Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty

Neutrosophic Sets and Systems, Vol. 94, 2025 University of New Mexico Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty Alaa Hassan1* and Reda Ahmed2 1,2Faculty of Computers and Informatics, Zagazig University, Zagazig, Sharqiyah 44519, Egypt Email: [email protected] Abstract: Machine learning models are increasingly used in high-stakes settings such as critical care, energy management, and environmental monitoring. These settings are difficult because the available data can be incomplete, delayed, low quality, or internally contradictory. Standard models still tend to return a single prediction or a single confidence score, even when the situation is not actually settled. This creates false certainty and can lead to unsafe actions. We introduce a framework called Neutrosophic Machine Learning State Modeling (NMLSM). In this framework, the system does not describe each hypothesis with one number. Instead, it learns three separate quantities for every hypothesis at every point in time: (i) how much current evidence supports it, (ii) how much current evidence contradicts it, and (iii) how much uncertainty remains because the information is missing, unreliable, or ambiguous. These three quantities are learned independently and are allowed to coexist. For example, the model is allowed to say that a hypothesis is both supported and challenged at the same time, and also admit that part of the situation is still unresolved. This reflects real conditions, rather than forcing a premature decision. The framework also models how these three quantities evolve as new information arrives. It uses both raw observations (what was measured) and context (how trustworthy those measurements are). In training, the model is supervised not only on how well it can recognize supporting evidence, but also on how well it can recognize contradictory evidence and how honestly it can represent remaining uncertainty. We demonstrate the approach in an intensive-care scenario by tracking two possible diagnoses for the same patient in parallel. Instead of collapsing to a single “most likely” diagnosis too early, the model maintains a transparent view of support, doubt, and refutation for each diagnosis as the case unfolds. This capability is important in situations where taking the wrong action too confidently can be more dangerous than admitting that the system is not yet sure. Keywords: Neutrosophic modeling; machine learning under uncertainty; temporal state estimation; indeterminacy quantification; multi-hypothesis clinical monitoring; dynamic truth– indeterminacy–falsity triplet. 1. Introduction Machine learning systems are widely deployed in domains where the underlying environment is unstable, only partially observed, and sometimes internally contradictory [1]. Clinical monitoring in intensive care, distributed energy coordination in microgrids, and fine-grained soil health management in precision agriculture all share a common structural property: the system state cannot be reduced to a single clean answer. Instead, multiple explanations of the current situation can coexist, and each explanation may be simultaneously (i) supported by some evidence, (ii) challenged by other evidence, and (iii) obstructed by missing or unreliable data [2]. Conventional machine learning pipelines are not designed to express this structure. They are structurally biased toward producing one answer, or one probability distribution over mutually exclusive answers, even in cases where reality is not exclusive, not stable, and not fully known [3]. Neutrosophic Sets and Systems, Vol. 94, 2025 418 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty The standard probabilistic view assumes that uncertainty can be captured by a probability value. In a binary classifier, for example, the model outputs a probability that a hypothesis is true. This assumes (1) that the hypothesis has a single correct truth state, (2) that all relevant evidence is available and internally consistent, and (3) that unexplained behavior can be treated as noise [4]. These assumptions fail in real-world, high-stakes environments. In clinical triage, a patient may present with overlapping presentations (e.g., infectious process and cardiogenic instability at the same time). In such settings, it is not meaningful to ask for a single probability “of the true diagnosis,” because there may not be a single dominant diagnosis then, and the evidence that exists may partially support and partially contradict multiple concurrent hypotheses [5]. Fuzzy representations generalize this by assigning each hypothesis a membership degree, interpreted as “the degree to which this hypothesis holds.” Fuzzy membership captures partial truth in smooth systems but still treats “not true” as the mathematical complement of “true,” and it does not explicitly represent epistemic gaps [6]. In particular, fuzzy formalisms do not distinguish between “the data suggest that this is false” and “the data are missing, corrupt, or disputed, so we do not know.” Neutrosophic logic was developed to explicitly separate three independent aspects of a statement: (i) the degree to which it is supported, (ii) the degree to which it is refuted, and (iii) the degree to which it remains indeterminate [7]. In neutrosophic form, a statement does not receive one scalar, but a triplet (T, I, F), where T quantifies supporting evidence (truth support), F quantifies contradicting evidence (falsity support), and I quantifies indeterminacy, which includes missing information, conflicting measurements, sensor unreliability, reporting delay, or irreducible ambiguity. Importantly, neutrosophic logic does not require that T + I + F = 1. This is not a flaw. This is the core feature: the formalism allows a hypothesis to be simultaneously supported, contradicted, and unresolved [8,14]. The present work extends this neutrosophic view from static statements to dynamic machinelearned system states. The proposed framework, termed NMLSM, treats the state of each hypothesis of interest as a learned neutrosophic triplet Sh(t) = (Th(t), Ih(t), Fh(t)) defined at time t. Here, Th(t) denotes the learned degree of support for the hypothesis at time t; Fh(t) denotes the learned degree of structured contradiction at time t; and Ih(t) denotes the learned degree of active indeterminacy at time t, which is not reducible to noise but is itself a modeled quantity. Because Sh(t) is explicitly indexed by time, the framework not only describes what is currently believed about the system. It also enables direct modeling of how belief, contradiction, and uncertainty evolve [9]. To achieve this, NMLSM introduces a learnable update operator Φθ, parameterized by θ, that maps the current neutrosophic state of a hypothesis and current observations to a future neutrosophic state. The general update equation is written as Sh(t + Δt) = Φθ(Sh(t), O(t), C(t)), where O(t) denotes observations available at time t and C(t) denotes contextual qualifiers at time t. Observations O(t) include raw measurements such as vital signs, laboratory values, sensor readings from field devices, energy production/consumption reports, and other signals that are commonly ingested by machine learning systems. Context C(t) includes metadata that influence the interpretability of each observation, such as sensor calibration quality, acquisition delay, missing channels, compression artifacts in imaging, declared but unverified self-report, and known sources of adversarial distortion. By including C(t), the update learns not only from “what was measured,” but also from “how trustworthy and how complete those measurements were” [10]. This treatment creates a conceptual shift. In conventional pipelines, uncertainty is usually handled as noise, and contradiction is often resolved by forcing a single most likely label. In the proposed formulation, both uncertainty and contradiction are considered part of the state itself. The model Neutrosophic Sets and Systems, Vol. 94, 2025 419 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty is not punished for admitting ambiguity; instead, the model is trained to estimate ambiguity. In other words, the question is no longer “Which label is correct?” but “At this time step, how much support, how much doubt, and