Marc Maibom

This note shows that a minimal order structure — termed structural time — arises as a necessary consequence of restricted transformation. Starting only from a state space and a set of admissible transformations, it is shown that if transformations are neither trivial nor universally mixing, the induced reachability relation cannot be total. This non-totality generates a structural non-simultaneity of states and induces an order relation. Structural time is thus not assumed and does not depend on identity, but emerges directly from the restriction of transformation. This result complements the emergence of identity as an invariant structure under the same transformation constraints.

This paper reconstructs the emergence of La Profilée as a structural principle of persistent systems. The starting point was not a theoretical model but a concrete design process: the development of an independent watch form. Only in the course of working with this form did it become apparent that it contains a structural organisation that reaches beyond the individual object. From the design process there emerged successively an object architecture, a generative design logic, and industrial, economic, and sustainable consequences. Only then was the underlying structure theoretically formulated: as the Structural Law of Persistence — the minimal condition for a system to preserve its identity under real transformation. The path led from form to theory. This paper documents that path.

Persistent systems face a fundamental structural paradox: they must change in order to remain viable, yet excessive change destroys the identity that persistence requires. While many disciplines study stability and transformation, the minimal structural conditions that allow systems to remain identifiable under real transformation remain insufficiently articulated. This paper introduces La Profilée, a structural theory of persistence under change. The theory proposes that persistent systems require a minimal architecture consisting of three functional roles: Frame, the carrier of identity and duration; Modules, the carriers of variation and adaptation; and Coupling, the mechanisms through which transformations are selectively integrated without destabilizing the system. From this architecture follow several structural consequences. Persistent systems must exhibit asymmetric dynamics, in which identity-carrying structures change more slowly than adaptive components. They must also operate under selective admissibility of transformations, since unrestricted transformation collapses all identity distinctions. Together these properties induce a directed transformation structure that generates Structural Time, a temporal ordering emerging from the system's own dynamics. The structural load acting on a system relative to its integration capacity — expressed as τ = R/I — determines whether the system operates within a coherence regime, approaches overload, or collapses. Persistent systems maintain stability within a bounded Persistence Zone between chaos and rigidity; gradual structural drift can move systems toward collapse regimes of chaos, rigidity, or fragmentation. The theory further provides an operational diagnostic layer, LP be[com]ing, which applies this structural architecture to the analysis of real systems. Originally derived from a modular design problem in watchmaking, La Profilée is presented as a general organizational principle observable across biological, technological, organizational, and social systems. La Profilée therefore offers a unified structural account of how complex systems sustain identity through change and provides a diagnostic framework for analyzing systemic stability, innovation capacity, and collapse.

This paper consolidates a series of results concerning the structural conditions under which identity can persist under real transformation. Starting from a minimal representation of a system as a state space equipped with admissible transformations and an identity assignment, it is shown that unrestricted transformation collapses all non-trivial identity distinctions. Persistent identity therefore requires the admissible transformation structure to be selectively restricted. This structural selectivity induces asymmetry in the reachability structure of states. The resulting transformation topology yields a directed acyclic condensation structure whose induced ordering constitutes the minimal structural form of temporal direction. These results together form the Structural Law of Persistence, describing the minimal structural architecture required for persistent identity under real transformation. The architecture corresponds to the structural principle termed La Profilée, consisting of stable identity carriers, transformable elements, and regulated coupling between them. An operational diagnostic layer, LP be[com]ing, applies this structural law to the analysis of real systems.

