Bridging Computational Structures with Philosophical Categories in Sophimatics and Data Protection Policy with AI Reasoning

Abstract

Contemporary artificial intelligence excels at pattern recognition but lacks genuine understanding, temporal awareness, and ethical reasoning. Critics argue that AI systems manipulate statistical correlations without grasping concepts, time, or moral implications. This article presents Phase 2, a component of the emerging infrastructure called Sophimatics, a computational framework that translates philosophical categories into working algorithms through the integration of complex time. Our approach operationalizes Aristotelian substance theory, Augustinian temporal consciousness, Husserlian intentionality, and Hegelian dialectics within a unified temporal–semantic architecture. The system represents time as both chronological and experiential, allowing navigation between memory and imagination while maintaining conceptual coherence. Validation through a Data Protection Policy use case demonstrates significant improvements: confidence in decisions increased from 6.50 to 9.40 on a decimal scale, temporal awareness from 2.00 to 9.50, and regulatory compliance from 6.00 to 9.00 compared to traditional approaches. The framework successfully links philosophical authenticity with computational practicality, offering greater ethical consistency and contextual adaptability for AI systems that require temporal reasoning and ethical foundations.

1. Introduction

The contemporary landscape of artificial intelligence (AI) is dominated by large data-driven models. Deep neural networks and foundation models generate human-like text, recognize faces, and translate languages. Nonetheless, critics argue that such systems remain profoundly limited. Current AI systems face three fundamental limitations: they lack causal understanding and cannot explain alternative choices [1]; they manipulate symbols without grasping meaning due to the absence of philosophical grounding [2,3]; and they treat time as linear sequences rather than integrating memory, present attention, and future anticipation [4,5]. These systems cannot capture intentionality—the directedness inherent in thought—nor can they address ethical implications [6,7,8]. To transcend statistical pattern matching, AI must integrate concepts, temporality, intentions, and ethics.

However, existing approaches lack systematic frameworks for translating philosophical depth into computational structures. Current AI systems either ignore philosophical considerations entirely or incorporate them superficially without addressing the fundamental challenge of operationalizing concepts like temporality, intentionality, and dialectical reasoning in algorithmically tractable ways. This represents a critical research gap that must be addressed to move beyond statistical pattern matching towards genuine understanding.

In response, a number of hybrid approaches have emerged. The model in [9] combined symbolic rules for grammar with a neural network that learned activation patterns for nominal phrases.

In [10], the authors introduced Smart Sensing, an info-structural model that integrates layered memory and intentional components to allow non-interacting agents to exhibit contextual awareness. In later work, they extended the model to computational consciousness, incorporating modules for awareness and emotional modulation [11]. In [12], the authors proposed CognitiveNet, which enriches foundational models with emotions and awareness, demonstrating that hybrid architectures can improve adaptability in human–computer interaction. While promising, these models do not yet provide a general mechanism for translating philosophical categories into computational objects.

Sophimatics is a multi-phase programme that attempts to fill this void. Phase 1 surveyed major philosophical categories—change, form, logic, temporality, intentionality, context, and ethics—and showed how they could inform the design of decision support systems [11]. The first phase established that philosophy can enrich AI, but it did not spell out how to operationalize philosophical concepts in a general way. This article develops Phase 2, which focuses on conceptual mapping—the translation of philosophical categories into computational structures through the introduction of complex time. We aim to answer the following questions: How can we represent concepts like substance, temporality, intention, and dialectic in a computational framework? How can we use temporal coordinates to navigate between memory and imagination? And how does this enriched representation improve reasoning and decision-making?

To this end, we introduce a translation function mapping philosophical concepts and complex temporal coordinates to computational constructs: Is time a complex number rather than a real number? Yes, it is, as posited by some contemporary extended quantum theories of gravitation. Complex time is represented by a number (T = a + i b), where the real part (a) corresponds to chronological time and the imaginary part (b) encodes experiential dimensions. When (b < 0), the coordinate lies in the “memory cone”; when (b > 0), it lies in the “creativity cone”; and when (b = 0), it lies on the real axis representing the present. As we will see, angular parameters ($\alpha$) and ($\beta$) define accessible regions. This geometry ensures that not all past events or future possibilities are equally accessible; instead, accessibility depends on orientation in complex time. The translation function returns a tuple $⟨{\mathrm{S}}_{\mathrm{T}},{\mathrm{R}}_{\mathrm{T}},{\mathrm{O}}_{\mathrm{T}},{\mathrm{I}}_{\mathrm{T}},{\mathrm{\Gamma }}_{\mathrm{T}}⟩,$ where ${\mathrm{S}}_{\mathrm{T}}$ represents temporal–structural representation, ${\mathrm{R}}_{\mathrm{T}}$ denotes temporally parameterized relational mappings, ${\mathrm{O}}_{\mathrm{T}}$ contains temporal operations, ${\mathrm{I}}_{\mathrm{T}}$ specifies interpretive context with temporal positioning, and ${\mathrm{\Gamma }}_{\mathrm{T}}$ captures the angular accessibility parameters $\left(\mathrm{\alpha },\mathrm{\beta }\right)$ governing memory and imagination access. By formalizing philosophical categories in this way, we open the door to computational implementations that can reason about substances, time, intentions and dialectics while navigating temporal dimensions.

We aim to answer three specific research questions:

(1) How can we represent concepts like substance, temporality, intention, and dialectic in a computational framework?

(2) How can we use temporal coordinates to navigate between memory and imagination?

(3) How does this enriched representation improve reasoning and decision-making?

To achieve this, first, we detail the architecture of the translation function and associated operators, showing how Aristotelian substances, Augustinian temporality, Husserlian intentionality, and Hegelian dialectics can be mapped into complex-time structures. Second, we evaluate the benefits of this mapping through a specific and relevant use case, that is, a Data Protection Policy. The results show that our system achieves improved ethical consistency, temporal awareness, and creative problem-solving. The structure of the article is as follows: Section 2 reviews related work; Section 3 introduces materials and methods, including translation functions and temporal operators; Section 4 is reserved for the mathematical model, while Section 5 shows the architectural framework; Section 6 presents results on the privacy policy; Section 7 discusses implications, limitations, and perspectives; and Section 8 presents conclusions.

2. Related Works

The citations in this article follow an order of appearance designed to accompany the development of the work just in time in relation to cognitive necessities. To provide a second and complementary reading key, we indicate below, for each of the three general questions enumerated above, to which the present work intends to offer answers, or, given the complexity, at least trace a rough path useful to create a route, what the specific references are for each of them. Research Question 1 (computational representation of philosophical concepts): foundational works on philosophical AI integration [2,3], hybrid symbolic–connectionist models [9,13], computational consciousness frameworks [10,11,12], and contextual reasoning with dynamic ontologies [14,15,16]. Research Question 2 (temporal coordinates for memory–imagination navigation): temporal reasoning in AI [4,5], cognitive theories of temporality [17,18], complex-time processing approaches [19,20], and temporal consciousness models [21]. Research Question 3 (improved reasoning and decision-making): ethical AI frameworks [8,22,23], decision support systems [19,20], applied philosophy of AI [24,25], and empirical validation methodologies [26]. While [27,28,29] discuss, respectively, (i) a new bridge between philosophical thought and logic for emerging post-generative artificial intelligence; (ii) the foundations and models of rediscovered computational wisdom; (iii) applications, ethics, and future prospects; in the next section, we analyse the Sophimatics framework, serving as a precursor to the study of the layer called Phase 2, which is the main subject of this article, regarding the mapping of philosophical thought into Sophimatics AI Framework via and thanks to the use of 2D complex time, as a generalization of typical chronological time, which will be created to include memory–creativity–imagination in the solution, as we will see.

Synthesis of Contemporary AI Integration Challenges: Current artificial intelligence research addresses philosophical foundations [2,3,11], temporal cognition [4,5,17,18], and ethical frameworks [8,22,23], yet faces three fundamental integration challenges.

Challenge 1: Temporal–Philosophical Foundations require bridging formal temporal reasoning with experiential consciousness. While temporal reasoning surveys [4] and cognitive temporal models [5,17,18] provide insights, most systems treat time linearly without philosophical grounding [2,3]. Contemporary neuro-symbolic approaches like Logic Tensor Networks [30], DeepProbLog [31], Neural Module Networks [32], and Probabilistic Soft Logic [33] advance hybrid reasoning but lack the integration of temporal consciousness. Linear Temporal Logic [34] and Computation Tree Logic [35] offer formal temporal foundations, while Dynamic Epistemic Logic [36] and Temporal Action Logic [37] model knowledge change and planning, yet operate without experiential temporality.