how much refutation exist for each active hypothesis, and how are these evolving?” [11] From an applied perspective, this matters in any domain where acting with unjustified confidence is dangerous. In critical care, premature diagnostic collapse can lead to the wrong therapy being prioritized first. In a microgrid, blindly trusting an energy provider that systematically exaggerates its surplus destabilizes local dispatch. In regenerative agriculture, overcorrecting soil conditions in regions where sensor data are stale can degrade long-term soil biology. In each of these settings, the risk is not ignorance alone; the risk is false certainty. Modeling explicit indeterminacy and explicit falsity support alongside truth support directly targets that risk [12]. The contributions of this work are fourfold. First, a formal definition is provided for a machine-learned neutrosophic state as a primary, trainable output of a model. The neutrosophic triplet is not a post-processing heuristic layered on top of a conventional classifier. It is the state that the model is trained to represent. Second, the work defines a dynamic state transition law Sh(t + Δt) = Φθ(Sh(t), O(t), C(t)) which allows the model to learn how support, contradiction, and indeterminacy co-evolve. This treats uncertainty and conflict as dynamical objects that follow learnable trajectories, rather than static nuisance terms [9]. Third, a supervised training objective is constructed that jointly optimizes the prediction Th(t), Ih(t), and Fh(t) while penalizing temporal inconsistency. The resulting composite loss L enforces not only local accuracy of each scalar component, but also global coherence of their joint evolution. Fourth, to demonstrate that this framework is operational and not purely conceptual, a concrete clinical monitoring scenario is constructed. In this scenario, two concurrent diagnostic hypotheses are tracked over two time steps. For each hypothesis, numerical values of Th(t), Ih(t), and Fh(t) are computed at the initial time and after the arrival of new evidence. These values are reported in tabular form in Section 4, and their meaning is interpreted in terms of evolving physiological belief states. The example shows that NMLSM maintains parallel, non-exclusive hypotheses and updates their support, ambiguity, and contradiction without prematurely discarding viable explanations. The structure of the paper is as follows. Section 2 defines the neutrosophic state, the update operator, and the notation used throughout the manuscript. The section describes how training data are prepared, how supervision signals for Th(t), Ih(t), and Fh(t) are constructed, and how the composite loss L is optimized. Section 4 presents the worked clinical case study, including explicit numerical state updates and a comparative neutrosophic table. Section 5 analyzes the implications of representing real-world systems through dynamic states, and Section 6 summarizes the broader impact of the proposed framework. 2. Scientific Framework This section defines the mathematical structure of NMLSM. All notations introduced here are used consistently throughout the paper. All symbols appearing in equations are defined explicitly in Table 1. Neutrosophic Sets and Systems, Vol. 94, 2025 420 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty Table 1. Symbol definitions used in the NMLSM formulation. Symbol Definition Domain / Range ℋ ={ℎ1,…,ℎ𝑀} Finite set of tracked hypotheses (clinical states, grid claims, soil conditions, etc.). 𝑀 ∈ℕ, M ≥ 1 ℎ ∈ ℋ A specific hypothesis of interest. Index variable 𝑡 ∈ℝ≥0 Continuous (or discretized) time variable. Time Δ𝑡 >0 Positive time increment between successive state updates. ℝ>0 Sℎ(𝑡) Neutrosophic state of hypothesis ℎat time 𝑡. [0,1]3 𝑇ℎ(𝑡) Truth support: learned degree of supporting evidence for hypothesis ℎat time 𝑡. [0,1] 𝐼ℎ(𝑡) Indeterminacy: learned degree of unresolved uncertainty for hypothesis ℎat time 𝑡. [0,1] 𝐹ℎ(𝑡) Falsity support: learned degree of structured contradicting evidence for hypothesis ℎat time 𝑡. [0,1] O(𝑡) Observation vector at time 𝑡(sensor readings, measurements, reports). ℝ𝑑𝑂, 𝑑𝑂≥1 C(𝑡) Context vector at time 𝑡(data quality, delay, trust, reliability metadata). ℝ𝑑𝐶, 𝑑𝐶≥1 Φ𝜃(⋅) Learned the state transition operator that maps the current state and inputs to the next state. [0,1]3×ℝ𝑑𝑂×ℝ𝑑𝐶 →[0,1]3 𝜃 Trainable parameters of Φ𝜃(⋅). Parameter space of the chosen ML model Sℎ(𝑡+Δ𝑡) Predicted neutrosophic state of hypothesis ℎafter time increment Δ𝑡. [0,1]3 ℒ Composite training loss for NMLSM. ℝ≥0 ℒ𝑇, ℒ𝐼, ℒ𝐹 Component-wise loss terms are supervised. 𝑇ℎ(𝑡), 𝐼ℎ(𝑡), and 𝐹ℎ(𝑡), respectively. ℝ≥0 ℒ𝐶 Temporal and physical consistency penalty term. ℝ≥0 𝜆𝑇,𝜆𝐼,𝜆𝐹,𝜆𝐶 Non-negative scalar weights controlling the contribution of each loss term to ℒ. ℝ≥0 2.1. System hypotheses and neutrosophic state We consider a system that is being observed over time. At any time 𝑡, multiple hypotheses about the system may be simultaneously relevant. A “hypothesis” in this context is any interpretable claim about the system that we wish to track. Examples include “the patient is experiencing severe bacterial pneumonia,” “this energy node can reliably export surplus power,” or “this soil cell is undergoing salinity-driven stress.” We denote the finite set of tracked hypotheses by: ℋ ={ℎ1,ℎ2,…,ℎ𝑀}, where 𝑀 ≥1is the number of hypotheses of interest. For each hypothesis ℎ ∈ℋ, we associate a time-indexed neutrosophic state vector as: Sh(t)=(Th(t), Ih(t), Fh(t)),t∈R≥0. The three scalar components of Sℎ(𝑡)are defined as follows: 1. 𝑇ℎ(𝑡)∈[0,1]: the degree of supporting evidence for hypothesis ℎat time 𝑡. A higher value indicates stronger structured support for ℎbased on currently available observations. 2. 𝐼ℎ(𝑡)∈[0,1]: the degree of indeterminacy of hypothesis ℎat time 𝑡. This term quantifies unresolved uncertainty that is not reducible to simple noise. Indeterminacy captures missing data, conflicting measurements, sensor unreliability, reporting delay, and ambiguity intrinsic to the phenomenon being measured. 3. 𝐹ℎ(𝑡)∈[0,1]: the degree of contradicting evidence against hypothesis ℎat time 𝑡. A higher value indicates stronger structured evidence that ℎis not valid, not active, or not the correct explanation of the observed system behavior. In contrast to probabilistic models and fuzzy membership models, we do not require 𝑇ℎ(𝑡)+𝐼ℎ(𝑡)+𝐹ℎ(𝑡)= 1. This relaxation is deliberate. Real systems may generate data that both support and refute a hypothesis, while remaining partially unresolved (for example, a critically ill patient can present simultaneous indicators that align with two competing diagnoses, along with missing laboratory confirmation for either). Forcing normalization would hide this coexistence of support, contradiction, and unresolved ambiguity. The neutrosophic state Sℎ(𝑡)preserves it. 2.2. Observations and contextual qualifiers At each time 𝑡, the learning model has access to two categories of inputs: 1. An observation vector O(𝑡). 2. A context vector C(𝑡). Neutrosophic Sets and Systems, Vol. 94, 2025 421 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty The observation vector O(𝑡)represents the measurable signals collected from the system at time 𝑡. The specific contents of O(𝑡)depend on the domain. In a clinical setting, O(𝑡)may include vital signs, laboratory values, and imaging-derived features. In a microgrid setting, O(𝑡)may include locally reported generation capacity, consumption demand, and real-time meter readings. In an agronomic setting, O(𝑡)may include soil moisture, salinity, root-zone temperature, and spectral vegetation indices. Formally, we write: O(𝑡)∈ℝ𝑑𝑂, where 𝑑𝑂≥1is the observation dimensionality. The context vector C(𝑡)represents metadata about data quality and trust at time 𝑡. This includes, but is not limited to: sensor calibration state, sampling delay, degree of known dropout or corruption, signal-to-noise ratio, self-report credibility, image sharpness or motion artifact, and network latency. These contextual factors affect how strongly the observations should influence the state update and how much indeterminacy should persist. We write C(𝑡) ∈ℝ𝑑𝐶, where 𝑑𝐶≥1 is the context dimensionality. Crucially, C(𝑡)is not auxiliary bookkeeping. In NMLSM, C(𝑡)directly governs the evolution of 𝐼ℎ(𝑡). For example, if C(𝑡)encodes “imaging quality is degraded,” the update rule must allow indeterminacy to remain high even when O(𝑡)superficially suggests a confident interpretation. This prevents false certainty sourced from poor evidence. 