Marc Maibom

The nature of time is one of the fundamental open questions of philosophy and the natural sciences. In physics, time is usually treated as a fundamental dimension of spacetime or as an external parameter of dynamical equations. Philosophical approaches often consider time as an ordering of events or as an emergent property of dynamical processes. This paper develops a structural approach to the emergence of time. The starting point is the analysis of systems that must preserve their identity under real transformation. This analysis was developed within the framework of the organizational principle La Profilée, which describes the structural conditions under which persistence and change can be coupled within a system. From this investigation there emerges a minimal architecture of dynamical systems with nontrivial identity: a state space, a transformation structure, and an identification of states. It is shown that complete transformation symmetry is incompatible with nontrivial identity under dynamics. Transformation under nontrivial identity cannot be fully symmetric. The resulting structural asymmetry necessarily generates a directed order in the transformation space. This directed transformation order constitutes the minimal structural form of time. Furthermore, it is shown that nontrivial identification under dynamics enforces a finite integration capacity. From this limitation arises a dimensionless load quantity τ = R / I which describes the intensity of identity-relevant transformation relative to the integration capacity of a system. This quantity is called structural time and measures the structural load imposed on a system by transformation, but not the existence of time itself. On this basis, a structural mechanics is developed that describes how integration deficits arise and lead to structural reorganizations. The resulting model distinguishes three structural levels of dynamical systems: • Time as directed transformation order • Structural time as the intensity of identity-relevant dynamics • Structural mechanics as the dynamics of integration capacity

The paradox of the Ship of Theseus addresses a classical problem: how identity can persist while components change. The complementary intuition that reality is fundamentally characterized by transformation appears to stand in tension with persistence. This paper approaches the problem structurally. If real transformation occurs and identity remains non-trivial, the transformation structure of the system cannot be fully symmetric. Transformation must therefore be selective. Selective transformability induces a reachability relation between states. This reachability structure generates a partial ordering of states. This ordering constitutes structural time. The Ship of Theseus thus illustrates not merely a paradox of identity but the minimal structural conditions under which persistence, transformation, and temporal order can coexist.

The paradox of the Ship of Theseus addresses a classical problem: how identity can persist while components change. The complementary intuition that reality is fundamentally characterized by transformation appears to stand in tension with persistence. This paper approaches the problem structurally. If real transformation occurs and identity remains non-trivial, the transformation structure of the system cannot be fully symmetric. Transformation must therefore be selective. Selective transformability induces a reachability relation between states. This reachability structure generates a partial ordering of states. This ordering constitutes structural time. The Ship of Theseus thus illustrates not merely a paradox of identity but the minimal structural conditions under which persistence, transformation, and temporal order can coexist.

This paper consolidates the minimal structural architecture of non-trivial persistence under real transformation. Assuming (i) a non-empty state space, (ii) admissible transformations containing at least one real transformation, and (iii) the meaningful assertion of non-trivial invariant identity, it is shown that complete transitivity of the transformation structure is incompatible with persistence. From this incompatibility follows a necessary structural asymmetry: transformation must act selectively. This selectivity induces orbit structure, a preorder of reachability (structural time), and a constraint on admissible transformation sequences. The argument is strictly conditional and does not claim a global ontology. It establishes only the structural conditions under which non-trivial identity under real transformation can coherently persist.

This work investigates whether structural direction — and thereby a minimal notion of time — can arise solely from the conditions required for coherent identity under real transformation. The starting point is deliberately minimal. No physical theory, no metric time, no thermodynamic assumptions, and no ontological substrate are presupposed. The framework consists only of: • a non-empty set of states, • a semigroup of realizable transformations, • and an induced identity relation obtained canonically via largest bisimulation. Within this minimal architecture it is shown that: (1) Non-trivial identity under real transformation excludes complete pair-collapseability. (2) The induced quotient dynamics generates strongly connected components. (3) Condensation of these components yields an acyclic directed graph. (4) A partial order necessarily arises. (5) If at least one transformation is non-permutational at identity level, structural direction is unavoidable. From this directed structure a composite notion of structural time is derived, consisting of an ordinal component (partial order), and an intensity component defined as a structural load ratio. Irreversibility is not derived from entropy, probability, or thermodynamic assumptions. Instead, structural asymmetry is shown to be a necessary condition of any system in which identity remains determinable under real transformation. The claim is strictly conditional: Any non-trivially evolving universe satisfying these structural conditions must instantiate structural asymmetry.

This work investigates the minimal structural conditions of determinable identity under real transformation. The starting point is neither a comprehensive ontology nor a physical model, but a deliberately reduced formal framework: a non-empty set of states, a set of realizable transformations, and a state-based identity mapping. Under these assumptions it is shown that non-trivial identity, insofar as it is to remain determinable under real dynamics, is incompatible with complete pair-collapseability. Total dynamic mixing of all states is structurally incompatible with well-defined identity. Furthermore, the algebraic structure of the identity mapping is made explicit: it induces a congruence relation of transformational action, enables a factorization of the dynamics on the quotient space, and generates—via strongly connected components—a partial order structure. Under additional intra-class dynamics, a minimal functional three-part structure arises from identity structure, variation, and merge-exclusion. In the further course, the ontological consequence is formulated: determinable existence under real transformation implies selective stabilization. The resulting directed order structure is interpreted as structural time and serves as the basis for a further independent paper. The claim is strictly conditional and purely structural. No derivation of physical laws is claimed; rather, a necessary condition for any form of determinable existence under change is formulated.

This paper presents La Profilée (LP). LP is described as an organisational ordering principle. It addresses the question under which structural conditions an entity that claims duration can remain identifiable as the same entity while undergoing change. LP does not describe how systems behave in practice. It does not attempt to predict outcomes. It does not provide a model of causation. It does not offer prescriptions for action. Its aim is narrower: it specifies structural conditions that must be present if identity is to be preserved while change occurs. LP applies only to entities that claim duration. Not every configuration falls within this scope. Some configurations are momentary. Some are purely event-based. Some do not claim continuity beyond a single occurrence. LP is not formulated for such cases. The structural insight that led to LP first appeared in the context of modular design architecture. While examining how an object could remain recognisable while allowing exchangeable components, a structural distinction became visible. That distinction is abstracted in this paper and formulated independently of its original design context. The following sections restate the reference definition of LP, clarify its scope, describe its structural roles, and develop clarifications concerning separation, coupling, duration, legibility, and structural time. A dedicated identification protocol is provided to prevent post-hoc assignment of roles. The paper concludes with a structurally bounded invitation to examination and counterexample. This preprint presents the system-level articulation of La Profilée. The formally derived minimal structural conditions of non-trivial persistence under real transformation (see related preprint) correspond structurally to the Frame–Module–Coupling distinction developed here.

This text establishes the minimal structural condition of non-trivial persistence under real transformation. Starting from a minimal existence assumption, it is shown that non-trivial identity presupposes difference and minimal invariance. Under the condition of real transformation, it is formally proved that persistence enforces the non-transitivity of the transformation structure. From this it follows necessarily that the state space is selectively partitioned into stability classes. Persistence under real transformation is therefore only possible if transformation acts in a structurally asymmetric manner. The statement is strictly conditional and claims no global ontology, but a necessary structure wherever non-trivial identity under real transformation is meaningfully asserted.