computational domains and resolves equilibrium, identity, and infinity/null paradoxes without invoking teleology.

1. Motivation

Most formal systems treat state as primitive and process as derivative. However:

  • Physical systems do not exhibit true rest.

  • Biological systems persist only through continuous transformation.

  • Computational systems exist only while executing.

This suggests that existence corresponds to continuation, not configuration. We therefore invert the standard ontology and treat trajectory as primitive.

2. Primitive Definitions

Definition 1 (Trajectory)

A trajectory is a functionally continuous process defined by repeated transformation under constraint.

Formally, a trajectory ( \mathcal{T} ) exists iff:

[
\exists f, C : x_{n+1} = C(f(x_n)) \quad \forall n \in \mathbb{N}
]

and the iteration does not terminate.

No claim is made about convergence or optimization.

Definition 2 (Baseline)

A baseline ( B ) is a reference condition such that:

[
\forall d \in \mathbb{R}, \quad |B + d| = |B - d|
]

The baseline is not an attractor, equilibrium, or null state. It exists solely to define symmetric deviation.

Definition 3 (Constraint)

A constraint ( C : \mathbb{R} \rightarrow \mathbb{R} ) is a shaping operator satisfying:

  • Symmetry: ( C(x) = -C(-x) )

  • Boundedness: ( |C(x)| \le k ) for some ( k \in \mathbb{R}^+ )

  • Non-teleology: ( C ) does not encode a preferred direction

Definition 4 (Termination)

A trajectory terminates when iteration halts:

[
\exists n : x_{n+1} ;\text{undefined}
]

Termination corresponds to non-existence.

3. Core Axiom

Axiom (Trajectory Primitivity)
Existence consists exclusively of non-terminating trajectories.
States are projections induced by observation.

4. Derived Structures

4.1 States as Observational Artifacts

Define an observation operator:

[
O_k(\mathcal{T}) = x_k
]

States exist only relative to ( O ). They have no persistence independent of the trajectory.

4.2 Identity

Two trajectories ( \mathcal{T}_1, \mathcal{T}_2 ) share identity iff:

[
\forall n < N,; \mathcal{T}_1(n) ;\text{and}; \mathcal{T}_2(n) ;\text{are both defined}
]

Identity is continuity, not value equality.

4.3 Laws as Local Invariants

A “law” is any invariant property preserved under constraint application along a trajectory. Laws are descriptive, not prescriptive.

5. Baseline Symmetry

No direction in trajectory space is privileged:

[
\Delta x = \pm d \quad \text{are equivalent relative to } B
]

Thus:

  • No inherent progress

  • No decay

  • No correction mechanism

6. Infinity and Nullity as Resolution Limits

Let resolution ( r ) define observational granularity.

[
\lim_{r \to 0} |\mathcal{T}| = \infty
]

[
\lim_{r \to \infty} |\mathcal{T}| = 0
]

Infinity and nothing are limits of observation, not ontological opposites.

7. Executable Semantics

7.1 Formal Correspondence

Formal Element Implementation
Trajectory Iterator
Constraint Bounding function
Baseline Zero-centered reference
Termination StopIteration

7.2 Reference Implementation (Python)

class Constraint:
    def __init__(self, bound):
        self.bound = abs(bound)

    def apply(self, x):
        if x > self.bound:
            return self.bound - (x - self.bound)
        if x < -self.bound:
            return -self.bound - (x + self.bound)
        return x


class Trajectory:
    def __init__(self, transform, constraint, initial=0.0):
        self.x = initial
        self.transform = transform
        self.constraint = constraint
        self.alive = True

    def step(self):
        if not self.alive:
            raise StopIteration
        self.x = self.constraint.apply(self.transform(self.x))
        return self.x

    def terminate(self):
        self.alive = False

Existence is the ability to call step() indefinitely.

8. Hierarchical Interpretation (Non-Normative)

The framework admits a structural hierarchy:

  1. Source (initiation capacity)

  2. Constraint (shaping structure)

  3. Manifestation (trajectory)

This hierarchy is causal, not evaluative.

9. Comparison to Existing Frameworks

Framework Primitive Limitation
State-space models State Requires rest
Process calculi Process Still state-reliant
Field theory Field Implicit equilibrium
This work Trajectory No static ontology

10. Competing Interpretations

Ontological Interpretation (~70%)

Trajectory is fundamental to reality itself.

Epistemic Interpretation (~30%)

Trajectory is the minimal structure accessible to finite observers.

Both interpretations preserve internal consistency.

11. Implications

  • Eliminates static being

  • Reframes death as termination

  • Removes teleology

  • Grounds meaning as constrained continuation

12. Conclusion

Existence is not what is, but what continues.

A system exists if and only if its trajectory persists. All other constructs—states, laws, identities—are secondary artifacts of observation.

Final Statement

Existence = non-terminating constrained transformation

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