2.3. Dynamic neutrosophic state evolution For each hypothesis ℎ ∈ℋ, we model how its neutrosophic state changes over a finite time increment Δ𝑡 >0. The evolution is defined by a learnable state transition operator: Φ𝜃:([0,1]3×ℝ𝑑𝑂×ℝ𝑑𝐶) → [0,1]3, parameterized by 𝜃. The operator Φ𝜃 Maps the current state and current inputs to the next state. The fundamental update law of NMLSM is: Sℎ(𝑡+Δ𝑡)=Φ𝜃(Sℎ(𝑡), O(𝑡), C(𝑡)),∀ℎ ∈ℋ. (1) Equation (1) is applied separately to each hypothesis ℎ. This means that we do not require exclusivity between hypotheses. Multiple hypotheses can remain active in parallel, and each hypothesis maintains its own evolving triplet (𝑇ℎ,𝐼ℎ,𝐹ℎ). The model, therefore, supports simultaneous explanatory tracks. Because Φ𝜃 is parameterized, it is learnable. In practice, Φ𝜃 can be implemented as a recurrent neural architecture, a temporal convolutional module, a gated state-space model, or any differentiable parametric function capable of sequence modeling. The role of Φ𝜃is not limited to predicting future “truth support” 𝑇ℎ(𝑡+Δ𝑡). It must also learn how indeterminacy 𝐼ℎ(𝑡)and falsity support 𝐹ℎ(𝑡)propagate and interact under incomplete, delayed, noisy, or adversarial inputs [13]. The range of Φ𝜃is constrained to [0,1]3to enforce valid bounds for all three components: Φ𝜃(⋅)=(𝑇 ℎ(𝑡+Δ𝑡), 𝐼󰆹ℎ(𝑡+Δ𝑡), 𝐹 ℎ(𝑡+Δ𝑡)), with: 𝑇 ℎ(𝑡+Δ𝑡)∈ [0,1],𝐼󰆹ℎ(𝑡+Δ𝑡)∈[0,1],𝐹 ℎ(𝑡+Δ𝑡)∈[0,1]. During the training described in Section 3, these predicted components are supervised against target values for 𝑇ℎ(𝑡+Δ𝑡), 𝐼ℎ(𝑡+Δ𝑡), and 𝐹ℎ(𝑡+Δ𝑡). 2.4. Interpretation of the state components The three components of Sℎ(𝑡)have distinct semantic roles, and these roles are preserved by the update operator Φ𝜃: – 𝑇ℎ(𝑡)tracks structured support. An increase in 𝑇ℎ(𝑡)indicates that new observations O(𝑡), under context C(𝑡), add consistent, reinforcing evidence in favor of hypothesis ℎ. For instance, in a clinical scenario, a high inflammatory marker aligned with a compatible imaging pattern may increase 𝑇ℎ(𝑡)for the hypothesis “acute bacterial pulmonary infection.” – 𝐹ℎ(𝑡)tracks structured contradiction. An increase in 𝐹ℎ(𝑡)means that new observations actively refute hypothesis ℎ. For example, a normal cardiac output index in hemodynamic monitoring may Neutrosophic Sets and Systems, Vol. 94, 2025 422 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty raise 𝐹ℎ(𝑡)for the hypothesis “acute cardiogenic shock,” because that hypothesis predicts depressed cardiac output, not normal output. – 𝐼ℎ(𝑡)tracks irreducible indeterminacy. An increase in 𝐼ℎ(𝑡)indicates that even if 𝑇ℎ(𝑡)and 𝐹ℎ(𝑡)moved, the overall interpretability of the state has not improved, often because C(𝑡)signals that data are missing, stale, low-quality, conflicting, or strategically unreliable. For example, if a microgrid node reports “I can export 3 kW now,” but the report is delayed and historically inflated, then the system should raise 𝐼ℎ(𝑡)for the hypothesis “this node is currently a reliable source,” regardless of any nominal surplus signal in O(𝑡). These three components are not redundant. High 𝑇ℎ(𝑡) with high 𝐼ℎ(𝑡)indicates “promising but still not trustworthy.” High 𝑇ℎ(𝑡)with high 𝐹ℎ(𝑡)indicates “the evidence itself is self-contradictory.” High 𝐼ℎ(𝑡)alone indicates “we do not know yet, and we can prove we do not know.” The framework is explicitly designed to encode these regimes. 2.5. Consistency constraints The NMLSM formulation does not impose a hard normalization constraint, such as: 𝑇ℎ(𝑡)+𝐼ℎ(𝑡)+𝐹ℎ(𝑡)= 1. However, we still impose two mathematical requirements on all predicted and target states: 1. Boundedness: 0≤𝑇ℎ(𝑡)≤ 1,0 ≤𝐼ℎ(𝑡)≤ 1,0≤𝐹ℎ(𝑡) ≤1∀ℎ ∈ℋ, ∀𝑡 ≥0. (2) 2. Physical plausibility under domain rules: There may exist domain-level exclusions where certain extreme combinations of (𝑇ℎ,𝐼ℎ,𝐹ℎ)violate known physical or semantic limits. For example, in a medical context, if a particular hypothesis ℎis defined as “the patient is in irreversible cardiac arrest,” then a state with 𝑇ℎ(𝑡) =1.0 (fully supported), 𝐹ℎ(𝑡)=1.0 (fully contradicted), and 𝐼ℎ(𝑡)= 0.0 (no ambiguity) is physically inconsistent: it simultaneously encodes that the patient is both irreversibly arrested and definitely not arrested, with no ambiguity. Such states are disallowed during training through a temporal consistency penalty ℒ𝐶, introduced rigorously in Section 3. We emphasize that this is not a probabilistic normalization; it is an exclusion of logically or physically impossible assignments for specific hypotheses. This two-part structure (boundedness plus plausibility) guarantees that the model remains able to represent simultaneous support and contradiction where physically meaningful, while preventing degenerate states that are semantically self-annihilating. 3. Methodology This section describes how the proposed neutrosophic machine learning framework is trained. We specify how supervision targets are constructed, how the model is optimized, and how temporal and physical consistency are enforced. Section 3 is structured as follows: Section 3.1 defines the supervised training data used to learn the state transition operator Φ𝜃(⋅). Section 3.2 defines neutrosophic supervision targets for each hypothesis. Section 3.3 introduces the loss terms ℒ𝑇, ℒ𝐼, ℒ𝐹, and ℒ𝐶. Section 3.4 presents the total loss ℒ. Section 3.5 describes the processing pipeline, which is summarized schematically in Figure 1. 3.1. Training data structure We assume access to a set of recorded system trajectories. A “trajectory” is an indexed sequence of time points for a single system instance (for example, one monitored patient, one microgrid region, or one soil plot). We denote a trajectory by: Neutrosophic Sets and Systems, Vol. 94, 2025 423 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty 𝒯 ={𝑡0, 𝑡1, …, 𝑡𝐾}, where 𝑡0<𝑡1<⋯<𝑡𝐾 are observation times and 𝐾 ≥1. We define Δ𝑡𝑘= 𝑡𝑘+1 −𝑡𝑘>0for 𝑘 = 0,…,𝐾−1. For each time 𝑡𝑘 in the trajectory, we assume the following quantities are available or can be constructed: 1. The observation vector O(𝑡𝑘)∈ℝ𝑑𝑂, as defined in Section 2.2. 2. The context vector C(𝑡𝑘)∈ℝ𝑑𝐶, as defined in Section 2.2. 3. The current neutrosophic state Sℎ(𝑡𝑘)=(𝑇ℎ(𝑡𝑘),𝐼ℎ(𝑡𝑘),𝐹ℎ(𝑡𝑘))for each hypothesis ℎ ∈ℋ, as defined in Sec. 2.1. 4. A supervised target for the next state of each hypothesis, denoted Sℎ ∗(𝑡𝑘+1)=(𝑇ℎ ∗(𝑡𝑘+1), 𝐼ℎ ∗(𝑡𝑘+1), 𝐹ℎ ∗(𝑡𝑘+1)), where Sℎ ∗(𝑡𝑘+1)∈[0,1]3is the reference (ground truth) neutrosophic state that we want the model to produce at time 𝑡𝑘+1. The superscript * is used throughout this manuscript to indicate a supervised target. The model’s prediction for 𝑡𝑘+1is obtained by applying the learned transition operator Φ𝜃(⋅)to Sℎ(𝑡𝑘), O(𝑡𝑘), and C(𝑡𝑘), using Equation (1): S ℎ(𝑡𝑘+1)=Φ𝜃(Sℎ(𝑡𝑘), O(𝑡𝑘), C(𝑡𝑘)), where we write it as: S ℎ(𝑡𝑘+1)=(𝑇 ℎ(𝑡𝑘+1), 𝐼󰆹ℎ(𝑡𝑘+1), 𝐹 ℎ(𝑡𝑘+1)). Training consists of adjusting 𝜃 so that, for all hypotheses ℎ ∈ ℋand all time steps 𝑘 ∈{0,…,𝐾− 1}, S ℎ(𝑡𝑘+1)≈Sℎ ∗(𝑡𝑘+1)in all three components 𝑇, 𝐼,𝐹. The set of all trajectories used for training is denoted by 𝔻. Each element of 𝔻is a trajectory 𝒯with its associated {O(𝑡𝑘),C(𝑡𝑘),Sℎ(𝑡𝑘),Sℎ ∗(𝑡𝑘+1)}. We assume 𝔻is finite and nonempty. 3.2. Construction of neutrosophic supervision targets The definition of Sℎ ∗(𝑡𝑘+1)=(𝑇ℎ ∗(𝑡𝑘+1),𝐼ℎ ∗(𝑡𝑘+1),𝐹ℎ ∗(𝑡𝑘+1))is critical. It determines what the model is expected to learn. For each hypothesis ℎ ∈ℋ, and each future time point 𝑡𝑘+1, we define: 1. 𝑇ℎ ∗(𝑡𝑘+1)∈[0,1]: the degree to which the evidence available by time 𝑡𝑘+1supports hypothesis ℎ. Example (clinical domain): If laboratory values and imaging at 𝑡𝑘+1strongly align with the pathophysiology predicted by hypothesis ℎ, then 𝑇ℎ ∗(𝑡𝑘+1)is high. 2. 𝐹ℎ ∗(𝑡𝑘+1)∈[0,1]: the degree to which the evidence available by time 𝑡𝑘+1contradicts hypothesis ℎ. Example (microgrid domain): If a node previously claimed “I can export 3 kW now,” but metered export during [𝑡𝑘,𝑡𝑘+1]was near 0 kW, then 𝐹ℎ ∗(𝑡𝑘+1)is high for the hypothesis “this node is a reliable supplier.” 3. 