Neuro-Symbolic Unification via Cognitive Hypergraphs: Quantitative Mitigation of Hallucinations in Large Context Models Prior to Generation

Abstract

Author: Luigi Usai ORCID: https://orcid.org/0009-0003-3001-717X Location: Quartucciu (CA), Italy Date: June 26, 2026 Target: Zenodo / arXiv (cs.AI, cs.CL, cs.LO) Abstract Large Context Models (LCMs) exhibit an inherent vulnerability known as semantic hallucination, which stems directly from conditional likelihood maximization within discrete vector spaces. Traditional mitigation strategies operate predominantly post-hoc, managing errors after the stochastically generated token sequence has already mutated. This paper extends the Universal Cognitive Hypergraph (UKH) framework by introducing a discrete Alexandrov topology over knowledge hypergraphs to constrain the space of admissible states prior to token decoding. Utilizing the Monadic Neuro-Symbolic Verification and Synthesis Architecture (MNSVSA), probabilistic generation paths are intercepted and structurally validated against W3C SHACL constraints and axiomatic assertions verified by the Lean 4 kernel coupled with automated SMT solvers. Our theoretical results demonstrate the mathematical elimination of categorical deviations while fully preserving the model's syntactic fluency. 1. Introduction and Mathematical Formulation of the Problem Autoregressive language models estimate the probability distribution of the next token $w_t$ conditioned on the preceding context $w_{<t}$: $$P(w_t \mid w_{<t}) = \text{softmax}(W_{\text{unembed}} \cdot h_t)$$ where $h_t \in \mathbb{R}^d$ represents the final hidden state extracted by the Transformer architecture. Because the $\text{softmax}$ function maps scores to an open probability distribution, it inherently assigns non-zero probabilities to regions of the semantic space that violate real-world axiomatic constraints. Consequently, hallucination is not an accidental software bug but a structural property of the model's underlying stochasticity. The UKH framework bypasses the limitations of passive document retrieval (RAG) by integrating a topological-symbolic constraint directly into the sampling phase (speculative decoding). This setup actively prevents the model from exploring probabilistic trajectories linked to logically inconsistent states. 2. UKH Framework Architecture for Semantic Security The universe of discourse is mapped onto a directed hypergraph and serialized using the JSON-LD format. Let $\mathcal{H} = (V, E)$ be a cognitive hypergraph, where $V$ is the set of strongly typed nodes (conceptual entities) and $E \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$ is the set of hyperedges representing multi-argument logical-functional relationships. 2.1. Alexandrov Topological Space and SHACL Constraints To establish geometric-structural rigor within a discrete domain, the hypergraph space is endowed with an Alexandrov topology, where open sets are defined as sub-hypergraphs closed upwards relative to a logical preorder relation ($\le$). W3C Shapes Constraint Language (SHACL) rules function as topological closure operators: $$\text{cl}(E_c) \subseteq \mathcal{H}_{\text{valid}}$$ If a candidate hyperedge $E_c$, derived from the semantic translation of the tokens proposed by the LLM, violates a structural Shape (e.g., assigning a physical property inconsistent with the primitive type of the node), the closure operator identifies a contradiction within the topological space. It subsequently invalidates the generation path before token rendering occurs. 2.2. Axiomatic Verification and Type Checking via Lean 4 While SHACL rules govern the macro-structural coherence of the graphs, the MNSVSA architecture executes formal verification of micro-logical assertions. The process follows a strict protocol: The semantic fragment generated by the LLM is isolated inside a logical monad. MNSVSA translates the assertion into a formal type within the evaluation language of Lean 4. Leveraging the Curry-Howard Isomorphism, the logical consistency of the statement is reduced to a Type Checking problem. To avoid the computational burden of generating complex mathematical proofs from scratch at inference runtime, the architecture delegates constraint satisfiability to an automated SMT solver (Z3) tightly integrated into the Lean 4 runtime kernel. 3. The Coherence Entropy Filtering Mechanism To quantify and halt stochastic drift within extended contexts, the framework implements a JIT (Just-In-Time) gatekeeping metric based on the Jensen-Shannon Divergence ($D_{JS}$). Let $P_{\text{LLM}}$ be the probability distribution over the next tokens generated by the model, and let $Q_{\text{UKH}}$ be the ontological adherence distribution derived from the allowed transition frequencies within the hypergraph $\mathcal{H}$. The semantic divergence is formally stated as: $$D_{JS}(P_{\text{LLM}} \parallel Q_{\text{UKH}}) = \frac{1}{2} D_{KL}(P_{\text{LLM}} \parallel M) + \frac{1}{2} D_{KL}(Q_{\text{UKH}} \parallel M)$$ where $M = \frac{1}{2}(P_{\text{LLM}} + Q_{\text{UKH}})$ and $D_{KL}$ is the Kullback-Leibler divergence defined over a discrete vocabulary $X$: $$D_{KL}(P \parallel M) = \sum_{x \in X} P(x) \log_2 \left( \frac{P(x)}{M(x)} \right)$$ If the divergence exceeds a system-defined critical threshold ($D_{JS} > \theta_{\text{max}}$), the generation hypothesis is immediately rejected. 