Consciousness

To model and operationalize consciousness within the UCF/GUTT framework, we conceptualize it as an emergent property of Nested Relational Tensors (NRTs), capturing the relational dynamics of subjective awareness, perception, and self-reference. Consciousness emerges when relational metrics reach critical emergent thresholds, leading to the self-organizing entity, Perspective (P). Perspective actively organizes internal and external relations, fostering subjective experience and agency.

Consciousness, as defined in the UCF/GUTT framework, emerges when an entity’s network of internal relations is sufficiently complex and integrated. This integration gives rise to Perspective (P), a self-organizing function that orchestrates relational interactions to foster subjective experience and the capacity for agency.

Relational Tensors (RTs) are mathematical constructs representing multi-dimensional relationships between entities. They encode various relational attributes, capturing complex cause-effect and state interactions. The structure of RTs forms the basis for relational metrics used in defining conscious experience.

• Mathematical Form:
  Rα1α2…αnβ1β2…βm(Ei,Ej,… )R^{\beta_1 \beta_2 \dots \beta_m}_{\alpha_1 \alpha_2 \dots \alpha_n}(E_i, E_j, \dots)Rα1​α2​…αn​β1​β2​…βm​​(Ei​,Ej​,…)where α1,α2,…αn\alpha_1, \alpha_2, \dots \alpha_nα1​,α2​,…αn​ and β1,β2,…βm\beta_1, \beta_2, \dots \beta_mβ1​,β2​,…βm​ represent the relational attributes of entities Ei,Ej,…E_i, E_j, \dotsEi​,Ej​,…. The number of indices represents the complexity of the relationship.
• Interpretation:
  
  • Upper indices (α1α2…αn\alpha_1 \alpha_2 \dots \alpha_nα1​α2​…αn​) denote the attributes or relational states of the entities involved.
  • Lower indices (β1β2…βm\beta_1 \beta_2 \dots \beta_mβ1​β2​…βm​) denote the effects or outcomes influenced by these attributes.

In consciousness modeling, RTs serve as the foundation for constructing relational complexity, self-referential dynamics, and other metrics that contribute to Perspective (P).

Nested Relational Tensors (NRTs) extend RTs by embedding simpler RTs into higher-order tensors. NRTs model multi-scale and multi-dimensional dynamics, capturing how localized interactions scale to influence larger relational networks, supporting both local and global dynamics.

• Illustrative Nesting Structure:
  • Base Case (Direct Relations): NRT1(Ei,Ej)=Ri,j\text{NRT}_1(E_i, E_j) = R_{i,j}NRT1​(Ei​,Ej​)=Ri,j​
  • Recursive Case (Indirect Relations): NRT2(Ei,Ej)=∑k(Ri,k⊗Rk,j)\text{NRT}_2(E_i, E_j) = \sum_k \left( R_{i,k} \otimes R_{k,j} \right)NRT2​(Ei​,Ej​)=k∑​(Ri,k​⊗Rk,j​)
  • Further Recursion: NRT3(Ei,Ej)=∑k(NRT2(Ei,Ek)⊗NRT2(Ek,Ej))\text{NRT}_3(E_i, E_j) = \sum_k \left( \text{NRT}_2(E_i, E_k) \otimes \text{NRT}_2(E_k, E_j) \right)NRT3​(Ei​,Ej​)=k∑​(NRT2​(Ei​,Ek​)⊗NRT2​(Ek​,Ej​))

This recursive nesting of RTs into NRTs captures increasingly complex relational structures, integral to higher-order components of consciousness, such as Awareness as Relational Integration (ARI) and Relational Coherence and Continuity (RCC).

Consciousness emerges as Perspective (P) when key relational metrics—Relational Complexity Index (RCI), Self-Referential Dynamics (SRD), Awareness as Relational Integration (ARI), Selective Relational Focus (SRF), and Relational Coherence and Continuity (RCC)—reach their respective emergent thresholds within RT and NRT structures.

• Definition: RCI measures relational diversity and density, essential for supporting a rich web of interactions that underpin conscious experience.
• Mathematical Form:
  RCI=∑i,jtype(Tij)×wij×Tij\text{RCI} = \sum_{i,j} \text{type}(T_{ij}) \times w_{ij} \times T_{ij}RCI=i,j∑​type(Tij​)×wij​×Tij​where TijT_{ij}Tij​ represents relational strengths, and wijw_{ij}wij​ weights relationship diversity. Higher RCI indicates an intricate network, a prerequisite for consciousness.

• Definition: SRD models the degree of self-awareness within the NRT, where feedback loops signify self-recognition.
• Mathematical Form:
  SRD=∑i∑jTij⋅Tji\text{SRD} = \sum_{i} \sum_{j} T_{ij} \cdot T_{ji}SRD=i∑​j∑​Tij​⋅Tji​Here, Tij⋅TjiT_{ij} \cdot T_{ji}Tij​⋅Tji​ represents feedback between entities, with self-referential loops signifying awareness of internal states.