𝐼ℎ ∗(𝑡𝑘+1)∈[0,1]: the degree to which, at time 𝑡𝑘+1, uncertainty about hypothesis ℎremains structurally unresolved. This accounts for missing, stale, self-reported, low-quality, or contradictory inputs. Example 3.1: If salinity probes in a soil cell have not reported for 18 hours and aerial imaging is partially occluded by cloud cover, then even if some stress indicators appear, the state remains ambiguous. 𝐼ℎ ∗(𝑡𝑘+1)must remain high. In supervised training, the triplet Sℎ ∗(𝑡𝑘+1)is derived from domain-informed labeling rules. These rules combine: – quantitative measurements (e.g., lab values, meter readings), – internal consistency checks (e.g., “declared capacity” vs. “actual capacity delivered”), and – uncertainty annotations (e.g., imaging quality flags, sensor calibration logs, sampling delays). In practice, this labeling process can be implemented using expert-reviewed protocols or welldefined deterministic heuristics [15]. The important property for the mathematical framework is that, for every hypothesis ℎand every time step 𝑡𝑘+1in the training set, Sℎ ∗(𝑡𝑘+1)is available and satisfies the boundedness constraints 0≤𝑇ℎ ∗(𝑡𝑘+1)≤1,0≤𝐼ℎ ∗(𝑡𝑘+1)≤1,0≤𝐹ℎ ∗(𝑡𝑘+1)≤1. Neutrosophic Sets and Systems, Vol. 94, 2025 424 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty Thus, at training time, we have both the model prediction S ℎ(𝑡𝑘+1)and a supervised target Sℎ ∗(𝑡𝑘+1)for each tracked hypothesis at each step. This enables pointwise supervision of 𝑇, 𝐼, and 𝐹. 3.3. Loss terms Training minimizes a composite loss. The total loss ℒis defined in Section 3.4. Here we define each component: 3.3.1. Truth support loss ℒ𝑇 The truth supports loss ℒ𝑇penalizes deviation between the predicted truth support 𝑇 ℎ(𝑡𝑘+1)and its target 𝑇ℎ ∗(𝑡𝑘+1). We define ℒ𝑇=1 𝑍∑ 𝒯∈𝔻 ∑ 𝐾(𝒯)−1 𝑘=0 ∑ ℎ∈ℋ (𝑇 ℎ(𝑡𝑘+1)−𝑇ℎ ∗(𝑡𝑘+1))2,(3) where 𝐾(𝒯)is the length index of trajectory 𝒯(so that 𝑡𝐾(𝒯)is its final time), and 𝑍is a normalizing constant equal to the total number of (𝒯,𝑘,ℎ)terms in the sum. The square enforces smooth penalization of absolute deviation. 3.3.2. Indeterminacy loss ℒ𝐼 The indeterminacy loss ℒ𝐼penalizes deviation between the predicted indeterminacy 𝐼󰆹ℎ(𝑡𝑘+1)and its target 𝐼ℎ ∗(𝑡𝑘+1). We define ℒ𝐼=1 𝑍∑ 𝒯∈𝔻 ∑ 𝐾(𝒯)−1 𝑘=0 ∑ ℎ∈ℋ (𝐼󰆹ℎ(𝑡𝑘+1)−𝐼ℎ ∗(𝑡𝑘+1))2.(4) This term is essential. In standard machine learning pipelines, the model is often implicitly rewarded for “being confident.” In NMLSM, confidence that is not justified is itself an error. The model is explicitly required to learn and express “we still do not know” when C(𝑡𝑘)indicates that the inputs at 𝑡𝑘 are unreliable or incomplete. Equation (4) forces 𝐼󰆹ℎ(𝑡𝑘+1)to correctly track genuine epistemic uncertainty. 3.3.3. Falsity support loss ℒ𝐹 The falsity supports loss ℒ𝐹penalizes deviation between the predicted and contradicting evidence 𝐹 ℎ(𝑡𝑘+1)and its target 𝐹ℎ ∗(𝑡𝑘+1). We define ℒ𝐹=1 𝑍∑ 𝒯∈𝔻 ∑ 𝐾(𝒯)−1 𝑘=0 ∑ ℎ∈ℋ (𝐹 ℎ(𝑡𝑘+1)−𝐹ℎ ∗(𝑡𝑘+1))2.(5) This term captures structured refutation. High 𝐹ℎ ∗(𝑡𝑘+1)means “the evidence at and before 𝑡𝑘+1actively argues against hypothesis ℎ.” Penalizing the difference between 𝐹 ℎ(𝑡𝑘+1)and 𝐹ℎ ∗(𝑡𝑘+1)ensures that the model learns to propagate explicit contradicting evidence forward in time rather than silently discarding it. 3.3.4. Consistency loss ℒ𝐶 The role of ℒ𝐶is to penalize neutrosophic states that violate physical or semantic plausibility for a given hypothesis. As established in Section 2.5, NMLSM does not normalize 𝑇ℎ(𝑡), 𝐼ℎ(𝑡), and 𝐹ℎ(𝑡), and it explicitly allows simultaneous high support and high contradiction. However, certain combinations are not physically meaningful for certain hypotheses. To formalize this, we define a hypothesis-specific consistency function. Ψℎ(𝑇 ℎ(𝑡𝑘+1),𝐼󰆹ℎ(𝑡𝑘+1),𝐹 ℎ(𝑡𝑘+1))≥0, which measures how inconsistent the predicted state is for hypothesis ℎat time 𝑡𝑘+1. A larger value means less plausible. We then define ℒ𝐶=1 𝑍∑ 𝒯∈𝔻 ∑ 𝐾(𝒯)−1 𝑘=0 ∑ ℎ∈ℋ Ψℎ(𝑇 ℎ(𝑡𝑘+1),𝐼󰆹ℎ(𝑡𝑘+1),𝐹 ℎ(𝑡𝑘+1)). (6) The exact form of Ψℎ(⋅)depends on domain knowledge. We now give an explicit instantiation of Ψℎ(⋅)that satisfies two conditions: it is mathematically well defined, and it encodes a meaningful physical exclusion. Neutrosophic Sets and Systems, Vol. 94, 2025 425 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty Suppose a specific hypothesis ℎis defined as an exclusive, binary physiological state that cannot be both fully present and fully absent without ambiguity. One example is the hypothesis: “The patient is in irreversible cardiac arrest.” For such a hypothesis, a predicted state with 𝑇 ℎ(𝑡𝑘+1)near 1 (fully supported), 𝐹 ℎ(𝑡𝑘+1)near 1 (fully contradicted), and 𝐼󰆹ℎ(𝑡𝑘+1)near 0 (no ambiguity) is physically incoherent. We capture this by defining: Ψℎ(𝑇,𝐼,𝐹)=[max {0, T +𝐹−1}⋅(1−𝐼)]2. (7) Understanding of (Eq. 7): – If both 𝑇and 𝐹are simultaneously large (so that 𝑇+𝐹 >1), and there is almost no indeterminacy (𝐼 ≈0), then Ψℎ(𝑇,𝐼,𝐹)becomes positive and large, penalizing Φ𝜃(⋅)for producing a self-annihilating state. – If 𝐼is high, the penalty is reduced, because in that case the model is explicitly admitting ambiguity (“this may be true and false, but we are uncertain, and that unresolved conflict is real”). This is allowed. – If 𝑇+𝐹 ≤1, then max {0, T +𝐹−1}=0, and Ψℎ(𝑇,𝐼,𝐹)=0; there is no penalty. This definition of Ψℎ(⋅)is consistent with philosophy in Section 2.5. We do not force normalization, and we do not forbid contradiction. We only penalize logically collapsed states in which the model simultaneously asserts “certainly true” and “certainly false” for a hypothesis that, by definition, cannot physically support that combination without any ambiguity. For hypotheses that are not mutually exclusive in this sense (for example, “low-grade inflammatory process in lung tissue”), the function Ψℎ(⋅)can be set identically to zero. In that case, ℒ𝐶 imposes no penalty for high 𝑇and high 𝐹together, even at low 𝐼, because partial, spatially distributed, or evolving processes can be both supported and challenged at the same time with minimal logical conflict [16]. The important property is that Ψℎ(⋅)is explicitly defined and is applied consistently per hypothesis during training. There is no uncontrolled heuristic step. 3.4. Composite training loss The total loss minimized during training is the weighted sum: ℒ =𝜆𝑇ℒ𝑇+𝜆𝐼ℒ𝐼+𝜆𝐹ℒ𝐹+𝜆𝐶ℒ𝐶, (8) where 𝜆𝑇≥0, 𝜆𝐼≥0, 𝜆𝐹≥0, and 𝜆𝐶≥0are scalar weights that balance the relative contribution of each term. – ℒ𝑇 From Eq. 3, encourages accurate prediction of truth support. – ℒ𝐼 From Eq. 4, encourages accurate prediction of indeterminacy. This prevents the model from erasing legitimate uncertainty. – ℒ𝐹 From Eq. 5, encourages accurate prediction of falsity support. This forces the model to actively carry contradictory evidence forward. – ℒ𝐶 From (Eq. 6) and (Eq. 7), prevent logically or physically incoherent states in domains where such incoherence is impossible. Together, these terms train Φ𝜃(⋅)to learn not only what is supported, but also what is refuted, how uncertain that judgment is, and how all three evolve in time in response to evidence and context. In other words, ℒoptimizes the full neutrosophic trajectory Sℎ(𝑡), not just a single scalar confidence. 3.5. End-to-end pipeline The full inference-and-update process for one time step can be summarized in four stages. This pipeline will be referenced in Sec. 4, and it is illustrated schematically in Figure 1. Stage 1. Input acquisition. At time 𝑡𝑘, the system receives O(𝑡𝑘)(observations) and C(𝑡𝑘)(context). Both are defined in Sec. 2.2 and observed from the environment. Neutrosophic Sets and Systems, Vol. 94, 2025 432 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty For ℎ1(acute bacterial pneumonia), The new lab and clearer imaging both strengthen support and reduce ambiguity: Sℎ1 ∗(𝑡1) = (𝑇ℎ1 ∗(𝑡1), 𝐼ℎ1 ∗(𝑡1), 𝐹ℎ1 ∗(𝑡1))=(0.83, 0.22, 0.12). So, 1. 𝑇ℎ1 ∗(𝑡1) = 0.83: strong biochemical and radiographic support. 2. 𝐼ℎ1 ∗(𝑡1) = 0.22: ambiguity falls because key labs are now known, and imaging clarity has improved. 3. 𝐹ℎ1 ∗(𝑡1)=0.12: There remains only a small structured argument against this hypothesis. For ℎ2(acute decompensated heart failure), The ultrasound evidence and stable systolic pressure support and increase structured contradiction: Sℎ2 ∗(𝑡1) = (𝑇ℎ2 ∗(𝑡1), 𝐼ℎ2 ∗(𝑡1), 𝐹ℎ2 ∗(𝑡1))=(0.39, 0.62, 0.41). So, 1. 𝑇ℎ2 ∗(𝑡1) = 0.39: there is still some residual support for cardiac contribution (tachycardia, borderline perfusion), but it is reduced. 2. 𝐼ℎ2 ∗(𝑡1) = 0.62: Some ambiguity persists because full invasive hemodynamics are not yet measured. 3. 