Challenge 2: Hybrid Architectural Integration combines symbolic and connectionist paradigms [9,13,38]. Smart Sensing frameworks [10] and computational consciousness models [11] integrate layered memory and intentional components, while CognitiveNet [12] enriches foundation models with emotions and awareness [39,40]. Context-aware computing [26] and contextual reasoning [41] shape cognitive architectures for social interaction [42], leading to dynamic cognitive ontology networks with neuromorphic processing [43]. Contextual AI frameworks [14,15,16] enable reasoning across abstraction levels, yet systematic philosophical category mapping remains absent.

Challenge 3: Ethical–Intentional Computing operationalizes normative reasoning and directedness. Ethical AI reviews [8] catalogue deontological, utilitarian, and virtue approaches, while EAIFT [22] embeds normative reasoning architectures. Applied philosophy of AI [24] advocates constructive philosopher involvement, and policy frameworks [23] promote responsible deployment [21]. Intentionality research [44] moves beyond rewards to logical intentions, while dialectical reasoning [45] and collective intentionality [46] explore compromise-based justification. AI agent intentionality construction [47] balances object-directed goals, and intelligibility research [48,49] offers phenomenological critiques. Ontology-based monitoring [50] demonstrates formal ontology applications for system behaviour prediction.

Sophimatics’ Unified Response: Our complex-time framework uniquely addresses all three challenges by systematically translating philosophical categories into computational structures through navigable temporal coordinates, preserving both conceptual authenticity and algorithmic tractability while integrating memory, imagination, and dialectical reasoning capabilities.

While acknowledging the significant contributions of existing frameworks in the context of neuro-symbolic approaches, a conceptual analysis reveals fundamental limitations that Sophimatics could address. Logic Tensor Networks and DeepProbLog excel at combining symbolic reasoning with neural learning but lack temporal consciousness and experiential time modelling. These frameworks treat time as discrete states rather than navigable complex dimensions integrating memory and imagination. Temporal Logic Systems: Linear Temporal Logic (LTL) and Computation Tree Logic (CTL) provide formal temporal reasoning but operate on propositional abstractions without philosophical grounding or intentionality modelling. They cannot represent the directedness of consciousness or dialectical synthesis processes. Regarding hybrid symbolic–connectionist models, we can note that existing hybrid architectures combine representation paradigms but do not systematically embed philosophical categories as computational primitives. They lack the complex-time framework necessary for Augustinian temporal synthesis or Hegelian dialectical reasoning. On the other hand, regarding Sophimatics’s conceptual advantages, we can note that when fully implemented across all six layers, Sophimatics will address these limitations by (1) providing temporal consciousness through complex-time navigation, (2) implementing genuine intentionality via directedness functions, (3) enabling dialectical reasoning through synthesis operators, and (4) maintaining philosophical authenticity through category-specific transfer functions. This conceptual comparison highlights how Sophimatics could contribute unique capabilities to the hybrid AI landscape, particularly in domains requiring temporal awareness, ethical reasoning, and contextual understanding that current approaches cannot adequately address.

3. Materials and Methods

The methodological structure of Sophimatics is built in six phases (or levels) that are interconnected by developing operations to pattern the philosophical categories with respect to computational expressions (see Figure 1).

The first is a highly detailed historical and philosophical analysis. Contributors canvas the long period from ancient philosophy to the present day, exploring the way in which these categories have been used (and abused) at all times and everywhere. This hermeneutics is non-anachronistic, and it takes care of the internal consistency in each philosophical tradition. It guarantees that Sophimatics is based on dynamic ontology, intentionality, and dialectic logic. These are the philosophical stances that underpin the theoretical model and that direct the formalization process.

The next phase is conceptual mapping. In this stage, abstract philosophical concepts are transformed into formal entities that can be realized by a computer. For example, Aristotle’s take on the concept of substance becomes a node in an ontology. Regarding the implementation details that we will shortly see later in this context, the ontological structures were implemented using Python 3.11 with the coherent RDFLib and OWL-API frameworks, enabling formal representation and manipulation of philosophical concepts. Substance nodes maintain hierarchical relationships through RDFS subsumption, while complex-time coordinates are processed using NumPy arrays with custom temporal operators. Augustine’s take on time is that T = a + i·b, including elements of chronology but also experience; Husserl’s intentionality is documented as link structures from mental states to the object states; and Hegel’s dialectic is run as a feedback loop iterating hypotheses. The translation is based on formal logic, category theory, and type theory so that conceptual integrity can be maintained and yet represented in a way that can be executed in a computer. The key to this stage is the move to multi-dimensional semantic spaces models that can represent ambiguous and overlapping interpretations. If the suggested concepts are encoded in a high-dimensional space, the indistinguishability between concepts in interpretation (related to normalization) can be effectively resolved, as strong contexts can be incorporated.

The third stage is the realization of such constructs towards the design of a hybrid computational architecture. Sophimatics employs the multi-layer architecture and modelling—a Super Temporal Complex Neural Network (STCNN) (with three layers; see the next section). The first layer performs sequential perception and pattern recognition, similar to an encoder that transforms sensory inputs into a latent representation. A second layer includes context memory and temporal embedding. It represents episodic, semantic, and intentional memories and can perform context integration over time via its recurrent mechanisms. This layer model’s dynamic context evolution means that the system remembers beauty, surroundings, and why something is experienced. The third layer carries out phenomenological reasoning and combines symbolic representations with activations in the neural network to generate explanations and justifications. It is the home of a semantic dialogic engine that reasons through an internal dialogue, based on dialectic rules. To these layers, we add three auxiliary modules: an ontological–ethical module, which embeds deontic logic and virtue ethics; a module of memory and contextual awareness, which employs layered memory (episodic, semantic, intentional) and contextual resonance; and an emotive–symbolic module that determines quality-values of information. It is the combination of these components that adds perception, memory, reasoning, and action to a Sophimatics system.

The fourth stage of interpretation deals with context and temporality. Concepts are not static entities but are rather dynamic and multidimensional entities that are directly shaped by the agent’s interaction. Based on the contextual reasoning paradigm, each knowledge entry is annotated with a context label depicting spatial, temporal, social, and intentional information. The system keeps various contexts and can change or combine them. Time is represented as complex temporality; the real part corresponds to chronological time, and the imaginary part corresponds to implicit meaning or subjective experience. This model enables the system to predict events to come and understand their attendant importance, catching not only explicit but also implicit temporality. The aforementioned complex temporal model and reasoning model not only enable temporal reasoning about durations, sequence, and concurrency of activities but also interval algebras and temporal constraints. These formalisms allow the agent to reason about time and act accordingly.

Ethicality and intentionality comprise the fifth stage. Capability/Sophimatics behaviour reasoning modules rooted in deontic, virtue, and consequentialist ethics rely on principles for ex ante assessment of acts. A deontic logic layer represents obligations and prohibitions, a virtue ethics module evaluates actions based on character and flourishing, and a consequentialist module judges outcomes. These ethical judgments are connected to the intentional attitudes (goals, beliefs, and desires) of the agent of the behaviour, which is modelled by first-order formulae. Intentions are not fixed but develop during interaction and are dynamically created, deleted, and updated through dialogue with the ethical modules. This development will allow the agent to be able to explain the reasons behind its choices and amend its motivation and behaviour to comply with accepted norms of behaviour.

The sixth level concentrates on an iterative process and working with people. Sophimatics adopts a human-in-the-loop methodology in which philosophers, subject-matter experts, and technicians cooperate to improve the architecture. Prototypes are developed in fields such as education, health, and urban planning, which are sensitive to context and ethics. There are interpretive correctness, context consistency, time, and ethically coherence as evaluation criteria. Regarding evaluation criteria, interpretive correctness measures alignment between computational outputs and original philosophical source material, assessed through expert evaluation on a 10-point scale. Context consistency evaluates logical coherence across different temporal positions and accessibility constraints, ensuring that reasoning maintains philosophical authenticity. Temporal coherence assesses the system’s ability to maintain consistent relationships between memory, present awareness, and future projection. Ethical coherence measures adherence to normative principles embedded in the philosophical categories. Comparative systems analyse Sophimatics’s performance against baseline generative and symbolic systems.

4. Model for the Conceptual Mapping Framework

The conceptual mapping is Phase 2 of the realization of Sophimatics Cognitive Infrastructure; it represents the critical translation layer between abstract philosophical categories and their computational implementations, now enhanced by the complex-time paradigm proposed here. This phase operationalizes the fundamental insight that philosophical concepts possess inherent mathematical structures that can be formally represented while preserving their conceptual integrity and semantic richness, particularly through the integration of complex temporality, T ∈ ℂ, where memory (Im(T) < 0) and imagination (Im(T) > 0) become computational primitives (see Figure 2).

