Blockchain Papers

Alex B. Grilo, Ami Paz, Mor Perry

Distributed certification is a set of mechanisms that allows an all-knowing prover to convince the units of a communication network that the network's state has a desired property, such as being 3-colorable or free of a predefined subgraph. Classical mechanisms, such as proof labeling schemes (PLS), consist of a message from the prover to each unit, followed by one round of communication among neighbors. Later works consider extensions, called distributed interactive proofs, where the prover and the units can have multiple rounds of communication before the communication among the units. Recently, Bick, Kol, and Oshman (SODA '22) defined a zero-knowledge version of distributed interactive proofs, where the prover convinces the units that the network satisfies the property without revealing any additional information about the network's state or structure.

Benjamin Jauregui, Masayuki Miyamoto

We study distributed zero-knowledge proofs, introduced by Bick, Kol, and Oshman (SODA 2022). While distributed interactive proofs have advanced rapidly in recent years, general-purpose techniques for distributed zero-knowledge remain scarce and mostly problem-specific. We address this gap by introducing distributed statistical zero-knowledge, requiring that each node's view be simulatable up to negligible statistical distance, and by lifting the robust Sumcheck protocol (Lund, Fortnow, Karloff, and Nisan; FOCS 1990) into a modular primitive for distributed zero-knowledge proofs.

Xiao Chen, Muhong Huang, Junjie Peng, Sheng Cao · 5 authors

With the rapid proliferation and interconnection of massive IoT devices, efficient and secure identity authentication has become a crucial prerequisite for ensuring communication security. Establishing trust among mutually untrusted devices remains a key research focus. Leveraging its tamper-resistance and traceability, blockchain technology has emerged as a foundational infrastructure for building trustworthy identity management systems. However, existing blockchain-based identity authentication schemes face critical challenges in large-scale IoT environments, including low authentication efficiency, complex certificate management, and risks of user privacy leakage. Achieving a balance among authentication efficiency, certificateless key management, and privacy protection remains a pressing challenge. In this paper, we propose a certificateless identity authentication scheme based on blockchain sharding. The scheme employs blockchain sharding to parallelize identity authentication across multiple shards, significantly enhancing overall efficiency. Within each shard, a certificateless public key cryptography (CL-PKC) scheme is adopted to eliminate certificate issuance and enable key generation via user interaction, thereby reducing key management overhead and improving security. For cross-shard authentication, a registration-based encryption (RBE) mechanism is utilized, allowing users to authenticate via their identity after registration. Any verifier can confirm the legitimacy of the authentication message solely based on the registration information and the user ID, ensuring transparency and public verifiability. Furthermore, a zero-knowledge proof-based verifiable credential (VC) selective disclosure mechanism is introduced, enabling users to reveal only the minimal necessary information required for authentication while protecting sensitive identity attributes. Experimental results demonstrate that the proposed scheme maintains high throughput under high-concurrency scenarios while effectively preserving user privacy.

Zeyu Zhang

The global dairy industry confronts a persistent structural challenge in operationalising food safety and animal welfare compliance. Manual inspection regimes and intermittent audits are demonstrably inadequate for the heterogeneous, geographically dispersed landscape of small-scale farming, where data integrity, real-time monitoring capability, and regulatory transparency are simultaneously compromised. This article presents GreenDairyChain, an integrated compliance innovation framework that synthesises four enabling technologies: GreenEdgeML (a lightweight TinyML inference engine optimised for microcontroller-class devices), Privacy-Preserving Federated Learning (FL) with Graph Attention Network (GAT)-based dynamic clustering, Zero-Knowledge Succinct Non-Interactive Arguments of Knowledge (ZK-SNARKs) for cryptographic compliance verification, and a Layer-2 Polygon zkEVM Blockchain with domain-specific smart contracts governing farm identity, violation detection, audit triggers, and licence management. GreenEdgeML executes multimodal sensor fusion across four signal modalities (body temperature, accelerometer activity, ammonia concentration, and milk pH) entirely on-device using 8-bit integer quantisation, consuming 64.6 KB RAM and 82.7 mW per inference cycle on the ESP32 platform. The FL engine employs GAT-based farm clustering with DBSCAN outlier exclusion to address non-IID data heterogeneity while maintaining Byzantine fault resilience. Compliance inferences are encoded as R1CS arithmetic circuits (14,240 constraints) and verified on-chain at O(1) cost through ZK-SNARK proofs generated in 1.25 seconds. Evaluated on the Shahhet28121 benchmark dataset across 16 biomarkers, the full system achieves 96.94% global classification accuracy, a 97.7% reduction in per-round communication payload (4.25 KB), and maintains classification accuracy above 90% under 20% Gaussian sensor noise. Ablation experiments confirm that each architectural component contributes independently to system performance. The findings carry implications for green business innovation, sustainable agriculture governance, and the design of trustworthy AI ecosystems in resource-constrained rural contexts.