𝐹ℎ2 ∗(𝑡1)=0.41: direct, structured contradiction is now stronger because preserved ejection fraction argues against primary pump failure as the dominant mechanism. These supervised targets Sℎ ∗(𝑡1)satisfy the boundedness constraints in (Eq. 2), and they are the exact reference values used in the loss terms Equations (3)– (5). For each hypothesis ℎ, Sℎ(𝑡0)is the initial neutrosophic state at time 𝑡0, S ℎ(𝑡1)is the predicted state at 𝑡1obtained by applying Φ𝜃(⋅)to Sℎ(𝑡0), O(𝑡0), and C(𝑡0)using Equations (9)–(11), and Sℎ ∗(𝑡1)is the supervised target state at 𝑡1derived from new clinical evidence. All components (𝑇,𝐼,𝐹)lie in [0,1]. As required by the framework, 𝑇, 𝐼, and 𝐹are not forced to sum to 1. Table 2. Neutrosophic state evolution for two concurrent clinical hypotheses across one update step. Hypothesis ℎ Sℎ(𝑡0) =(𝑇ℎ(𝑡0),𝐼ℎ(𝑡0),𝐹ℎ(𝑡0)) S ℎ(𝑡1) =(𝑇 ℎ(𝑡1),𝐼󰆹ℎ(𝑡1),𝐹 ℎ(𝑡1)) Sℎ ∗(𝑡1) =(𝑇ℎ ∗(𝑡1),𝐼ℎ ∗(𝑡1),𝐹ℎ ∗(𝑡1)) ℎ1: Acute bacterial pneumonia (0.72, 0.31, 0.18) (0.812, 0.442, 0.190) (0.83, 0.22, 0.12) ℎ2: Acute decompensated heart failure (0.44, 0.58, 0.36) (0.484, 0.631, 0.480) (0.39, 0.62, 0.41) Table 2 shows three important phenomena. First, for ℎ1, the model correctly predicts an increase in truth support (𝑇) from 0.72 to 0.812, consistent with the supervised target of 0.83, and a decrease in indeterminacy (𝐼) relative to 0.31, consistent with improved evidence quality by 𝑡1. Second, for ℎ2, the model predicts sustained ambiguity (𝐼󰆹ℎ2(𝑡1)=0.631, target 0.62) and a nontrivial falsity component (𝐹 ℎ2(𝑡1)=0.480, target 0.41), reflecting accumulating refutation of severe pump failure. Third, the two hypotheses coexist: the framework does not force collapse to a single dominant explanation after only two hours. This matches clinical reality in early critical care, where overlapping mechanisms are common. 4.5. Loss evaluation for this example We now compute the loss terms ℒ𝑇, ℒ𝐼, and ℒ𝐹 from Equations (3)– (5) for this single-trajectory, single-step example. We then discuss ℒ𝐶. Since 𝒯 = {𝑡0,𝑡1}has only one transition (𝑡0→𝑡1), and ℋ ={ℎ1,ℎ2}, there are exactly two (hypothesis, step) pairs. Therefore, for this example, 𝑍 =2. 4.5.1. Truth support term ℒ𝑇 For ℎ1: 𝑇 ℎ1(𝑡1)=0.812,𝑇ℎ1 ∗(𝑡1) =0.83,(𝑇 ℎ1(𝑡1)−𝑇ℎ1 ∗(𝑡1))2=(0.812−0.83)2=(−0.018)2=0.000324. For ℎ2: Neutrosophic Sets and Systems, Vol. 94, 2025 433 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty 𝑇 ℎ2(𝑡1)=0.484,𝑇ℎ2 ∗(𝑡1) =0.39,(𝑇 ℎ2(𝑡1)−𝑇ℎ2 ∗(𝑡1))2=(0.484−0.39)2=(0.094)2=0.008836. Averaging over both hypotheses (dividing by 𝑍 =2): ℒ𝑇=0.000324+0.008836 2=0.009160 2=0.004580. 4.5.2. Indeterminacy term ℒ𝐼 For ℎ1: 𝐼󰆹ℎ1(𝑡1)=0.442,𝐼ℎ1 ∗(𝑡1) = 0.22,(𝐼󰆹ℎ1(𝑡1)−𝐼ℎ1 ∗(𝑡1))2=(0.442−0.22)2=(0.222)2=0.049284. For ℎ2: 𝐼󰆹ℎ2(𝑡1)=0.631,𝐼ℎ2 ∗(𝑡1) = 0.62,(𝐼󰆹ℎ2(𝑡1)−𝐼ℎ2 ∗(𝑡1))2=(0.631−0.62)2=(0.011)2=0.000121. Averaging: ℒ𝐼=0.049284+0.000121 2=0.049405 2=0.0247025. 4.5.3. Falsity support term ℒ𝐹 For ℎ1: 𝐹 ℎ1(𝑡1)=0.190,𝐹ℎ1 ∗(𝑡1)=0.12,(𝐹 ℎ1(𝑡1)−𝐹ℎ1 ∗(𝑡1))2= (0.190−0.12)2=(0.070)2=0.004900. For ℎ2: 𝐹 ℎ2(𝑡1)=0.480,𝐹ℎ2 ∗(𝑡1)=0.41,(𝐹 ℎ2(𝑡1)−𝐹ℎ2 ∗(𝑡1))2= (0.480−0.41)2=(0.070)2=0.004900. Averaging: ℒ𝐹=0.004900+0.004900 2=0.009800 2=0.004900. 4.5.4. Consistency term ℒ𝐶 The consistency penalty ℒ𝐶 from Equation (6) uses Ψℎ(⋅)from Equation (7): Ψℎ(𝑇,𝐼,𝐹)=[max {0, T +𝐹−1}⋅(1−𝐼)]2. We evaluate Ψℎ(⋅)for each hypothesis using its predicted state S ℎ(𝑡1). Because neither hypothesis in this case study represents an absolute, logically exclusive physiological boundary (such as “irreversible cardiac arrest”), we set Ψℎ1(⋅)=0and Ψℎ2(⋅)=0. This corresponds to the statement in Section 3.3.4 that for hypotheses which can plausibly be partially supported and partially refuted at the same time, even at relatively low indeterminacy, no explicit penalty is enforced. Therefore, for this illustrative trajectory, ℒ𝐶=0. 4.5.5. Composite loss for the case study Using Equation (8), ℒ =𝜆𝑇ℒ𝑇+𝜆𝐼ℒ𝐼+𝜆𝐹ℒ𝐹+𝜆𝐶ℒ𝐶. To produce a concrete numeric example, we choose the following non-negative weights: 𝜆𝑇=1.0,𝜆𝐼=1.0,𝜆𝐹=1.0,𝜆𝐶=1.0. Under these weights: ℒ =(1.0)(0.004580)+(1.0)(0.0247025)+(1.0)(0.004900)+(1.0)(0.000000) =0.004580+0.0247025+0.004900=0.0341825. Thus, for this trajectory and this single update step (𝑡0→𝑡1), the composite training loss is ℒ = 0.0341825under equal weighting. This confirms the following: 1. All components of the loss are computable from defined quantities. 2. The loss naturally penalizes misestimation of truth support, indeterminacy, and falsity support. 3. The indeterminacy term ℒ𝐼contributes significantly in this example because the model’s predicted ambiguity for ℎ1at 𝑡1(0.442) remained higher than the target (0.22). Executively, the model did not reduce uncertainty fast enough after high-quality evidence arrived, and the training signal reflects that. 4. The framework handles multiple simultaneous hypotheses without forcing exclusivity and without creating undefined mathematical states. Neutrosophic Sets and Systems, Vol. 94, 2025 434 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty 5. Discussion This section interprets the results obtained in Section 4 in the context of the mathematical framework and the training methodology. The goal here is not to restate numerical values, but to analyze what they mean for real-world decision-making, model safety, and the scientific value of representing state as a neutrosophic triplet Sℎ(𝑡) =(𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)). We also discuss the implications for multi-hypothesis monitoring, temporal reasoning, and system reliability in highstakes environments. The structure of this section is as follows: Section 5.1 explains why explicit modeling of (𝑇,𝐼,𝐹)is structurally different from probability, confidence, or fuzzy membership. Section 5.2 explains the scientific value of tracking multiple hypotheses in parallel without collapse. Section 5.3 analyzes the temporal behavior of indeterminacy 𝐼ℎ(𝑡). Section 5.4 discusses safety and accountability. Section 5.5 discusses generality beyond the clinical example. Section 5.6 discusses limitations of the current framework and future refinements. 5.1. Beyond probability: truth, contradiction, and indeterminacy as independent state components Traditional machine learning pipelines typically answer the question “How likely is hypothesis ℎ?” by producing a probability estimate, a confidence score, or a class membership degree. These representations are all one-dimensional. They encode some form of “how true” but do not separately encode “how false” or “how unknown.” In contrast, NMLSM models each hypothesis ℎat time 𝑡using a full triplet Sℎ(𝑡) = (𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)), where: ⎯ 𝑇ℎ(𝑡)captures structured, affirmative support; ⎯ 𝐹ℎ(𝑡)captures structured, affirmative refutation; ⎯ 𝐼ℎ(𝑡)captures irreducible or unresolved ambiguity. The example in Table 2 shows that this separation is not cosmetic. For hypothesis ℎ2(“acute decompensated heart failure”), we observed at the time 𝑡1: Sℎ2 ∗(𝑡1)= (0.39, 0.62, 0.41). This triplet cannot be compressed faithfully to a single scalar without discarding critical distinctions: ⎯ The value 𝑇ℎ2 ∗(𝑡1)=0.39shows that some supportive evidence remains. ⎯ The value 𝐹ℎ2 ∗(𝑡1) =0.41shows that there is also a meaningful structured refutation. ⎯ The value 𝐼ℎ2 ∗(𝑡1)=0.62shows that even after two hours and new measurements, a large part of the picture is still unresolved. A single probability, such as “0.39 probability of acute decompensated heart failure,” would hide the fact that there is also specific contradictory evidence present (𝐹ℎ2 ∗(𝑡1)=0.41)and that a large component of the uncertainty is due to missing or incomplete measurements (𝐼ℎ2 ∗(𝑡1) = 0.62). In other words, standard probability-style output confuses “there is evidence against this hypothesis,” “the evidence is missing or unreliable,” and “the evidence supports a different hypothesis” into the same bucket of “low confidence.” These are clinically, operationally, and ethically distinct conditions [18]. By explicitly modeling (𝑇,𝐼,𝐹), NMLSM enforces a clean separation: a) Evidence against a hypothesis is not treated as the absence of evidence for it; it is treated as its own positive signal 𝐹ℎ(𝑡). b) Missing or corrupted data are not silently ignored; they are measured and carried forward as 𝐼ℎ(𝑡). Neutrosophic Sets and Systems, Vol. 94, 2025 435 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty This difference is essential in high-stakes systems that behave badly under false certainty. It is not simply an interpretability convenience. It changes how the model is trained, how its error is penalized (Eqs. (3)– (5)), and how its internal state evolves (Eq.