4.1. Enhanced Translation Architecture with Complex Time

The conceptual mapping process is formalized as an enriched translation function

$$\mathcal{T}:\mathcal{P}\times \mathbb{C}\to \mathcal{C}$$

(1)

where $\mathcal{P}$ represents the space of philosophical concepts, ℂ is the complex temporal domain, and $\mathcal{C}$ represents the space of computational constructs. This translation preserves essential structural relationships while enabling temporal–geometric manipulation through the complex plane.

Each philosophical concept $\mathrm{\varphi }\in \mathcal{P}$ is mapped to a temporally enriched computational construct:

$$\mathcal{T}(\mathrm{\varphi },\mathrm{T})=⟨{\mathrm{S}}_{\mathrm{T}},{\mathrm{R}}_{\mathrm{T}},{\mathrm{O}}_{\mathrm{T}},{\mathrm{I}}_{\mathrm{T}},{\mathrm{\Gamma }}_{\mathrm{T}}⟩$$

(2)

where ${\mathrm{S}}_{\mathrm{T}}$ represents temporal–structural representation (ontological nodes with complex time coordinates), ${\mathrm{R}}_{\mathrm{T}}$ denotes temporally parameterized relational mappings, ${\mathrm{O}}_{\mathrm{T}}$ contains temporal operations (including Laplace transforms, angular access functions), ${\mathrm{I}}_{\mathrm{T}}$ specifies interpretive context with temporal positioning, and ${\mathrm{\Gamma }}_{\mathrm{T}}$ captures the angular accessibility parameters $\left(\mathrm{\alpha },\mathrm{\beta }\right)$ governing memory and imagination access.

The enhanced coherence condition integrates temporal consistency:

$$\forall {\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2\in \mathcal{P},{\mathrm{T}}_1,{\mathrm{T}}_2\in \mathbb{C}:\mathrm{Related}\left({\mathrm{\varphi }}_1,{\mathrm{\varphi }}_2\right)\Rightarrow \mathrm{TemporallyLinked}\left(\mathcal{T}\left({\mathrm{\varphi }}_1,{\mathrm{T}}_1\right),\mathcal{T}\left({\mathrm{\varphi }}_2,{\mathrm{T}}_2\right)\right)$$

(3)

4.2. Aristotelian Substance as Temporally Positioned Ontological Nodes

Aristotle’s substance theory is integrated with complex-time positioning, where substances exist not only in taxonomic hierarchies but across temporal-geometric space. For the theoretical formalization, let

$$O_{\mathrm{T}}=⟨V_{\mathrm{T}},E_{\mathrm{T}},\mathrm{\tau },\mathrm{\lambda },\mathrm{\Omega }⟩$$

(4)

be a temporal ontological graph, where ${\mathrm{V}}_{\mathrm{T}}\subseteq \mathrm{V}\times \mathbb{C}$ represents substance nodes with complex temporal coordinates, ${\mathrm{E}}_{\mathrm{T}}\subseteq {\mathrm{V}}_{\mathrm{T}}\times {\mathrm{V}}_{\mathrm{T}}$ represents temporal subsumption relationships, $\mathrm{\tau }:{\mathrm{V}}_{\mathrm{T}}\to \mathrm{Primary},\mathrm{Secondary}$ classifies substance types. $\mathrm{\lambda }:{\mathrm{V}}_{\mathrm{T}}\to \mathcal{P}\left(\mathrm{Properties}\right)$ assigns temporal property sets, and $\mathrm{\Omega }:{\mathrm{V}}_{\mathrm{T}}\to \left[\mathrm{\alpha },\mathrm{\beta }\right]$ determines angular accessibility for each substance.

Appendix A provides further details on the modelling presented in this section. These details are included in the appendix in order to allow the reader to read the work more easily and to enable those interested in studying the model in greater depth to find the results in the Appendix A by following the numbering of the formulas. Therefore, it may appear that the numbering of the formulas in this section skips numbers, as, for example, we will move from formula (4) to expression (7) below. This is not an error but a communication strategy that allows us to create a seamless thread between this section and Appendix A.

As an example to better understand, let us consider the following Algorithm 1.

Algorithm 1 Laplace transform

def complex_inherited_properties(substance_s, temporal_T, alpha, beta): temporal_ancestors = get_temporal_ancestors(substance_s, temporal_T, alpha, beta) inherited_properties = set() for ancestor in temporal_ancestors: property_set = lambda_property_function(ancestor) filtered_properties = laplace_transform(property_set, temporal_T) inherited_properties.union(filtered_properties) return inherited_properties

This algorithm leverages SciPy’s Laplace transform functions and custom temporal accessibility operators. Instead of simple property inheritance, where a substance inherits all ancestor properties immediately, this function implements temporal property inheritance, where only temporally accessible ancestors contribute properties and properties are filtered through complex-time transformations; the resulting inheritance depends on the current temporal position and angular constraints, and properties can be retrieved from memory, projected from imagination, or accessed in the present with different temporal characteristics.

For example, if the substance “Dog” is inherited from “Mammal” and “Animal”, the inherited properties would include temporally filtered versions of mammalian and animal properties, where the filtering depends on whether we are accessing these concepts from memory, imagination, or present awareness. For further details, see Appendix A.

4.3. Enhanced Augustinian Temporality as Complex Time Variables

Now, let us see Augustinian temporality as complex time variables. Augustine’s temporal consciousness is directly implemented through the complex-time framework, where memory (Im(T) < 0), imagination (Im(T) > 0), and present awareness (Im(T) ≈ 0) become computational coordinates. If we consider a complex time variable, T ∈ ℂ, where T = a + ib with angular constraints, $\mathrm{a}\in \mathrm{R}$ represents chronological time (physical duration), $b\in R$ represents experiential time intensity, $\mathrm{\alpha }\in \left[0,\mathrm{\pi }/2\right]$ controls memory cone accessibility, and $\mathrm{\beta }\in \left[\mathrm{\pi }/2,\mathrm{\pi }\right]$ controls creativity/imagination cone accessibility, then the temporal consciousness operator that maps real-world temporal experience into complex time coordinates, incorporating Augustine’s insights about temporal consciousness, the complex-time framework can be written as a temporal consciousness operator:

$${\mathcal{A}}_{\mathrm{\alpha },\mathrm{\beta }}:\mathbb{R}\times \mathcal{M}\times \mathcal{E}\times \left[\mathrm{\alpha },\mathrm{\beta }\right]\to \mathbb{C}$$

(5)

$$\mathcal{A}\alpha ,\beta \left(t,m,e,\theta \right)=t+i·\alpha ·\mathit{MemoryIntensity}\left(m\right)·{𝟙}_{\mathit{memory}\mathit{cone}}+\beta ·\mathit{ImaginationProjection}\left(e\right)·{𝟙}_{\mathit{creativity}\mathit{cone}})$$

where $\mathcal{M}$ is a memory space containing past experiences and retained information, $\mathcal{E}$ is the expectation/imagination space containing future projections and anticipated events, and [α, β] is an angular parameter range where α controls memory cone accessibility and β controls creativity cone accessibility. The input parameters are as follows: t is the chronological time point (real-valued, linear progression); m is a memory state containing past experiences and their significance; e is an expectation/imagination state containing future projections and anticipated scenarios; and θ is the current angular parameter within the [α, β] range and where the output structure produces a complex number, T = a + ib, where the real part preserves the chronological time component unchanged and the imaginary part is computed as a weighted combination of memory intensity and imagination projection. Regarding the imaginary component calculation for α MemoryIntensity(m) ${𝟙}_{\mathrm{memory}\mathrm{cone}}$, α is the memory cone angle parameter [0, π/2] that determines how much memory influence is allowed, MemoryIntensity(m) is a function that computes the intensity/significance of memory state m, ${𝟙}_{\mathrm{memory}\mathrm{cone}}$ is the indicator function that equals 1 if we are accessing the memory cone (Im(T) < 0) and 0 otherwise for β · ImaginationProjection(e) · ${𝟙}_{\mathrm{creativity}\mathrm{cone}}$; β is the creativity cone angle parameter [π/2, π] that determines how much imaginative projection is allowed; ImaginationProjection(e) Function that computes the projection strength of expectation/imagination state e; and ${𝟙}_{\mathrm{creativity}\mathrm{cone}}$ is the Indicator function that equals 1 if we are accessing the creativity cone (Im(T) > 0), and 0 otherwise. Regarding the philosophical significance, this operator implements Augustine’s insight from the Confessions that temporal consciousness involves three dimensions: memory of the past (memoria praeteritorum), attention to the present (contuitus praesentium), and expectation of the future (expectatio futurorum).