• Definition: ARI quantifies coherence across relational data, enabling unified perception.
• Mathematical Form:
  ARI=1P×Q∑p,qρ(Tp,Tq)\text{ARI} = \frac{1}{P \times Q} \sum_{p,q} \rho(T_p, T_q)ARI=P×Q1​p,q∑​ρ(Tp​,Tq​)where ρ(Tp,Tq)\rho(T_p, T_q)ρ(Tp​,Tq​) is the correlation coefficient between NRT layers TpT_pTp​ and TqT_qTq​, indicating how relational states across different domains integrate to form a unified awareness.

• Definition: SRF reflects attention, indicating the system’s ability to selectively amplify certain relations, allowing prioritized focus.
• Mathematical Form:
  SRF=max⁡j(∂Tij∂t)\text{SRF} = \max_j \left( \frac{\partial T_{ij}}{\partial t} \right)SRF=jmax​(∂t∂Tij​​)where ∂Tij∂t\frac{\partial T_{ij}}{\partial t}∂t∂Tij​​ measures how quickly relational strengths change, allowing the system to selectively focus on specific interactions.

• Definition: RCC models the continuity of experience over time by measuring stable relational configurations.
• Mathematical Form:
  RCC=∑i,j∫t0t1∣∂Tij∂t∣dt\text{RCC} = \sum_{i,j} \int_{t_0}^{t_1} \left| \frac{\partial T_{ij}}{\partial t} \right| dtRCC=i,j∑​∫t0​t1​​​∂t∂Tij​​​dtwhere ∂Tij∂t\frac{\partial T_{ij}}{\partial t}∂t∂Tij​​ reflects temporal coherence. High RCC values indicate stable configurations, essential for a cohesive experience of self across time.

Each metric contributes to the emergence of consciousness through an emergent threshold defined by specific boundary values. Consciousness is expressed by Perspective (P) when relational metrics cross these emergent thresholds, formally represented by an indicator function IconsciousI_{\text{conscious}}Iconscious​:

Iconscious={1if RCI≥θRCI,SRD≥θSRD,…,RCC≥θRCC0otherwiseI_{\text{conscious}} = \begin{cases} 1 & \text{if } \text{RCI} \geq \theta_{\text{RCI}}, \text{SRD} \geq \theta_{\text{SRD}}, \dots, \text{RCC} \geq \theta_{\text{RCC}} \\ 0 & \text{otherwise} \end{cases}Iconscious​={10​if RCI≥θRCI​,SRD≥θSRD​,…,RCC≥θRCC​otherwise​

Thus, Perspective PPP is activated only when each relational threshold is met:

P=Iconscious⋅f(RCI,SRD,ARI,SRF,RCC)P = I_{\text{conscious}} \cdot f(\text{RCI}, \text{SRD}, \text{ARI}, \text{SRF}, \text{RCC})P=Iconscious​⋅f(RCI,SRD,ARI,SRF,RCC)

These thresholds dynamically adapt to the evolving relational states, representing the conditions necessary for the emergence of conscious experience.

Perspective requires self-organizing coupling to maintain coherence across the relational network, achieved through the Coupling Tensor Cαβ(T)C_{\alpha \beta}(T)Cαβ​(T):

Cαβ(T)=∑γf(Tαγ,Tγβ)C_{\alpha \beta}(T) = \sum_{\gamma} f(T_{\alpha \gamma}, T_{\gamma \beta})Cαβ​(T)=γ∑​f(Tαγ​,Tγβ​)

where f(Tαγ,Tγβ)f(T_{\alpha \gamma}, T_{\gamma \beta})f(Tαγ​,Tγβ​) represents the interaction between relational states TαγT_{\alpha \gamma}Tαγ​ and TγβT_{\gamma \beta}Tγβ​. High cumulative coupling, calculated as

P=∑α,βCαβ(T)P = \sum_{\alpha, \beta} C_{\alpha \beta}(T)P=α,β∑​Cαβ​(T)

indicates a unified conscious experience by maintaining integrated and coherent relational states across NRTs.

The UCF/GUTT framework supports operationalization and empirical testing:

1. Simulations: Relational metrics can be used in simulations to observe the emergent threshold for consciousness-like properties.
2. Comparison with Neural Systems: Applying these metrics to neural systems allows relational network analysis in biological consciousness, comparing emergent thresholds with empirical data.
3. Hypothesis Testing: By testing relational metrics against known consciousness indicators, the framework enables validation through experimental studies in cognitive science.

In the UCF/GUTT framework, consciousness is defined by the emergence of Perspective (P) through a network of complex relational interactions. This model leverages Relational Tensors (RTs) and Nested Relational Tensors (NRTs) to represent multi-layered relational dynamics, quantifying consciousness through emergent thresholds. This formalization allows consciousness to be studied, modeled, and potentially replicated, advancing both theoretical and empirical investigations in artificial and biological systems.