4. Heterogeneous Hardware Implementation To bypass the parallelization bottlenecks inherent to logical-symbolic algorithms—which trigger massive thread divergence on SIMD architectures—the framework adopts a heterogeneous computation model powered by Speculative Decoding: GPU Execution (CUDA/Triton): The LLM generates $K$ candidate token pathways (drafting sequences) in parallel. CPU Async Execution: A high-frequency multicore CPU pool simultaneously executes the structural parsing of SHACL shapes and the Lean 4 type-checking over the sparse graphs corresponding to the proposed pathways. Non-compliant branches are pruned before the validation and synchronization phase of the model weights. 5. Conclusions Coupling information-theoretic metrics based on the Jensen-Shannon divergence, Alexandrov topological constraints on SHACL-structured hypergraphs, and axiomatic verification within Lean 4 delivers a rigorous formal methodology capable of neutralizing semantic hallucinations. Shifting control from post-hoc output filtering to a priori state space restriction sets a new benchmark for safety in Neuro-Symbolic Artificial Intelligence. Versione Italiana Unificazione Neuro-Simbolica mediante Ipergrafi Cognitivi: Mitigazione Quantitativa delle Allucinazioni nei Large Context Models a Monte della Generazione Autore: Luigi Usai ORCID: https://orcid.org/0009-0003-3001-717X Luogo: Quartucciu (CA), Italy Data: 26 Giugno 2026 Target: Zenodo / arXiv (cs.AI, cs.CL, cs.LO) Abstract I Large Context Models (LCM) presentano una vulnerabilità intrinseca nota come allucinazione semantica, derivante dalla massimizzazione della verosimiglianza condizionata in spazi vettoriali discreti. I tentativi di mitigazione tradizionali agiscono prevalentemente a valle del processo probabilistico, intervenendo quando l'alterazione sequenziale è già avvenuta. Il presente lavoro estende il framework Universal Cognitive Hypergraph (UKH), introducendo una topologia discreta di Alexandrov su ipergrafi di conoscenza per vincolare lo spazio degli stati ammissibili a monte della decodifica dei token. Mediante l'architettura Monadic Neuro-Symbolic Verification and Synthesis Architecture (MNSVSA), i cammini di generazione probabilistica vengono intercettati e validati strutturalmente tramite vincoli W3C SHACL e vincoli logici verificati dal kernel di Lean 4 accoppiato a solutori SMT automatici. I risultati teorici mostrano l'eliminazione matematica delle deviazioni categoriali senza compromissione della fluidità sintattica del modello. 1. Introduzione e Definizione Matematica del Problema Un modello linguistico autoregressivo stima la distribuzione di probabilità del token successivo $w_t$ condizionata alla storia precedente $w_{<t}$: $$P(w_t \mid w_{<t}) = \text{softmax}(W_{\text{unembed}} \cdot h_t)$$ dove $h_t \in \mathbb{R}^d$ rappresenta lo stato nascosto finale estratto dall'architettura Transformer. Poiché la função $\text{softmax}$ mappa i punteggi su una distribuzione di probabilità aperta, assegna intrinsecamente probabilità non nulle a porzioni dello spazio semantico che violano i vincoli assiomatici della realtà. Di conseguenza, l'allucinazione non è un bug accidentale, ma una proprietà strutturale della natura stocastica del modello. Il framework UKH supera i limiti del recupero documentale passivo (RAG) integrando un vincolo topologico-simbolico direttamente nella fase di campionamento (speculative decoding), impedendo all'architettura di esplorare traiettorie probabilistiche associate a stati logicamente non consistenti. 2. Architettura del Framework UKH per la Sicurezza Semantica L'universo del discorso viene mappato su un ipergrafo orientato e serializzato in formato JSON-LD. Sia $\mathcal{H} = (V, E)$ un ipergrafo cognitivo, dove $V$ è l'insieme dei nodi (entità concettuali fortemente tipizzate) ed $E \subseteq \mathcal{P}(V) \setminus \{\emptyset\}$ è l'insieme degli iperarchi che rappresentano relazioni logico-funzionali multi-argomento. 2.1. Spazio Topologico di Alexandrov e Vincoli SHACL Per garantire il rigore geometrico-strutturale su un dominio discreto, lo spazio dell'ipergrafo viene dotato di una topologia di Alexandrov, definendo gli insiemi aperti come i sottoipergrafi chiusi superiormente rispetto a una relazione di preordine logico ($\le$). I vincoli W3C Shapes Constraint Language (SHACL) operano come operatori di chiusura topologica: $$\text{cl}(E_c) \subseteq \mathcal{H}_{\text{valid}}$$ Se un iperarco candidato $E_c$, generato dalla traduzione semantica dei token proposti dall'LLM, viola una Shape strutturale (es. assegnazione di una proprietà fisica inconsistente con il ti