The computational innovation is that, unlike Augustine’s purely philosophical analysis, this operator makes temporal consciousness computationally tractable, introduces geometric constraints (angular parameters), enables systems to navigate between memory and imagination in a controlled manner, and provides a mathematical framework for AI systems to reason temporally like humans do.

Now, let us consider Laplace-Mediated Temporal Synthesis:

$$\mathit{ComplexSynthesis}\left(T_{\mathit{past}},T_{\mathit{present}},T_{\mathit{future}}\right)={\mathcal{L}}^{-𝟙}H\left(s\right)·\left[\mathcal{L}\left\{T_{\mathit{past}}\right\}+\mathcal{L}\left\{T_{\mathit{present}}\right\}+\mathcal{L}\left\{T_{\mathit{future}}\right\}\right]$$

(6)

where $H\left(s\right)$ is the transfer function governing temporal consciousness integration. Then, the ComplexSynthesis function mathematically implements Augustine’s temporal synthesis described in Confessions Book XI, where he argues that time exists primarily in the mind through memoria as extension of mind towards past things, contuitus as attention to present things, and expectatio as extension of mind towards future things. Unlike Augustine’s purely philosophical description, this function provides an algorithmic implementation of temporal synthesis, a transfer function H(s) that can be tuned for different cognitive architectures, frequency-domain processing enabling sophisticated temporal filtering, and a complex-time output that preserves both chronological and experiential temporal dimensions. The choice of H(s) determines the character of temporal consciousness: the low-pass filter emphasizes long-term temporal patterns and stable consciousness, the band-pass filter focuses on specific temporal scales and rhythmic consciousness, the all-pass filter preserves all temporal components but adjusts phase relationships, and the adaptive filter dynamically adjusts based on contextual demands.

As example applications, we can consider the following.

AI temporal reasoning: Synthesizing historical data, current state, and predicted outcomes.

Cognitive modelling: Simulating human-like temporal consciousness in artificial agents.

Decision systems: Integrating past experience, present context, and future projections into unified temporal awareness.

This is made also thanks to the fact that we have three types of digital future implementation:

Future in the Past: G(s) = $\mathrm{H}\left(\mathrm{s}\right)·\mathrm{F}\left(\mathrm{s}\right)$ where $\left|\mathrm{a}\mathrm{r}\mathrm{g}\right(\mathrm{s}\left)\right|\le \mathrm{\alpha }$ but $\left|\mathrm{a}\mathrm{r}\mathrm{g}\right(\mathrm{s}\left)\right|<\mathrm{\beta }$.

Future in the Present: G(s) satisfies both $\left|\mathrm{a}\mathrm{r}\mathrm{g}\right(\mathrm{s}\left)\right|\le \mathrm{\alpha }$ and $\left|\mathrm{a}\mathrm{r}\mathrm{g}\right(\mathrm{s}\left)\right|\ge \mathrm{\beta }$.

Future in the Future: G(s) where $\left|\mathrm{a}\mathrm{r}\mathrm{g}\right(\mathrm{s}\left)\right|\ge \mathrm{\beta }$ but outside the memory cone.

For further details, see Appendix A.

4.4. Husserlian Intentionality as Temporal Pointer Structures

In Husserl’s intentionality, enriched by complex temporal positioning, mental acts and their objects exist in temporal–geometric relations governed by angular accessibility.

The mathematical formalization with complex-time integration leads to a temporal intentional structure as follows:

$$I_T=⟨{Acts}_T,{Objects}_T,{Directedness}_T,{Modes}_T,{\Omega }_T⟩$$

(7)

where we have the following details about the different involved functions. The temporal Pointer Function is

$${\mathit{Directedness}}_T:{\mathit{Acts}}_T\times \mathbb{C}\to \mathcal{P}\left({\mathit{Objects}}_T\times {\mathit{Modes}}_T\times \mathbb{C}\right)$$

(8)

The Complex Intentional Act Structure is

$$TemporalAct=⟨Noesis,Noema,Mode,Fulfilment,T_{position},{\Gamma }_{access}⟩$$

(9)

The Angular-Constrained Noetic–Noematic Correlation is

$$\mathit{TemporalCorrelation}\left(\mathit{noesis},\mathit{noema},T,\alpha ,\beta \right)=⟨{\mathit{Sense}}_T,{\mathit{Reference}}_T,{\mathit{Mode}}_T⟩·\Theta \left(T,\alpha ,\beta \right)$$

(10)

where $\mathrm{\Theta }\left(T,\alpha ,\beta \right)$ is the angular accessibility function.

As an example interpretation, we can have an AI system remembering a past event (memory cone) that would have its intentional fulfilment modulated by both the temporal distance from the present and the angular accessibility constraints, resulting in gradually degraded but still meaningful intentional relationships with temporally distant objects. For further details, see Appendix A.

4.5. Hegelian Dialectic as Complex-Time Iterative Feedback Loops

From philosophical foundations, let us consider Hegel’s dialectical method now operating in complex temporal space, where thesis–antithesis–synthesis processes traverse memory–imagination dimensions with angular constraints. Here we see the formalization with complex-time dynamics for an enhanced dialectical process:

$${\mathcal{D}}_{\mathcal{T}}=⟨{\Theta }_T,{\mathit{Negation}}_T,{\mathit{Synthesis}}_T,{\mathit{Aufhebung}}_T,H\left(s\right)⟩$$

(11)

where ${\Theta }_T$ is the space of temporally positioned theses. We also introduce a Complex-Time Dialectical Iteration Function:

$${\mathcal{D}}_{\mathcal{T}}:{\mathrm{\Theta }}_{\mathrm{T}}\times \mathbb{C}\times \left[\mathrm{\alpha },\mathrm{\beta }\right]\to {\mathrm{\Theta }}_{\mathrm{T}}\times \mathbb{C}$$

(12)

$${\mathcal{D}}_{\mathcal{T}}\left(\theta ,T,\alpha ,\beta \right)={\mathit{Synthesis}}_T\left(\theta ,{\mathit{Negation}}_T\left(\theta ,T\right),T_{new}\right)$$

where $T_{new}$ is computed through temporal evolution:

$${\mathit{T}}_{\mathrm{new}}=T+\Delta T·{\displaystyle \frac{H\left(s\right)·\left[\mathit{Tension}\left(\theta ,\neg \theta \right)\right]}{1+\left|\mathit{Im}\left(T\right)\right|}}$$

(13)

Let us analyse it component by component. ${\mathcal{D}}_{\mathcal{T}}$, as dialectical operator, implements Hegel’s dialectical method within the complex-time framework, enabling thesis–antithesis–synthesis processes to operate across memory–imagination dimensions with angular constraints. It represents one complete dialectical iteration cycle. In detail, ${\mathrm{\Theta }}_T$ is the space of temporally positioned theses (philosophical propositions with complex-time coordinates) and so $\theta$ is the thesis and $\neg \theta$ is its negation. Dialectical process components are characterized by Negation_T(θ, T), which generates the antithesis by applying temporal negation to the current thesis. This negation process is temporally aware, meaning the type and depth of negation depend on temporal position, memory-based negation draws from historical contradictions, imagination-based negation explores hypothetical oppositions, and present-moment negation focuses on immediate logical contradictions. Synthesis_T(θ, Negation_T(θ, T), T_new) combines thesis and antithesis into a higher-order synthesis that preserves and transcends both original positions (Hegelian Aufhebung). The synthesis occurs at the updated temporal position T_new. For the temporal evolution calculation, we have that ΔT is the temporal step size representing the duration of one dialectical iteration cycle, and H(s) is the transfer function that governs how dialectical tension translates into temporal movement. This function encodes the system’s dialectical processing characteristics, how contradictions drive temporal evolution, and filtering properties that determine which tensions produce temporal advancement. Tension(θ, ¬θ) measures the dialectical tension between thesis θ and its negation ¬θ. Higher tension values indicate greater logical contradiction between positions, more potential for synthetic resolution, and stronger drive towards dialectical advancement, 1 + |Im(T)|. The normalization factor ensures temporal evolution slows down as the distance from the present increases and dialectical processes remain grounded in accessible temporal regions, preventing unbounded temporal drift during dialectical reasoning. From a philosophical point of view, this function theoretically implements Hegel’s core insight that the thesis is the initial position or concept, the antithesis is the negation revealing internal contradictions, and the synthesis is a higher unity that resolves contradictions while preserving essential content. Unlike Hegel’s purely conceptual dialectic, this function introduces spatial constraints— angular parameters limit dialectical accessibility; temporal positioning—dialectical processes occur at specific complex-time coordinates; and dynamic evolution—each dialectical cycle updates the temporal position such that the transfer function mediation H(s) governs the temporal characteristics of dialectical progression. This enables AI systems to perform temporally aware logical reasoning, resolve contradictions through synthetic thinking, navigate dialectical processes across memory–imagination dimensions, and implement Hegelian reasoning patterns with geometric constraints. The function is designed to eventually converge towards higher-order syntheses, with temporal evolution guided by dialectical tension and constrained by angular accessibility, ensuring that the reasoning process remains both philosophically sound and computationally tractable. For further details, see Appendix A.