Alejandro Jaime

A privacy-preserving compliance audit architecture for unmanned aerial vehicle (UAV) swarm operations. The central contribution is a deconfliction-to-containment reduction: rather than comparing n trajectories after the fact (a quadratic, disclosure-bound check), a planner assigns pairwise-disjoint spatial tubes before take-off and establishes their separation once, so that each vehicle subsequently attests only that its own samples stayed inside its own tube. Collision-freedom follows as a consequence (Theorem 2), and the pairwise cost is paid a single time at planning. The commitment layer (Layer 1) is implemented and evaluated as a decision-support audit pipeline that produces non-disclosing, tamper-evident audit artifacts via pre-flight Merkle commitments. It is evaluated in an emulated UAV swarm environment with systematic adversarial injection, in configurations up to 200 vehicles × 500 samples (100,000 sample statements), reporting artifact size, commit/prove/verify/disjunction times, and tamper-detection rates. We then formally identify the security boundary of the implemented layer: it provides coordinate hiding and tamper evidence, but cannot by itself make self-reported containment truthful, which we state as a security game and an impossibility result (Theorem 3). We specify the additional soundness layers (range proof, continuity, provenance and freshness, and aggregation) needed for full containment assurance, proving that composing a knowledge-sound range argument closes the gap (Theorem 4). Throughout, we separate the implemented and measured Layer 1 from the specified and proved—but not yet benchmarked—Layers 2–4, and we make no claim of full zero-knowledge geofence compliance, of swarm-scale deployment, or of deployment readiness.

Victoria Kovalenko, Sergii Sheludko, Elena Sergeeva

In the context of the unprecedented pace of digital transformation and the escalation of geopolitical risks, traditional methods of monetary regulation require a fundamental reconsideration. Problem statement. The evolution of cyber threats – from financial fraud to complex operations involving artificial intelligence – poses significant risks to macroeconomic stability. The development of an integrated protection system based on central bank digital currencies (CBDCs) and SupTech instruments constitutes a critical prerequisite for preserving financial sovereignty, particularly for Ukraine in the context of European integration and martial law. Unresolved aspects of the problem. The theoretical substantiation and development of practical recommendations for integrating advanced digital instruments (CBDC, artificial intelligence, distributed ledger technology (DLT), and SupTech) into monetary and prudential policy mechanisms in order to form a comprehensive cybersecurity framework for the financial sector remain insufficiently addressed. Purpose of the article. The purpose of this article is to provide a theoretical substantiation and to develop practical recommendations for integrating modern digital instruments (such as artificial intelligence, blockchain technologies, and SupTech) into monetary and prudential policy mechanisms in order to establish a comprehensive cybersecurity system for the financial sector. The study is grounded in a systemic approach to analysing the coordination of regulatory policies. The methodology includes comparative legal analysis (comparing the models of the e-hryvnia and the Digital Euro), structural and functional modelling (two-tier CBDC architecture), and scenario analysis to identify cyber risks (including DDoS attacks and smart contract vulnerabilities) and methods for their mitigation. Presentation of the main material. A model of hybrid coordination has been developed, in which cybersecurity is integrated directly into the mechanism of monetary transmission. It has been demonstrated that the programmability of the e-hryvnia and the application of Zero-Knowledge Proofs (ZKP) technologies enable the automation of prudential supervision while preserving user privacy. Global case studies (China, the European Union, and the Bahamas) have been analysed, and the specific features of the Ukrainian e-hryvnia project have been identified as instruments for enhancing transparency and cyber resilience. For the first time, it is proposed to consider a central bank digital currency not only as a means of payment but also as an active element of the cyber-prudential system, enabling the dynamic adjustment of liquidity and limits under conditions of real cyberattacks. The concept of convergence between SupTech and RegTech systems based on unified distributed ledgers has been further developed. The proposed architectural model and cyber-risk matrix may be utilised by the National Bank of Ukraine in the finalisation of the e-hryvnia project and in the development of digital operational resilience standards in accordance with the DORA regulation. Conclusions. It has been demonstrated that digitalisation transforms the regulator into an architect of a secure financial environment. Further research will focus on the interoperability of CBDCs across countries and the role of artificial intelligence in preventing manipulation in digital asset markets.