Physical theories describe dynamical evolution but typically treat time as a fundamental parameter. This work develops a structural account in which temporal direction and persistence arise from the topology of admissible transformations in persistent systems. Persistent systems are represented as (S,G,I), where S is a state space, G⊆End(S) the admissible transformations, and I an identity relation invariant under admissible transformations. From these minimal assumptions it follows that persistent differentiation cannot be maintained in fully symmetric transformation structures. Asymmetric reachability relations necessarily generate irreversible transitions between transformation domains. These domains form strongly connected components whose condensation graph induces a partial order that defines structural temporal direction. Irreversible transitions progressively reduce the set of reachable future domains, generating a structural entropy gradient. Together with energy-driven transformation dynamics this leads to a universal persistence boundary that constrains the stability of physical systems. The theory introduces a multidimensional concept of structural time consisting of ordering time, structural load time, and integration time. These quantities determine the conditions under which physical systems can maintain identity while undergoing transformation.

Time is central to nearly all scientific descriptions of the world, yet its structural origin remains unclear. Most physical theories treat time either as a primitive parameter or as a geometric dimension of spacetime. While these approaches successfully describe temporal dynamics, they rarely explain why temporal order exists at all. This paper develops a structural account of time derived from the minimal requirements of systems capable of persistence under transformation. The analysis begins with a minimal representation of persistent systems consisting of a state space, a set of admissible transformations, and an identity relation. It is shown that persistence requires selective admissibility of transformations: if every transformation is allowed, identity distinctions collapse and persistent structure becomes impossible. Admissible transformations induce a reachability topology whose strongly connected components represent regions of reversible transformation. Collapsing these components produces a directed acyclic graph that induces a partial order representing temporal direction. Persistent systems must additionally balance transformation with their capacity to integrate change. Together, the ordering of transformation space and the structural load of transformation define structural time: the ordered and structurally integrable transformation of persistent identity.

Time is central to nearly all scientific descriptions of the world, yet its structural origin remains unclear. Most physical theories treat time either as a primitive parameter or as a geometric dimension of spacetime. While these approaches successfully describe temporal dynamics, they rarely explain why temporal order exists at all. This paper develops a structural account of time derived from the minimal requirements of systems capable of persistence under transformation. The analysis begins with a minimal representation of persistent systems consisting of a state space, a set of admissible transformations, and an identity relation. It is shown that persistence requires selective admissibility of transformations: if every transformation is allowed, identity distinctions collapse and persistent structure becomes impossible. Admissible transformations induce a reachability topology whose strongly connected components represent regions of reversible transformation. Collapsing these components produces a directed acyclic graph that induces a partial order representing temporal direction. Persistent systems must additionally balance transformation with their capacity to integrate change. Together, the ordering of transformation space and the structural load of transformation define structural time: the ordered and structurally integrable transformation of persistent identity.

This paper derives the minimal structural condition required for the existence of persistent reality. Instead of beginning with physical quantities such as matter, spacetime, or energy, the analysis starts from a weaker requirement: the existence of distinguishable states and admissible transformations between them. It is shown that if all logically possible transformations of the state space are admissible, persistent identity becomes impossible. Under unrestricted transformation any state can be mapped into any other state, forcing every identity assignment invariant under transformation to become trivial. Therefore, any system in which non-trivial identity persists under real transformation must restrict its admissible transformations. This result expresses the minimal structural condition underlying the structural architecture developed in the La Profilée theory.

This work excavates a structural architecture that reality must instantiate wherever persistent systems exist under transformation. It does not propose a hypothetical model, but derives minimal structural conditions that any domain of persistent relational reality must in fact satisfy. Starting from the weakest possible structural assumptions – the existence of distinguishable states and transformations between states – the analysis derives the structural requirements necessary for the persistence of identifiable entities. The analysis shows that unrestricted transformation leads to identity collapse. Persistent systems therefore require constrained transformation structures, structural asymmetry, and finite integration capacity. From these conditions emerges a transformation topology characterized by strongly connected components and an acyclic condensation structure. This topology generates a partial ordering that defines structural time. The resulting framework describes the minimal architecture required for persistence under transformation. This architecture is expressed as the La Profilée structure, consisting of a stable frame, transformable modules, and constrained coupling relations. The theory applies across multiple domains, including physical systems, biological systems, and complex social systems. Persistence under real transformation requires structural asymmetry.

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.