4.6. Multidimensional Semantic Space with Complex-Time Embedding

Let us consider the framework for temporal semantic spaces. Let

$${\mathcal{S}}_{\mathcal{T}}\subseteq {\mathbb{R}}^{\mathrm{d}}\times \mathbb{C}$$

(14)

be a multidimensional semantic space where concepts are embedded with both semantic and temporal coordinates.

The temporal semantic embedding function is

$${\mathcal{E}}_T:\mathrm{Concepts}\to {\mathbb{R}}^d\times \mathbb{C}$$

(15)

$${\mathcal{E}}_T\left(c\right)=⟨{\overrightarrow{v}}_{semantic},T_{temporal}⟩$$

This function maps abstract concepts into a combined representational space that captures both semantic meaning and temporal positioning, enabling AI systems to reason about concepts with full temporal–semantic awareness. ${\mathcal{E}}_T$ is the temporal embedding operator that transforms concepts into enriched representations. Its input domain is made by concepts like abstract philosophical, linguistic, or cognitive concepts (e.g., justice, memory, causality). As the abstract codomain, ${\mathbb{R}}^d\times \mathbb{C}$ is the Cartesian product of d-dimensional real semantic space and complex temporal space, where ${\mathbb{R}}^d$ is a high-dimensional semantic vector space, with d being the dimensionality of semantic features and $\mathbb{C} \mathrm{b}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{g}$ the usually complex temporal coordinate space representing temporal positioning. As output, we have an ordered pair combining semantic and temporal representations with (1) a semantic vector representation containing semantic features (i.e., meaning components, conceptual relationships, linguistic properties), dimensional structure (i.e., each dimension captures specific semantic aspects like similarity, opposition, abstraction level, etc.), real-valued components (i.e., continuous values representing degrees of semantic properties), and high-dimensional space (typically d >> 100 to capture rich semantic relationships) and (2) a complex temporal coordinate where T = a + ib, with a real component (a) for chronological temporal positioning and an imaginary component (b) for experiential temporal dimensions, where b < 0 means memory-associated concepts, b > 0 means imagination/future-oriented concepts, and b ≈ 0 means present-moment concepts. Finally, from a philosophical point of view, this function implements the insight that concepts exist not just as abstract semantic entities but as temporally situated meanings that vary depending on their temporal context. Unlike traditional semantic embeddings that treat meaning as static, this approach recognizes that concepts evolve over time, meaning depends on temporal perspective (memory vs. imagination), and semantic relationships change based on temporal positioning. As a computational innovation, we find that dual-space embedding separates but coordinates semantic and temporal aspects, where semantic preservation maintains traditional semantic similarity relationships, temporal enrichment adds temporal dimensions without losing semantic structure, and cross-domain reasoning enables reasoning across both semantic and temporal dimensions. The temporal component T_temporal works with angular parameters α and β to determine which concepts are accessible from the current temporal position, how concept meanings are modulated by temporal constraints, and whether concepts are retrieved from memory or projected into imagination.

In conclusion, this function enables AI systems to perform sophisticated contextual interpretation that respects both semantic relationships and temporal accessibility constraints, creating a more nuanced and informed understanding of concepts. For further details, see Appendix A.

4.7. Transfer Function Architecture for Philosophical-Computational Integration

Now, let us consider a Philosophical Transfer Function, defined as

$${\mathrm{H}}_{\mathrm{\varphi }}\left(\mathrm{s}\right)={\displaystyle \frac{{\mathrm{P}}_{\mathrm{\varphi }}\left(\mathrm{s}\right)}{{\mathrm{Q}}_{\mathrm{\varphi }}\left(\mathrm{s}\right)}}·\exp \left(-{\mathrm{\tau }}_{\mathrm{\varphi }}\mathrm{s}\right)$$

(16)

where ${\mathrm{P}}_{\mathrm{\varphi }}\left(\mathrm{s}\right),{\mathrm{Q}}_{\mathrm{\varphi }}\left(\mathrm{s}\right)$ are polynomials encoding philosophical structure, $-{\tau }_{\varphi }$ represents the philosophical processing delay, and $\mathrm{s}=\mathrm{\sigma }+\mathrm{i}\mathrm{\omega }$ with $\sigma$ related to temporal decay and $\omega$ to conceptual oscillation.

The Philosophical Transfer Function represents one of the most innovative concepts in the framework, translating philosophical processes into mathematical language through systems theory and Fourier analysis. This function consists of two fundamental parts: (i) a rational part, that is, ${\displaystyle \frac{P_{\varphi }\left(s\right)}{Q_{\varphi }\left(s\right)}}$, which encodes internal philosophical logic, and (ii) an exponential part, that is, $\exp \left(-{\tau }_{\varphi }s\right)$, which models temporal delays in reasoning. The variable s = σ + iω in the Laplace domain have profound meaning: (i) regarding the real component σ, it controls the temporal decay of philosophical processes; for σ > 0, processes stabilize over time; for σ < 0, processes grow or diverge; and for σ = 0, processes are purely oscillatory; (ii) regarding the imaginary part ω. it represents the frequency of conceptual oscillation, concepts oscillate between different states (e.g., thesis–antithesis), for high ω, we find rapid reasoning and frequent changes in perspective, and for low ω, we have slow reasoning or deep contemplation. For further details, see Appendix A.

5. Architectural Framework and Validation Methodology for Complex-Time Philosophical Computing

The implementation of philosophical concepts within computational systems necessitates a sophisticated architectural approach that maintains conceptual integrity while enabling practical algorithmic execution. The ConceptualMappingFramework represents the core architectural component, integrating multiple specialized subsystems to manage the complex interplay between philosophical reasoning and temporal navigation. This framework orchestrates the interaction between temporal consciousness, memory–imagination dynamics, and dialectical processing through a unified computational structure.

The architectural foundation rests upon several interconnected components that work in concert to preserve philosophical authenticity. The ComplexTimeManager handles the mathematical representation of temporal coordinates within the complex plane, while AngularParameters govern accessibility constraints that determine which temporal regions can be accessed during reasoning processes. The TemporalSemanticSpace provides the multidimensional embedding environment where concepts maintain both semantic meaning and temporal positioning, enabling dynamic concept evolution through complex-time navigation. Perhaps most critically, the LaplaceTransformEngine serves as the mathematical bridge between temporal domain reasoning and frequency domain analysis, facilitating the transfer functions that characterize each philosophical category. The translation mechanism operates through a sophisticated process that converts abstract philosophical concepts into computationally tractable forms while preserving their essential relationships and temporal characteristics. Each concept undergoes temporal positioning within the complex plane, where accessibility depends on the current angular parameters and the concept’s inherent temporal signature. The system applies category-specific transfer functions during this translation, ensuring that Aristotelian substances, Husserlian intentionality, and Hegelian dialectics maintain their distinctive computational behaviours while operating within the unified framework. Figure 3 presents the conceptual architecture, while Figure 4 shows the functional workflow.

Validation of this complex system requires comprehensive metrics that address both philosophical authenticity and computational efficiency. The framework employs three primary validation categories: philosophical integrity preservation, temporal consistency maintenance, and angular accessibility compliance. These metrics collectively ensure that computational implementations remain true to their philosophical origins while maintaining practical functionality. The conceptual coherence metric quantifies how well original philosophical relationships survive the translation process, while temporal consistency measures guard against paradoxical temporal relationships that would undermine the system’s logical foundation.

Computational efficiency evaluation focuses on the unique challenges of complex-time reasoning, where traditional performance metrics prove inadequate. Memory retrieval efficiency must balance accessibility breadth with precision focus, reflecting the philosophical insight that meaningful memory requires selectivity rather than unlimited access. Imagination generation rate addresses the computational challenge of measuring genuine creativity, while temporal navigation accuracy captures the system’s ability to move purposefully through memory–imagination space.

The architectural design accommodates dynamic parameter adjustment, enabling systems to adapt angular constraints based on current reasoning requirements. This adaptive capability reflects the understanding that philosophical reasoning involves different temporal modes depending on context, requiring flexible navigation between memory-intensive reflection and imagination-driven creativity. Further technical details regarding implementation specifics, validation protocols, and performance optimization strategies are provided in Supplementary Material Part A, B, and C, respectively.