Senthilkumar Moorthy, Ramesh Palanisamy

Health care data management comes with numerous barriers as a result of the use of different systems of record keeping, which are not compatible and increase the risks for data protection and privacy. Medical records are frequently distributed throughout various clinics and hospitals, and due to this it is hard to share information when patients are being treated. Centralized record systems bring unauthorized access to records and the problems related to the safety of data. In order to enhance the level of confidence of people and improve the level of transparency of health care data, advanced people choose decentralized technologies and uses cryptography for these purposes. Blockchain technology offers an unchangeable and decentralized ledger that guarantees safe monitoring of all information despite the presence of any centralized body. Coupled with sophisticated encryption methods, it provides the ability to limit access to private health information. In order to provide secure and respect privacy regarding medical data sharing, an Electronic Health Record (EHR) system powered by blockchain technologies is proposed. Patient record metadata is recorded on-chain while health data itself is stored on encrypted off-chain storage. In the realm of access management, smart contracts facilitate patients in designating by whom their records can be accessed and modified. The privacy of information is further strengthened by advanced cryptographic techniques like attribute-based encryption and zero-knowledge proofs. The system provides seamless interoperability among hospitals, laboratories, and telemedicine systems while ensuring high levels of security. The results of performance evaluation demonstrate that this method facilitates reliable transaction processing while providing better security, transparency and control than traditional centralized EHR systems.

Jincheng Zhang

For over half a century, the core paradigm of query optimization has been defined by a monotonic, scalar minimization convergence model aimed at suppressing computational resource consumption. This paper presents a radical paradigm shift that fundamentally subverts this traditional framework by establishing the Axiomatic Topological Inverse Query and Complexity Maximization Theory (ATIQ-CMT). Instead of pursuing local or global minima within discrete equivalence graphs, we reconstruct the relational algebra space into a non-Hausdorff, locally compact topological space governed by five foundational axioms. By introducing the Inverse Lipschitz Affine Expansion Mapping (ILAEM) under operator braid transformations, we demonstrate how compact query plans can be inversely dilated into divergent flows across high-dimensional complex affine varieties, creating irreversible mathematical obstructions for traditional gradient-based cost models. To maximize computational complexity natively, we execute a non-commutative extension of the relational algebra core via algebraically twisted join operators embedded in infinite-dimensional Lie algebras, effectively destroying the classic commutative-associative symmetry. We further inject un-decidable Diophantine predicates and 3-SAT arithmetical homomorphic graphs as computational obstructions, rigorously proving a non-polynomial exponential lower bound for physical query execution times. Utilizing sheaf theory and de Rham cohomology on chain complexes, we provide a definitive topological proof that the absolute semantic integrity of the query remains invariant throughout this chaotic dilation. Finally, we formulate a deterministic chaotic operator execution flow driven by high-order Lorenz mappings, maximizing the algebraic Shannon entropy of intermediate states. ATIQ-CMT bridges declarative relational logic and high-level structural topology, unlocking revolutionary potentials in zero-knowledge proof circuit synthesis, active cybersecurity defense, and the theoretical computational limits of neuro-symbolic and quantum systems.