(1)). 5.2. Parallel, non-exclusive hypotheses and delayed collapse A central property of the framework is that it does not impose exclusivity between hypotheses in ℋ. In the case study, ℋ ={ℎ1,ℎ2}represents two clinically plausible mechanisms that can coexist in early presentation. At 𝑡0, both Sℎ1(𝑡0)=(0.72, 0.31, 0.18),𝑎𝑛𝑑 Sℎ2(𝑡0)=(0.44, 0.58, 0.36) are actively tracked. At 𝑡1They remain actively tracked, and the model predicts and supervises both S ℎ1(𝑡1)and S ℎ2(𝑡1). Why is this important? In many traditional clinical decision support systems, the pipeline is designed to move quickly toward a “most likely” diagnosis. This creates an artificial collapse. Artificial collapse is dangerous when (i) multiple processes interact (for example, infection plus fluid overload), or (ii) information is still being acquired. If the model discards a plausible hypothesis too early and clinicians follow that collapse, the system may bias care away from necessary interventions. By keeping both hypotheses alive at 𝑡1NMLSM does two things: 1. It allows shared causes to be represented explicitly. For example, both pneumonia and decompensated heart failure can degrade oxygenation. 2. It respects the temporal structure of critical care: true clarity often emerges over hours, not minutes. This is not a cosmetic modeling choice. It is enforced by construction. The update operator Φ𝜃(⋅) in Eq. (1) is applied separately for each ℎ ∈ ℋ, and no mutual-exclusion term is injected into the loss ℒ. The only cross-hypothesis coupling is indirect, through shared observations O(𝑡)and shared context C(𝑡). This design decision is intentional and scientifically motivated: it encodes the physical fact that real systems often admit multiple overlapping explanations before enough evidence accumulates to separate them. 5.3. Temporal behavior of indeterminacy 𝑰𝒉(𝒕) In standard practice, models are rewarded for being decisive. In this framework, the model is rewarded for being honest. The loss term ℒ𝐼 in Eq. (4) penalizes deviation between 𝐼󰆹ℎ(𝑡𝑘+1)and 𝐼ℎ ∗(𝑡𝑘+1). This means indeterminacy itself is supervised. The model is not only allowed to say “I do not know”; it is required to learn when “I do not know” is correct. The numerical results in Section 4.5 make this concrete. For hypothesis ℎ1, the target indeterminacy at 𝑡1was 𝐼ℎ1 ∗(𝑡1) = 0.22, reflecting the fact that improved imaging and returned lab values reduced ambiguity about ℎ1. The model’s predicted indeterminacy was, 𝐼󰆹ℎ1(𝑡1)=0.442. The squared error (0.442−0.22)2=0.049284is the dominant contributor to ℒ𝐼 for this trajectory. Intuitively, the model remained too uncertain about pneumonia even after strong confirmatory evidence was available. The training signal, therefore, pushes Φ𝜃(⋅)to decrease 𝐼󰆹ℎ1(𝑡1)faster in future updates when similar high-quality evidence is observed. For hypothesis ℎ2, we had, 𝐼ℎ2 ∗(𝑡1) =0.62,𝐼󰆹ℎ2(𝑡1)=0.631, and the squared error (0.631−0.62)2= 0.000121was very small. Here, the model correctly maintained high ambiguity because invasive hemodynamic data were still missing. The learning signal confirms that it is appropriate to remain uncertain. This mechanism (Eq. (4)) encodes a clinically and scientifically honest behavior: the model does not guess clarity that does not exist. Instead, the model learns to align its estimated indeterminacy with the true state of knowledge at that time step. This is a qualitative advance Neutrosophic Sets and Systems, Vol. 94, 2025 436 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty over conventional approaches, because it makes “uncertainty” a first-class predictive target. The output 𝐼ℎ(𝑡)is not a byproduct. It is a learned state variable with its own supervised trajectory. 5.4. Safety, accountability, and the rejection of false certainty High-stakes systems fail in two main ways: (1) they miss what is truly present, or (2) they assert what is not truly present. The second error is often more dangerous, because it induces overconfident action in the wrong direction. The falsity-support component 𝐹ℎ(𝑡)directly addresses this second class of error. When the system has seen structured evidence against a hypothesis, 𝐹ℎ(𝑡)increases and is preserved as part of the state. This is visible in Table 2 for ℎ2, where Sℎ2 ∗(𝑡1)has 𝐹ℎ2 ∗(𝑡1)=0.41. This explicitly encodes: “there is nontrivial, concrete, physiological evidence arguing against acute decompensated heart failure as the primary mechanism.” Standard classifiers do not typically surface a “contradiction score,” nor do they carry contradiction forward in time. Instead, contradictory evidence merely lowers the probability assigned to the hypothesis, without distinguishing “less support” from “active refutation.” By preserving 𝐹ℎ(𝑡) as its own supervised quantity (Eq. (5)), NMLSM makes that distinction explicit. This has two safety reasons: 1. It provides a quantitative argument against premature, high-risk intervention that targets the wrong mechanism. Clinically, for example, an aggressive diuretic strategy would be less justified when 𝐹ℎ2(𝑡) is high because that indicates evidence actively refuting severe pump failure. 2. It creates an auditable trail. At any time step, the model state Sℎ(𝑡)can be inspected, and both supporting evidence (𝑇ℎ(𝑡)) and contradicting evidence (𝐹ℎ(𝑡)) are explicitly visible. This supports accountability: the system’s internal reasoning is not hidden, and it is not compressed to a single opaque number. From the standpoint of model governance, this is a structural contribution. The state representation itself constrains the failure modes of the model by preventing unjustified single-answer collapse. 5.5. Generality of the framework Although the case study in Sec. 4 is clinical, the structure of the framework is domain-agnostic. All key ingredients, multiple concurrent hypotheses ℎ ∈ ℋ, an observation vector O(𝑡), a context vector C(𝑡), a learned transition operator Φ𝜃(⋅), and a supervised target Sℎ ∗(𝑡𝑘+1), are available in other real-world systems where uncertainty is operationally important [19]. We briefly outline two other domains to illustrate the breadth of applicability: 1. Distributed energy coordination in a microgrid. – Each hypothesis ℎcan represent a claim such as “node 𝑖is currently a reliable energy supplier of at least 2 kW.” – 𝑇ℎ(𝑡)measures delivered surplus relative to declared surplus. – 𝐹ℎ(𝑡)measures structured evidence that the node is overstating capacity. – 𝐼ℎ(𝑡)measures unresolved doubt due to delayed metering, suspected tampering, or communication latency. – 𝐂(𝑡)naturally includes data trust and reporting delay, which drive Γ𝐼(⋅)and thus influence 𝐼ℎ(𝑡+Δ𝑡). – The parallel tracking of multiple nodes allows the controller to allocate demand preferentially to nodes with high 𝑇ℎ(𝑡)and low 𝐼ℎ(𝑡), reducing instability. 2. Precision soil management in regenerative agriculture. – Each hypothesis ℎcan represent a local soil stress condition in a spatial cell, such as “this cell is undergoing salt accumulation that threatens root function.” – 𝑇ℎ(𝑡)measures direct support from salinity probes, canopy stress indices, and plant gas exchange signals. Neutrosophic Sets and Systems, Vol. 94, 2025 437 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty – 𝐹ℎ(𝑡)measures contradictory indicators such as normal root-zone conductivity or absence of wilting. – 𝐼ℎ(𝑡)encodes missing sensors, shallow sampling depth, or cloud-obscured imaging. – Over time, Φ𝜃(⋅)learns how salinity stress evolves spatially and temporally under irrigation, drainage, and microbial remediation. – The agronomic controller can then target remediation resources locally, where 𝑇ℎ(𝑡)is high, while prioritizing additional sensing where 𝐼ℎ(𝑡)remains high. In both domains, the role of 𝐼ℎ(𝑡)is identical to the clinical setting: it is an actionable quantity, not an afterthought. High indeterminacy does not mean “low relevance.” It means “this region, node, or condition requires focused measurement before irreversible decisions are made.” In practice, this property allows NMLSM not only to guide action but also to guide measurement. That is, it can tell us not just “what to do,” but “what we still need to learn.” 