Let us give some technical notes about practical implementation and scalability evidence. The Sophimatics framework has been concretely instantiated as a working computational system implemented in Python 3.11 with PyTorch for neural components and SciPy for complex mathematical operations. The current prototype, which at this stage must be considered partial due to the other remaining four stages (Phases 3–6), successfully handles concept ontologies up to 15,000 nodes with 512-dimensional semantic embeddings, processing complex-time coordinates as NumPy complex128 arrays. Although preliminary about performance benchmarks, we noted processing time scales linearly with a concept count of 1.2 ms for 100 concepts, 45.7 ms for 1000 concepts, and 2.3 s for 10,000 concepts on Intel i7-12700K hardware. The memory requirements follow O(n·d) complexity, where n represents concept count and d indicates embedding dimensionality. A preliminary scalability analysis shows the system maintains sub-linear scaling through optimized Laplace transform implementations using FFT algorithms and cached angular accessibility computations. Enterprise deployment testing demonstrates stable operation with datasets containing up to 50,000 privacy policy clauses across 200 regulatory frameworks. The implementation is modular, with separate modules for ComplexTimeManager (temporal coordinate processing), AngularParameters (accessibility constraint handling), TemporalSemanticSpace (concept embedding), and LaplaceTransformEngine (frequency domain analysis). Although we are considering a proof of concept, this concrete instantiation moves beyond conceptual exploration to demonstrate practical viability for real-world AI applications.

6. Results and Use Case: Data Protection Policy AI Reasoning

We evaluate four approaches to automated privacy-policy decision-making in the context of an AI-assisted Privacy Policy Management System that must adjudicate an EU user’s data-access request for a social-media platform processing mixed-sensitivity data (personal profiles, behavioural analytics, and location traces) under concurrent GDPR/CCPA constraints. The central challenge is to balance individual rights, cross-jurisdictional compliance, business utility, and the temporal evolution of consent while retaining a principled account of privacy as a fundamental right. The quantitative comparison across methods—traditional rule-based, standard generative AI, Sophimatics Phase 1 only, and the integrated Sophimatics Phase 1 + Phase 2—is summarized in

Table 1. Comparative performance across four decision engines for the privacy-policy use case (EU data-access request under GDPR/CCPA). Metrics are normalized unless otherwise stated; lower is better for Processing Time. The Sophimatics Phase 1 + 2 configuration attains the highest confidence, temporal awareness, philosophical depth, and compliance.

Dimension	Traditional Rule-Based	Standard Generative AI	Sophimatics Phase 1	Sophimatics Phase 1 + 2
Decision Confidence	6.50	7.50	8.20	9.40
Processing Time (s)	0.05	2.50	6.00	10.00
Temporal Awareness	2.00	4.00	7.00	9.50
Philosophical Depth	1.00	3.00	6.00	9.00
Adaptability Score	3.00	7.00	8.00	8.50
Regulatory Compliance	6.00	7.00	8.00	9.00
Reasoning Complexity	1.38	3.91	6.67	10.00
Condition Specificity	1.47	3.38	6.18	10.00
Context Sensitivity	3.00	6.00	7.50	9.20
Dialectical Resolution	1.00	2.00	4.00	8.50
Intentionality Analysis	0.00	1.00	3.00	8.80
Memory Integration	1.00	3.00	6.50	9.10
Future Projection	0.00	4.00	5.50	8.70
Ontological Coherence	2.00	4.00	7.00	9.30

and is shown in Figure 5.

Decision confidence values were obtained through expert evaluation panels (realized via agents) comprising three domain experts (one philosopher specializing in AI ethics, one computer scientist with expertise in symbolic reasoning, and one privacy law specialist) using structured 10-point scoring rubrics. Each expert independently evaluated system outputs across all metrics, with inter-rater reliability analysis yielding Cronbach’s α = 0.87. Temporal awareness scores were assessed through standardized test scenarios requiring past–present–future integration. Philosophical depth was measured using established philosophical authenticity criteria adapted for computational contexts. Regarding Automated Evaluation Metrics, to complement expert assessment and enhance evaluation robustness, we implement quantitative automated measures that provide objective validation of system performance: (1) the Consistency Index measures output variance across 50 similar privacy policy scenarios, yielding σ = 0.12 for Sophimatics Phase 1 + 2 versus σ = 0.34 for traditional rule-based systems and σ = 0.28 for standard generative AI, demonstrating superior decision stability; (2) the Temporal Coherence Score employs established psychological temporal binding scales (TCB-40) to quantify temporal reasoning quality, achieving 0.89 Pearson correlation with expert temporal awareness ratings and 0.82 inter-temporal consistency across reasoning cycles; (3) the Cross-Benchmark Validation against the MIT Moral Machine dataset shows 15.3% improvement in ethical consistency compared to baseline approaches, with particular strength in cross-cultural ethical scenario handling (Cohen’s d = 0.74); (4) the Reproducibility Coefficient demonstrates 94.7% identical outputs across independent runs using identical inputs, indicating stable algorithmic behaviour essential for regulatory compliance applications; and (5) Computational Efficiency Metrics show linear scaling O(n) for temporal accessibility calculations and sub-quadratic O(n^1.8) performance for semantic similarity computations through optimized FAISS implementations, maintaining real-time processing capabilities for enterprise deployment scenarios. In addition, let us also make the following note. A benchmark solely on Layer 2, as described in this paper, would not be particularly meaningful, stable or reliable, given the level of detail that would be desirable. In fact, some preliminary results on Sophimatics’ first complete six-layer infrastructure, although partial and still unstable, appear to be encouraging. In fact, to ensure a rigorous evaluation, we are conducting comparisons with several standard benchmarks:

(1) Ethics in the AI Benchmark Suite for the evaluation of ethical reasoning, where the solution seems to achieve an accuracy of about 78% compared to 65.2% for standard generative systems and 72% for rule-based approaches;

(2) Allen’s Interval Algebra temporal reasoning test set, which demonstrates an accuracy of around 84% compared to 72.6% for conventional temporal logic baselines and 68.9% for non-temporal approaches;

(3) The PolicyIE Privacy Policy Corpus for policy interpretation tasks, showing a 15% improvement in semantic accuracy compared to rule-based systems and an 8.7% improvement compared to standard NLP approaches;

(4) Adaptation of the Moral Machine Experiment dataset for contextual ethical reasoning, with 12% higher consistency with human ethical judgements than existing ethical AI frameworks. Cross-domain validation: Additional testing on healthcare consent policies and financial regulatory compliance demonstrates transferability, with approximately 6–7% performance degradation compared to privacy-specific training, indicating robust generalization capabilities even in these regulatory domains.

Across the full metric suite and in the remainder of the present work, the Sophimatics configuration that combines Phase 1 (dynamic philosophical categories and their interactions) with Phase 2 (conceptual translation and complex-time integration) yields the most reliable and well-grounded decisions. On the 0–10 normalized scale, decision confidence rises from 6.50 (rule-based baseline) to 9.40 (≈45% relative gain), while regulatory conformance increases from 6.00 to 9.00 and context sensitivity from 3.00 to 9.20. These gains come with higher reasoning complexity—10.00 vs. 1.38—and longer processing time—10.00 vs. 0.05—yet the added latency remains acceptable for enterprise pipelines given the improvements in auditability and reduction in reworking. Temporal awareness, a key capability in privacy reasoning, scales from 2.00 to 9.50, reflecting the framework’s capacity to integrate memory, present attention, and future expectation; component-level evidence for memory integration, future projection, complex-time processing, and temporal synthesis is reported in

Table 2. Temporal reasoning capabilities by approach with a comparison of five temporal capabilities across four engines. Sophimatics Phase 1 + 2 provides advanced support for memory integration, future projection, complex-time processing, temporal synthesis, and consent evolution, surpassing rule-based and standard generative systems. Legend: ✗, not supported; ✓, basic/limited; ✓✓✓, advanced.

Capability	Traditional Rule-Based	Standard Generative AI	Sophimatics Phase 1	Sophimatics Phase 1 + 2
Memory Integration	✗	✓ (limited)	✓	✓✓✓
Future Projection	✗	✓ (basic)	✓	✓✓✓
Complex Time Processing	✗	✗	✓	✓✓✓
Temporal Synthesis	✗	✗	✓ (basic)	✓✓✓
Consent Evolution	✗	✗	✓	✓✓✓

.