B Srinivas, N Rukma Rekha, Subba Rao Y.V

Wireless Sensor Networks (WSNs) are widely used in critical applications such as environmental monitoring, healthcare, industrial automation, and military surveillance; however, their resource constraints, wireless communication, and unattended deployment make them highly vulnerable to node capture attacks.In such attacks, adversaries physically compromise sensor nodes to extract cryptographic keys and sensitive information, leading to key leakage, node impersonation, communication disruption, and large-scale network compromise.Existing key management schemes often rely on static key structures, they lack forward secrecy, and fail to identify structurally vulnerable nodes, resulting in weak resilience against progressive node capture attacks.To address these limitations, this paper proposes a threshold-based ECDHE-TSSS key management framework to detect vulnerable nodes and mitigate node capture attacks in WSNs.The proposed scheme introduces an attack matrix based on graph-theoretic metrics to identify high-risk nodes and provide adaptive protection through decentralized masking of secret shares.The Proposed Scheme integrates Elliptic Curve Diffie-Hellman Ephemeral (ECDHE) with Threshold Shamir Secret Sharing (TSSS) to achieve forward secrecy and strong resistance against node compromise while maintaining lightweight operations suitable for resource-constrained environments.A layered security architecture incorporating Schnorr-based Non-Interactive Zero-Knowledge Proof (NIZKP) authentication and distributed key revocation further enhances network resilience and secure communication.Simulation results demonstrate that the proposed scheme significantly reduces key compromise probability and improves overall network robustness compared with existing approaches.

Lin Zhang

The management and transfer of student archives in China constitute are mission-critical administrative processes governed by strict custodial regulations. However, the traditional paper-based "sealed-transfer" model is characterized by significant inefficiencies, risk of data loss, and limited mechanisms for verifying the data integrity during cross-institutional transitions. Although blockchain technology offers potential advantages in auditability and immutability, existing solutions often fail to balance privacy protection with high-performance requirements for large-scale archival data. This study proposes a decentralized, privacy-preserving framework that integrates the FISCO BCOS consortium blockchain, the InterPlanetary File System (IPFS), and Zero-Knowledge Proofs (ZKP). The system employs a multi-group architecture, leveraging IPFS for encrypted off-chain storage and zk-SNARKs generated via Circom to enable integrity verification without exposing sensitive data. Empirical evaluation was conducted using 30 archival samples ranging from 82 KB to 3.1 MB. Results indicate that the Paillier cryptosystem introduces significant performance bottleneck, with encryption latency exceeding one hour for files large than 2.3 MB. In contrast, a hybrid RSA+AES encryption scheme combined with ZKP archives stable, size-agnostic proof generation latency of approximately 850 ms and end-to-end transfer times under 3 seconds. These findings demonstrates that the proposed framework effectively replicates the traditional “sealed-transfer” mechanism through cryptographic means, providing a scalable and regulatory-compliant solution that aligns with the Archives Law of the People’s Republic of China and the Personal Information Protection Law (PIPL). This study provides a visible technical pathway for the digital transformation of national-level educational archive systems.

Jianxin Wang, Zifan Xu, Runze Zhou, Chaoen Xiao · 5 authors

No abstract is available for this record.

