Topological–Integration–Energetic Regime Theory (TIER): Phenomenology as an Integrated Topological Dynamical Regime

Abstract

Phenomenology remains one of the central open problems in the study of complex dynamical systems. Existing theoretical frameworks often attempt to define consciousness through informational or computational constructs whose combinatorial complexity prevents direct empirical identification in realistic systems.

This work introduces and empirically validates a different formulation. Within the Topological–Integrability–Energetic Regime Theory (TIER), phenomenological organization is identified as a structural property of a specific dynamical regime of a physical system.

From multiscale dynamical data of independent human subjects, the system’s activity is projected into a structural space capturing energetic and integrative organization. A topological structure is constructed over this space through a family of proximity graphs, allowing the analysis of connectivity regimes across scales. The empirical geometry of this space defines a stable structural manifold organizing the system’s trajectories.

Across all analyzed subjects, the constructed topology exhibits robust signatures of global integration, including macroscopic percolation, the emergence of a giant connected component, the formation of a persistent structural core dominating the connectivity of the graph, and a well-defined spectral dimension supporting diffusive dynamics over the structural manifold. Dynamical analysis shows that trajectories remain confined to this integrated structural domain, forming a closed dynamical organization over the manifold.

Within this regime, the system systematically reproduces classical phenomenological properties — unity, graded organization, integration, interiority, and privacy — as invariants of its structural dynamics.

Crucially, structural intervention experiments that selectively disrupt the relational topology of the system collapse these phenomenological invariants while preserving marginal signal statistics, establishing a direct dependence between the integrity of the structural regime and the presence of phenomenological organization.

The central result is therefore a structural identification:

phenomenological organization corresponds to the operation of a physical system within a topologically integrated dynamical regime of its structural state space.

The central result is therefore a structural identification:

Phenomenological organization corresponds to the operation of a physical system within a topologically integrated dynamical regime of its structural state space. This regime is empirically identifiable from multiscale dynamical data through the structural projection operator Φθ, which induces the observable structural state space (E, U).

Under this formulation, phenomenological organization corresponds to a physically realized structural regime defined by the joint geometry, topology, and dynamics of the system’s state space.

Resumen

La fenomenología continúa siendo uno de los problemas abiertos centrales en el estudio de los sistemas dinámicos complejos. Los marcos teóricos existentes intentan con frecuencia definir la conciencia mediante construcciones informacionales o computacionales cuya complejidad combinatoria impide su identificación empírica directa en sistemas realistas.

Este trabajo introduce y valida empíricamente una formulación diferente. Dentro de la Topological–Integrability–Energetic Regime Theory (TIER), la organización fenomenológica se identifica como una propiedad estructural de un régimen dinámico específico de un sistema físico.

A partir de datos dinámicos multiescala de sujetos humanos independientes, la actividad del sistema se proyecta en un espacio estructural que captura la organización energética e integrativa. “Sobre este espacio se construye una estructura topológica mediante una familia de grafos de proximidad, lo que permite analizar regímenes de conectividad a diferentes escalas. La geometría empírica de este espacio define un manifold estructural estable que organiza las trayectorias del sistema.

En todos los sujetos analizados, la topología construida exhibe firmas robustas de integración global, incluyendo percolación macroscópica, la aparición de un componente gigante conexo, la formación de un núcleo estructural persistente que domina la conectividad del grafo y una dimensión espectral bien definida que soporta dinámica difusiva sobre el manifold estructural. El análisis dinámico muestra que las trayectorias permanecen confinadas dentro de este dominio estructural integrado, formando una organización dinámica cerrada sobre el manifold.

Dentro de este régimen, el sistema reproduce sistemáticamente propiedades fenomenológicas clásicas — unidad, organización gradada, integración, interioridad y privacidad — como invariantes de su dinámica estructural.

De manera crucial, experimentos de intervención estructural que perturban selectivamente la topología relacional del sistema provocan el colapso de estos invariantes fenomenológicos mientras se preservan las estadísticas marginales de la señal, estableciendo una dependencia directa entre la integridad del régimen estructural y la presencia de organización fenomenológica.

El resultado central es, por tanto, una identificación estructural:

La organización fenomenológica corresponde a la operación de un sistema físico dentro de un régimen dinámico topológicamente integrado de su espacio estructural de estados. Este régimen es empíricamente identificable a partir de datos dinámicos multiescala mediante el operador de proyección estructural Φθ, el cual induce el espacio de estados estructural observable (E, U).

Bajo esta formulación, la organización fenomenológica corresponde a un régimen estructural físicamente realizado definido por la geometría, la topología y la dinámica conjuntas del espacio de estados del sistema.

Version 2.0 — Description of changes

This version substantially expands and consolidates the original manuscript.

Major changes include:

• Clarification of the structural operator Φθ and the definition of the induced structural space (E,U).

• Expanded analysis of the topological organization of the system.

• Addition of dynamical analysis demonstrating confinement of system trajectories.

• Inclusion of spectral graph properties (spectral dimension).

• Integration of structural intervention experiments.

• Substantial revision of the theoretical interpretation.

Version 2.1 — Description of changes

This update introduces minor conceptual clarifications to the

theoretical formulation of the TIER framework without altering

the empirical results, structural invariants, or the central

theorem of the manuscript.

Changes include:

• Clarification that the structural embedding (E,U) should be

interpreted as an observational structural projection induced

by the operator Φθ rather than a privileged parametrization

of the system.

• Clarification of the relationship between global structural

properties of the regime and their dynamical manifestations

within the structural manifold.

These changes improve conceptual precision and theoretical

interpretability without modifying the mathematical structure

or empirical basis of the framework.