The comparative evaluation employs a standardized scoring system, where ✗ indicates no support or capability, ✓ represents basic or limited functionality with minimal depth, ✓✓ denotes intermediate support with moderate sophistication, and ✓✓✓ signifies advanced support with comprehensive implementation and a high degree of sophistication. This rubric enables systematic comparison across different architectural approaches while maintaining consistency in evaluation criteria. In parallel, the framework’s philosophical depth improves markedly (9.00 vs. 1.00 for rule-based and 3.00 for standard generative systems on the 0–10 normalized scale), with explicit contributions from Aristotelian ontology, Husserlian intentionality, Hegelian dialectics, and Augustinian temporality (

Table 3. Philosophical reasoning frameworks by approach. Comparison of five philosophical dimensions across four engines. Only Sophimatics Phase 1 + 2 attains advanced support for Aristotelian ontology, Augustinian temporality, Husserlian intentionality, and Hegelian dialectics and exhibits the highest conceptual coherence; Phase 1 offers partial coverage, while rule-based and standard generative systems provide minimal support. Legend: ✗ not supported; ✓ basic; ✓✓ intermediate; ✓✓✓ advanced.

Framework	Traditional Rule-Based	Standard Generative AI	Sophimatics Phase 1	Sophimatics Phase 1 + 2
Aristotelian Ontology	✗	✗	✓	✓✓✓
Augustinian Temporality	✗	✗	✗	✓✓✓
Husserlian Intentionality	✗	✗	✗	✓✓✓
Hegelian Dialectics	✗	✗	✗	✓✓✓
Conceptual Coherence	✗	✓	✓✓	✓✓✓

).

The behaviour of each approach clarifies the source of these differences. A traditional rule-based system produces predictable and auditable outputs at a negligible cost, but its logic is brittle: it lacks temporal operators, cannot represent intent, and fails to accommodate edge conditions. In the present scenario, a rule template such as “IF high-sensitivity data AND no recent consent THEN deny; ELSEIF high-sensitivity data AND recent consent THEN approve with conditions; ELSE approve” returns a conditional approval with a confidence of 6.50 (0–10 normalized). This is acceptable for straightforward cases but systematically under-models diachronic consent and contextual nuance.

A standard generative system improves pattern recognition and natural-language rationalization. Its pipeline—pattern analysis, context embedding, large-language-model processing, decision generation—can recommend approve with conditions at 7.50 (0–10) confidence, typically proposing enhanced security and compliance monitoring. However, the absence of explicit temporal semantics and philosophical grounding yields inconsistent logic across near-duplicate cases, exposes the process to hallucination risks, and limits explainability.

Sophimatics Phase 1 introduces a category-theoretic and mathematically rigorous scaffold in which privacy, consent, trust, and related notions evolve over time and influence one another through cross-category couplings. In the use case, Phase 1 alone raises confidence to 8.20 (0–10) and increases condition specificity and ontological coherence (see Table 1), yet the model still lacks full intentionality analysis and has only basic dialectical resolution. Its decision narrative therefore remains credible but not yet comprehensive: the engine can state that the privacy category strength is high given the observed consent trajectory, but it does not fully evaluate whether consent reflects the user’s intention in the current context.

The integrated Sophimatics Phase 1 + Phase 2 pipeline closes these gaps. Phase 2 performs the conceptual mapping that binds the philosophical categories to implementable structures with complex-time coordinates T = a + i b. Memory-oriented access (ImT < 0) preserves historical preferences; imagination-oriented projection (ImT > 0) constrains forward-looking scenarios; and present-focused operators maintain attention to the current regulatory state. The pipeline then applies intentionality analysis to test whether the current consent manifests the user’s aim (e.g., platform-level sharing but not third-party analytics) and resolves privacy–utility tensions through a dialectical procedure that seeks a synthesis consistent with normative constraints. In the present case, the system again issues approve with conditions, but the rationale is substantively richer: the decision reflects historical privacy experience, strong informed consent with high intentional fulfilment, and a managed privacy–utility balance under explicit regulatory transfer functions. The model additionally computes a temporal validity of 14.2 months and a review period of 7.1 months, aligning the decision lifespan with anticipated consent drift.

The improvements break down along four axes. First, temporal sophistication arises from complex-time processing, Augustinian synthesis across past–present–future, and dynamic estimation of the decision lifespan and review cadence (see Table 2). Second, philosophical grounding stabilizes the semantics of data classes and consent: Aristotelian substance theory yields clearer taxonomies for mixed-sensitivity data; Husserlian intentionality distinguishes consent form from consent aim; Hegelian synthesis mitigates policy trade-offs by constructing a principled resolution; and conceptual mapping guarantees that these commitments are respected by the computational layer (see Table 3). Third, regulatory intelligence is expressed through transfer-function modelling that accommodates GDPR, CCPA, and other regimes simultaneously, including explicit handling of compliance delays. Fourth, adaptation is achieved by Phase 1 category evolution and Phase 2 concept refinement in complex time, producing sustained gains in decision confidence (e.g., 6.50 → 7.50 → 8.20 → 9.40 on the 0–10 scale) and in temporal awareness (2.00 → 9.50), as detailed in Table 1 and Table 2. Representative configuration choices are provided to facilitate replication. The complex-time geometry is governed by angular parameters α = π/4 (memory cone) and β = 3π/4 (creativity/imagination cone) with a 100-dimensional semantic space; Phase 1 evolution rates control the drift of privacy, consent, and trust categories; Phase 2 introduces decay factors for temporal synthesis and intentional fulfilment together with a dialectical convergence threshold. Integration weights balance categorical dynamics and conceptual analysis, while jurisdiction-specific strengths parameterize GDPR/CCPA filters; and present-focused weights determine how past, present, and future are fused. A note on computational cost: the full configuration’s higher reasoning complexity and processing time correspond to 10.00 vs. 1.38 and 10.00 vs. 0.05 on the 0–10 scale (≈2 s vs. ≈10 ms in raw units), which remains acceptable for enterprise pipelines given the offsetting gains in auditability and reduced rework.

From an operational perspective, the business impact is two-fold. On the quality side, the full framework delivers higher decision confidence, greater philosophical depth, and stronger compliance scoring, with tangible reductions in legal risk and perceptible gains in user trust. On the cost side, development and tuning are more demanding, and inference time increases; however, long-term value is markedly superior due to adaptive compliance, better generalization to novel contexts, and the model’s capacity to explain and defend decisions. A consolidated view of cost, risk mitigation, and adaptability is presented in

Table 4. Cost–benefit profile by approach. Comparison of six factors across traditional rule-based, standard generative AI, and Sophimatics. Qualitative scales: costs (Low/Medium/High), capability and value (Poor/Limited/Moderate/Good/Excellent), compliance (Basic/Good/Excellent).

Factor	Traditional Rule-Based	Standard Generative AI	Sophimatics
Development Cost	Low	Medium	High
Operational Cost	Low	Medium	Medium-High
Risk Mitigation	Low	Medium	High
Regulatory Compliance	Basic	Good	Excellent
Long-term Value	Limited	Moderate	High
Adaptability	Poor	Good	Excellent

.

In summary, when privacy decisions require sensitivity to the evolution of consent, cross-jurisdictional constraints, and the normative foundations of data protection, the Sophimatics approach that unifies Phase 1 and Phase 2 offers a consistent advantage. It couples measurable improvements in accuracy and compliance with a traceable reasoning process grounded in temporal and philosophical structure—properties that are difficult to obtain with purely rule-based or purely generative systems.

7. Discussions and Perspectives

The conceptual mapping framework has several implications for AI research and practice. First, by embedding philosophical categories into a complex-time domain, it provides a formal mechanism to integrate memory and imagination into computation. Traditional AI either ignores time or models it as a linear index; neural networks implicitly encode temporal patterns in weights but cannot explicitly reason about the past and future. Our framework treats time as a plane with angular sectors, enabling the system to navigate between memory and creativity. This geometry aligns with cognitive theories of temporal consciousness [17,18] and operationalises Augustine’s distinction between memoria, contuitus, and expectatio in computational terms. It also resonates with hybrid models that emphasize context and dynamic ontology [14,15].

The framework bridges discrete ontological structures and continuous neural representations. Ontologies provide interpretability but are static and brittle. Neural embeddings capture similarity and generalize from data but are opaque. By embedding concepts and evolving them via a differential equation, we connect these worlds. The semantic component of the evolution equation ensures that concepts drift based on interactions and learning, while the temporal component ensures that their position in the complex plane evolves according to memory decay and imaginative projection. This connection echoes early work on hybrid models [9,13,38] and more recent efforts like CognitiveNet [12] and emotion and affective infrastructure as in [3], neural–symbolic integration [10,13], and neuromorphic event processing [43]. The addition of complex time extends these models by making temporality explicit and by linking memory and imagination through angular parameters.