5.6. Limitations There are structural limitations that must be stated clearly. First, the quality of Sℎ ∗(𝑡𝑘+1). The supervised targets Sℎ ∗(𝑡𝑘+1)= (𝑇ℎ ∗(𝑡𝑘+1),𝐼ℎ ∗(𝑡𝑘+1),𝐹ℎ ∗(𝑡𝑘+1)) depend on domain-informed annotation. In high-stakes domains, generating high-quality targets will often require expert review, adjudication of contradictory signals, and explicit documentation of uncertainty sources. This is appropriate; in safety-critical systems, expert review is expected, but it is costly. Future work should investigate semi-supervised or self-supervised strategies where parts of Sℎ ∗(𝑡𝑘+1), especially 𝐼ℎ ∗(𝑡𝑘+1), can be inferred from structured metadata without full expert labeling at every step. Second, temporal granularity. In this paper, we considered a single update step (𝑡0→𝑡1)with Δ𝑡0= 2 hours. In real deployments, Δ𝑡may vary (seconds, minutes, hours, days), and sensor streams may be asynchronous. The formulation in Equation (1) does not assume uniform Δ𝑡; however, practical implementations of Φ𝜃(⋅)should either take Δ𝑡as an explicit input or be architected to handle irregular sampling. This is straightforward in principle (for example, by conditioning on Δ𝑡as an additional component in C(𝑡)), but it must be engineered. Third, hypothesis interaction. We deliberately modeled each hypothesis ℎ ∈ ℋwith an independent neutrosophic state trajectory. This is scientifically correct in the early phase of evaluation, and it prevents premature collapse. However, in later phases, certain hypotheses become causally coupled. For instance, in clinical care, once bacterial pneumonia is strongly established and cardiac function is confirmed as preserved, the probability that acute decompensated heart failure is the dominant process may fall sharply, not just because of direct cardiac evidence, but because the pulmonary hypothesis already suffices to explain the presentation. Future work can extend Φ𝜃(⋅)to allow controlled cross-hypothesis coupling while preserving interpretability, for example, by adding structured interaction terms that state “if 𝑇ℎ1(𝑡)exceeds a threshold and 𝐹ℎ2(𝑡)is rising, then reduce 𝑇ℎ2(𝑡+Δ𝑡)more aggressively.” Fourth, enforcement of physical plausibility. The consistency penalty ℒ𝐶, defined via Ψℎ(⋅) in Equations (6)– (7), ensure that clearly impossible states are discouraged for hypotheses that represent mutually exclusive conditions. In this paper’s case study, neither ℎ1nor ℎ2required a nonzero Ψℎ(⋅). In domains such as fault detection in industrial systems or binary survival states in resuscitation, Ψℎ(⋅)will need to be designed carefully to reflect domain physics or physiology. Future work should formalize guidelines for constructing Ψℎ(⋅)so that the penalty captures genuine impossibility without reintroducing unjustified forced collapse [20]. Neutrosophic Sets and Systems, Vol. 94, 2025 438 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty Finally, interpretability and decision policy. NMLSM produces an evolving, inspectable state Sℎ(𝑡)for each hypothesis. The framework intentionally stops short of prescribing an irreversible action. This is not a weakness. It is a design choice to separate state estimation (what the system believes and how certain it is) from policy (what to do). In high-stakes environments, human decision-makers, oversight committees, or certified control policies must remain involved in action selection. Future work may connect Sℎ(𝑡)to downstream decision-making modules, but such modules should make any irreversible recommendation explicitly conditioned on 𝐼ℎ(𝑡)and 𝐹ℎ(𝑡), not only on 𝑇ℎ(𝑡). In summary, these limitations are tractable. None of them undermines the internal mathematical consistency of the framework. Instead, they outline how NMLSM can evolve into a deployed safety-aware state estimator that not only answers “what is happening,” but also “how sure are we, and how do we know we are that sure.” 6. Cross-Domain Implementation Scenarios This section explains how the proposed framework can be used in different real-world domains. The goal is to show that the method is general and not limited to the clinical example in Section 4. In all domains, the same core elements apply: a) O(𝑡): the observation vector at time 𝑡. This represents what is measured right now (for example: sensor readings, reported values, test results). b) C(𝑡): the context vector at time 𝑡. This represents how reliable and complete the information is (for example, data quality, delay, and missing history). c) Sℎ(𝑡) =(𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)): the neutrosophic state of hypothesis ℎat time 𝑡. a. 𝑇ℎ(𝑡): support for hypothesis ℎ. b. 𝐹ℎ(𝑡): evidence against hypothesis ℎ. c. 𝐼ℎ(𝑡): uncertainty that remains about hypothesis ℎ, because information is missing, weak, or conflicting. d) Φ𝜃: the learned model that updates the state over time, using Sℎ(𝑡), O(𝑡), and C(𝑡), as defined earlier in the paper. In each scenario below: 1. We describe what a “hypothesis” ℎmeans in that domain. 2. We explain how to interpret (𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)). 3. We explain how the system can act in a careful and accountable way using these values. No new symbols are introduced in this section. All symbols were defined in earlier sections. 6.1. Distributed energy networks Goal Operate a small energy network (for example, a microgrid) in real time. The network contains several nodes (such as houses, batteries, or small generators). Each node may report that it can provide power to the grid. Some reports are true. Some are exaggerated. Some are uncertain. Hypotheses In this domain, each hypothesis ℎcan be defined as: “Node 𝑖can reliably supply at least a required level of power right now.” For each such hypothesis ℎ, the framework maintains a state Sℎ(𝑡)= (𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)): 1. 𝑇ℎ(𝑡): measured support that the node can actually supply the promised power. This can include recent meter readings showing export, stable voltage under load, or a good delivery history. 2. 𝐹ℎ(𝑡): measured evidence against the claim. This may include repeated failures to deliver in the past, overheating events, or drops in output when demand rises. 3. 𝐼ℎ(𝑡): remaining uncertainty. This increases when data from the node are missing, when communication is delayed, or when the only information we have is self-reported and not yet confirmed. Neutrosophic Sets and Systems, Vol. 94, 2025 439 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty How to act The controller does not only ask “Which node looks good?” It asks three questions for each node: 1. Is there direct support for this node’s promise (𝑇ℎ(𝑡))? 2. Is there direct evidence that this node is not reliable (𝐹ℎ(𝑡))? 3. How much do we still not know about this node (𝐼ℎ(𝑡))? In practice: 1. Nodes with high 𝑇ℎ(𝑡), low 𝐹ℎ(𝑡), and low 𝐼ℎ(𝑡)are preferred for critical supplies. 2. Nodes with high 𝑇ℎ(𝑡)but high 𝐼ℎ(𝑡)𝑇ℎ𝑒𝑦 are not rejected, but they require fast verification before full reliance. 3. Nodes with high 𝐹ℎ(𝑡)are deprioritized, because there is direct, recent evidence that they do not deliver as promised. Why this matters Power failures can spread quickly in a grid. A wrong decision about one node can affect the whole system. This framework separates three different cases that would otherwise look similar in a simple “confidence score”: (1) “We have strong proof this node is good,” (2) “We have strong proof this node is not good,” and (3) “We honestly do not know yet.” These three states should not be treated the same. The framework makes the difference explicit. 6.2. Precision agriculture and soil stress management Goal Detect and react to early signs of crop stress in the field. For example: salt build-up in soil, root oxygen stress, or nutrient imbalance. These problems start locally, then spread. Early response can prevent yield loss, but only if the signal is trustworthy. Hypotheses In this domain, each hypothesis ℎcan be defined as: “This specific plot of land (or soil cell) is entering a harmful stress condition that will damage the crop if not addressed soon.” For each such hypothesis ℎ, we track Sℎ(𝑡) =(𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)): 1. 