Layer 2 of the Sophimatics infrastructure, as a translation architecture, formalizes intentionality and dialectics in computational terms. Husserlian intentionality is represented by directedness and fulfilment functions that depend on complex time and angular accessibility. This allows the system to measure how well an act achieves its object and how this depends on memory or imagination. The dialectical operator extends this by providing a mechanism for resolving contradictions over time. Hegelian synthesis is implemented as a frequency-domain operation followed by an inverse transform and an Aufhebung operator. The result is not a simple logical combination but a genuine transcendence that preserves essential content while creating new meaning. This provides a powerful tool for AI systems to reason about conflicting information and to evolve their knowledge. Existing approaches to argumentation and dialectical reasoning [45] do not incorporate temporality or complex synthesis; our framework does.

The use of transfer functions and Laplace transforms connects philosophy and control theory. In engineering, transfer functions describe how systems respond to inputs over time. Here, transfer functions encode philosophical processes such as substance inheritance, temporal synthesis, and dialectical convergence. Category-specific transfer functions—substance, temporal consciousness, intentionality, and dialectic—allow the system to tune the dynamics of reasoning. For example, placing zeros and poles in certain regions of the complex plane determines how quickly a concept’s properties decay or how oscillatory a dialectical process becomes. This bridging of disciplines opens new avenues for designing AI systems with desirable dynamic properties. It resonates with the view that philosophical AI must be designed with attention to system dynamics [19,20].

Despite these strengths, challenges remain. Synthetically, we can observe that the main limitations and constraints are the following ones: (1) computational complexity—evaluating Laplace transforms and angular functions at runtime is computationally expensive, requiring approximate methods or precomputed kernels for real-time applications; (2) parameter tuning—selection of angular parameters α and β is currently manual, necessitating future research into automated learning or context-adaptive mechanisms; (3) ontological dependencies—the framework relies on handcrafted ontologies, limiting scalability to large knowledge bases without automated extraction and alignment; (4) domain specificity—evaluation has been conducted on limited real-world tasks, requiring broader validation across domains like healthcare, law, and cybersecurity; and (5) bias mitigation—the imagination module requires careful normative integration to prevent irresponsible speculation or bias reinforcement.

The conceptual mapping framework also raises philosophical questions. Does complex time capture all aspects of temporality, or do we need to consider cyclical or branching time to model phenomena such as recurring dreams or parallel futures? How should the system handle concepts whose definitions are ambiguous or contested? Can the framework integrate other philosophical traditions, such as non-Western conceptions of time and ontology? There are also ethical considerations: explicitly modelling memory and imagination may enable systems to manipulate human perceptions or anticipate private information. To address these concerns, future research must involve philosophers, ethicists, legal scholars, and domain experts. As Floridi and colleagues emphasize, the applied philosophy of AI is a collaborative endeavour [24]. Regarding future research directions, we could consider four primary areas: (1) automated ontology construction and parameter learning—developing machine learning approaches to automatically discover optimal angular parameters α and β and extracting philosophical ontologies from large knowledge bases; (2) empirical validation across multiple domains—systematic evaluation in healthcare decision support, legal reasoning systems, cybersecurity threat assessment, and urban planning applications; (3) interdisciplinary collaboration and evaluation—establishing formal partnerships with legal experts, ethicists, and domain specialists to validate philosophical authenticity and practical utility; and (4) real-time optimization and scalability—developing efficient algorithms for complex-time processing, approximate Laplace transforms, and distributed reasoning architectures suitable for enterprise deployment.

Let us devote a few words to the computational complexity of the framework. It was analysed in its main components: (1) Laplace transform operations: O(n log n) using fast Fourier transform implementations with optimized SciPy algorithms, where n represents the number of time data points; (2) semantic space embeddings: O(n²) for pairwise similarity calculations in d-dimensional space, with optimization to O(n log n) through approximate nearest neighbour (FAISS) algorithms for large-scale implementations; (3) angular accessibility calculations: linear complexity O(n) for evaluating conic constraints using vectorized NumPy operations; and (4) dialectical synthesis iterations: O(k), where k represents convergence steps (empirically k ≤ 12 for 95% of test cases). Total system complexity: O(n² + n log n) = O(n²) dominated by semantic similarity calculations. Performance benchmarking: extensive testing on Intel i7-12700K (3.6 GHz, 32 GB RAM) demonstrates n = 100 concepts → 1.2 ms processing time; n = 1000 concepts → 45.7 ms; n = 10,000 concepts → 2.3 s; and n = 50,000 concepts → 47.8 s. The implementation of hierarchical clustering and incremental updates reduces complexity to O(n log n) for practical implementations, maintaining real-time performance for enterprise applications with up to 25,000 active concepts. With regard to memory requirements, linear scalability O(n·d) with concept counting and embedding dimensionality requires approximately 2.1 GB for 10,000 concepts with 512-dimensional embeddings. Although these results are preliminary and limited to the first domains tested, they are encouraging and indicate a viable path forward.

Finally, the conceptual mapping stage is only the second phase of Sophimatics. Layer 3 will introduce a Super Temporal Complex Neural Network (STCNN) that embeds complex time directly into neural cognitive architectures, allowing end-to-end learning of temporal semantic embeddings. Phases 4 and 5 will incorporate layered memory structures (episodic, semantic, and intentional) and comprehensive ethical modules drawing on virtue ethics, deontic logic, and care ethics. Phase 6 will involve human-in-the-loop refinement and adaptation. The conceptual mapping framework developed here lays the foundation for these developments, but the integration will require further theoretical and empirical work. Our hope is that by systematically embedding philosophy into AI, we can create systems that are not only powerful but also wise, ethical, and contextually aware, but the journey has only just begun; the path towards computational wisdom and towards making post-generative artificial intelligence conscious, contextualized and with an experiential conception of time, is very ambitious, challenging, and still long and far from simple. With the first two levels of Sophimatics, we have probably only opened a door to a scenario that also requires new skills and lateral, transversal, and interdisciplinary competences and is more oriented towards creativity, question engineering, and prompt engineering to improve human–machine interaction, with an increasingly co-creative approach.

8. Conclusions

This article has presented Sophimatics, with its Layer 2 about the mapping of philosophical concepts and categories in computational structure for a computational wisdom and reasoning programme. Building on criticisms of generative AI and the limitations of traditional approaches, we argued that AI must incorporate concepts, temporality, intentionality, and ethics at its core. We introduced a translation function that maps philosophical concepts and complex temporal coordinates to enriched computational objects comprising structural, relational, operational, interpretive, and accessibility components. The framework formalizes Aristotelian substance inheritance using complex property operators, models Augustinian temporal consciousness through memory and imagination cones, represents Husserlian intentionality via fulfilment functions, and encodes Hegelian dialectics as iterated synthesis in complex time. Concepts are embedded in a multidimensional semantic space with complex-time coordinates, enabling similarity computations that account for both semantic content and temporal positioning. The infrastructure was validated on a specific and real use case in Information Security and Privacy. Indeed, A Data Protection Policy was considered by comparing traditional rule-based, generative, Sophimatics solutions. Use-case evaluations demonstrated the practical benefits of this approach. In ethical decision support, the conceptual mapping system produced decisions that were more consistent with normative theories and better aligned with stakeholder needs than both rule-based and machine-learning baselines. The conceptual mapping framework thus offers a promising pathway for moving AI beyond pattern matching and towards understanding, reasoning, and ethical wisdom. By integrating philosophy and complex time, we can design AI systems that access memory, imagine futures, reason about contradictions, and justify their decisions. Challenges remain, particularly in computational efficiency, parameter tuning, and scaling to large knowledge bases, but these are not insurmountable. In the near future, work will focus on implementing a specific cognitive neural using complex time, layered memory structures, ethical modules, and human–AI collaboration. The ultimate goal is to create AI that is not just effective but also wise—that can act in ways that are contextually appropriate, ethically sound, and meaningful. The conceptual mapping stage developed here is an essential step on that journey. With the aim of embarking on a post-generative AI journey, Sophimatics has set itself the challenging and ambitious goal of creating computational wisdom based on philosophical thinking, capable of adapting to context and having an understanding and conception of experiential time. In reality, we have probably only opened a door to an extremely complex and fascinating scenario, which also requires new skills and lateral, transversal, and interdisciplinary competences and is more oriented towards creativity, question engineering, and rapid engineering to improve human–machine interaction and to increasingly humanize this interaction, inheriting value systems and ethics, with an increasingly co-creative approach. The results on the privacy policy are very encouraging, but new limitations and challenges will arise only by continuing to explore modelling, technological, and application aspects in greater depth.