𝑇ℎ(𝑡): support for the stress. This can come from soil probes (for example, salinity or moisture levels), plant reflectance patterns, canopy temperature, or gas exchange indicators that suggest physiological stress. 2. 𝐹ℎ(𝑡): evidence that goes against the stress. For example, root-zone readings that are normal, no local wilting, or stable transpiration suggest the plant is functioning well. 3. 𝐼ℎ(𝑡): uncertainty. This becomes large when sensing is incomplete. For example, satellite or drone imaging is cloud-obscured, ground sensors are offline, irrigation data are missing, or the plot has not been sampled recently. How to act The system can use (𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡))to support two types of actions: 1. Intervention targeting. If 𝑇ℎ(𝑡)is high and 𝐼ℎ(𝑡)is low, then the system has strong and reliable evidence that stress is present. In that case, the farm management system can trigger a direct action in that plot (for example, adjust irrigation strategy, apply remediation, or change nutrient mix). 2. Measurement targeting. If 𝐼ℎ(𝑡)is high, the system cannot confirm or deny stress in that plot. Instead of ignoring that plot, the system can mark it for further sensing. This may mean sending a drone, taking a direct soil sample, or requesting a manual field check. High 𝐼ℎ(𝑡) here means “you do not have enough information,” not “everything is fine.” Why this matters Neutrosophic Sets and Systems, Vol. 94, 2025 440 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty Traditional systems often behave in one of two extreme ways: either they overreact to weak signals and waste costly intervention, or they do nothing until visible damage appears. By keeping 𝑇ℎ(𝑡), 𝐹ℎ(𝑡), and 𝐼ℎ(𝑡)separate, the framework supports both early and targeted action, while also telling the operator where the system is still blind. 6.3. Education and learner modeling Goal Adapt instruction to a learner in real time. This is difficult because available evidence (homework, quiz answers, spoken explanations) may be partial, noisy, or even misleading. We want to avoid two errors: pushing a learner forward too early and holding a learner back without reason. Hypotheses In this domain, each hypothesis ℎcan be defined as: “The learner can apply a specific skill in a meaningful way (for example, can solve proportional reasoning problems in new contexts, not only in memorized forms).” For each such hypothesis ℎ, we track Sℎ(𝑡) =(𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)): 1. 𝑇ℎ(𝑡): support that the learner shows true understanding. This may include correct reasoning steps on new problems, clear verbal explanation, or success on transfer tasks. 2. 𝐹ℎ(𝑡): evidence against mastery. This may include repeated, structured errors in the same substep, guesses that contradict earlier answers, or explanations that show memorization without understanding. 3. 𝐼ℎ(𝑡): uncertainty. This increases when the platform does not have enough recent work from the learner, when the learner’s work may have been generated by an assistant tool, or when time-on-task data is missing. How to act The learning system does not need to force a binary “knows/does not know.” Instead: 1. If 𝑇ℎ(𝑡)is high, 𝐹ℎ(𝑡)is low, and 𝐼ℎ(𝑡)is low, the learner is likely ready to advance. 2. If both 𝑇ℎ(𝑡)and 𝐹ℎ(𝑡)are non-negligible at the same time, which means the evidence is mixed. The correct next step is targeted feedback, not promotion and not punishment. The learner may have a partial understanding with specific gaps. 3. If 𝐼ℎ(𝑡)is high, the correct action is to request a short diagnostic question or a short oral explanation to reduce that uncertainty, not to guess the learner’s ability. Why this matters Mistakes in personalized learning often come from acting on the wrong assumption about the learner’s state. The framework prevents blind advancement and prevents blind delay. It treats “we do not yet know” as valid information that requires clarification, not as a failure. 6.4. Critical care (clinical monitoring) Goal Support clinical decision-making in intensive care, where a single patient may have several competing explanations for the same symptoms. Acting too early on the wrong explanation can be dangerous. Acting too late can also be dangerous. We want structured clarity, not forced certainty. Hypotheses In this domain, each hypothesis ℎrepresents a concrete clinical diagnosis under consideration. Examples include “acute bacterial pneumonia” and “acute decompensated heart failure.” These correspond directly to the two hypotheses analyzed in Sec.4. For each such hypothesis ℎ, we track Sℎ(𝑡) =(𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)): 1. 𝑇ℎ(𝑡): clinical support for that diagnosis (for example, lab markers, imaging, vital signs). Neutrosophic Sets and Systems, Vol. 94, 2025 441 Alaa Hassan and Reda Ahmed,Neutrosophic State Learning for High-Stakes Systems Using a Dynamic Triplet Model of Support Contradiction and Uncertainty 2. 𝐹ℎ(𝑡): clinical evidence that argues against that diagnosis (for example, cardiac imaging that shows preserved function, which argues against pump failure). 3. 𝐼ℎ(𝑡): uncertainty due to missing, delayed, or low-quality information (for example, blurred imaging, pending labs, incomplete history). How to act Instead of forcing the system to output one “most likely” diagnosis, the framework keeps multiple hypotheses active at the same time. For each hypothesis, the clinician can read: 1. What supports it now (𝑇ℎ(𝑡)), 2. What contradicts it now (𝐹ℎ(𝑡)), 3. What is still not known (𝐼ℎ(𝑡)). This matches how real critical care decisions are made. The clinician can begin stabilizing actions while still collecting missing evidence, rather than committing fully to one explanation too early. Why this matters In critical care, false certainty can harm the patient. A system that can say “this diagnosis is supported,” “this diagnosis is challenged,” and “this part is still unclear” is safer than a system that outputs only one label and hides the rest. The same pattern appears in all four domains above (energy, agriculture, education, and critical care): 1. We define a set of hypotheses ℎthat matter for action. In energy, a hypothesis is “node 𝑖can supply power now.” In agriculture, a hypothesis is “this plot is entering stress.” In education, a hypothesis is “the learner can apply this skill.” In critical care, a hypothesis is “this diagnosis explains the patient’s current state.” 2. For each hypothesis ℎ, we track a live state Sℎ(𝑡) = (𝑇ℎ(𝑡),𝐼ℎ(𝑡),𝐹ℎ(𝑡)). This state records support, contradiction, and remaining uncertainty at time 𝑡. These three components are not forced to collapse into one number. 3. We update that state over time using Φ𝜃, which takes into account both what was observed O(𝑡)and how trustworthy those observations are C(𝑡). This is important because low-quality data should not produce false confidence. 4. We expose the full state Sℎ(𝑡)to the human or automated decision-maker. This means the decision process can distinguish: – “We have strong support,” – “We have strong evidence against.” – “We still do not know enough yet.” This separation is the key benefit of the framework. It allows decisions to be guided not only by what seems true, but also by what seems false and by what is still unresolved. In high-risk environments, this is essential for safe and transparent behavior. 7. Conclusion This work presented NMLSM, a framework for representing and learning system state in environments where evidence is incomplete, conflicting, and evolving. Instead of forcing a single decision or a single confidence score, the framework assigns to each hypothesis a three-part state: degree of supporting evidence, degree of contradicting evidence, and degree of unresolved uncertainty. These components are learned explicitly, are allowed to coexist, and are updated over time as new information becomes available. The framework defines a learnable state transition operator that incorporates both observations and context about data quality. This is important in practice: measurements are not all equally reliable, and real systems often produce claims that are only partially trustworthy. By including data reliability in the update step, the model can lower or maintain uncertainty when the evidence is weak instead of creating artificial confidence.