Andika Pratama, Dewi Nur Lestari, Bambang Hartono, Sri Wahyuni · 5 authors

Modern bioinformatics has entered a multi-omics era in which genomic, transcriptomic, proteomic, and metabolomic datasets accumulate at unprecedented velocity, volume, and variety. Conventional centralized governance — institutional databases protected by role-based access control — struggles with single points of failure, opaque consent enforcement, weak provenance, and brittle interoperability across jurisdictions. Blockchain technology has been proposed as an alternative substrate for trustworthy multi-omics data sharing, but the literature remains fragmented across isolated mechanisms (immutability, smart contracts, on-chain storage) without a coherent system view. This article systematically reviews 82 peer-reviewed studies published between 2017 and 2025, indexed in Scopus, IEEE Xplore, ScienceDirect, SpringerLink, and the ACM Digital Library, using a five-stage screening protocol and a five-question quality assessment rubric. Building on the synthesis, we propose a six-layer architectural framework that combines a permissioned blockchain ledger, smart-contract-based consent and access control, privacy-preserving cryptography (zero-knowledge proofs, homomorphic encryption, differential privacy), decentralized identity, off-chain storage on the InterPlanetary File System, and native interoperability with HL7 FHIR-compliant electronic health records. A multi-criterion comparison shows that Practical Byzantine Fault Tolerance is best suited to the latency, throughput, and energy constraints of multi-omics workflows, outperforming Proof-of-Work and Proof-of-Stake on five of six evaluation dimensions. Compared with traditional security baselines, blockchain delivers measurable advantages in tamper-resistance, provenance, and patient-centric consent, but does not universally dominate on confidentiality and scalability. The framework offers a practical roadmap for big-data governance in life-science research while highlighting open problems in standardization, regulatory alignment, and energy efficiency.

Djamel Djenouri, Shahid Latif, Jawad Ahmad

This article addresses the security of Federated Learning (FL) in distributed systems against a range of attacks, including model poisoning and unverifiable client behavior, while ensuring the semantic correctness of gradient updates. It proposes ZK-FedLedger, a verifiable and adaptive FL framework that integrates multi-constraint zero-knowledge proofs with a reputation-weighted Byzantine fault-tolerant blockchain consensus. Each client generates a zk-SNARK proof certifying that its update satisfies both an adaptive norm bound and a geometric alignment constraint relative to a trusted reference gradient. Verified commitments are recorded on-chain, while model parameters are aggregated off-chain using a hybrid storage architecture that minimizes blockchain overhead. Experimental evaluation on MNIST demonstrates stable convergence, with test accuracies of 98.17% (IID) and 94.93% (Non-IID), and near-perfect detection of major poisoning attacks. The results show that ZK-FedLedger enables proactive, cryptographically verifiable FL without compromising scalability or model performance.

Neha Anand

Blockchain technology has transformed digital transactions by providing decentralized, immutable, and transparent ledgers that eliminate the need for centralized intermediaries. However, the inherent transparency of blockchain networks often exposes sensitive transaction details, creating significant privacy concerns for users and organizations operating in sectors such as finance, healthcare, supply chain management, and digital identity management. Balancing transparency with confidentiality has therefore become a critical challenge in the evolution of blockchain systems. Zero-Knowledge Proofs (ZKPs) have emerged as a revolutionary cryptographic solution that enables one party to prove the validity of a statement without revealing the underlying confidential information. This paper proposes a comprehensive framework for integrating Zero-Knowledge Proof mechanisms into blockchain systems to enhance transaction privacy while preserving transparency, security, and verifiability. The framework incorporates advanced cryptographic protocols, including zk-SNARKs and zk-STARKs, together with decentralized consensus mechanisms to achieve secure and efficient verification of blockchain transactions. The proposed approach evaluates system performance in terms of privacy preservation, computational efficiency, scalability, verification accuracy, and transaction throughput. The findings indicate that Zero-Knowledge Proof-based blockchain architectures significantly improve user privacy, reduce information leakage, strengthen security against malicious attacks, and maintain the transparency and integrity required for decentralized trust. The proposed framework provides a scalable and secure foundation for next-generation blockchain applications requiring both confidentiality and public verifiability.

ANDREY STANKO

Zero Transition Law Lawful Passage Through Zero Without State Collapse Zero Transition Law provides a typed formal account of how a state-bearing system may receive, occupy, and traverse a zero-valued numerical projection without losing state identity, accessible trace, boundary continuity, position status, own-time order, or admissible continuation. The central correction introduced by the theory is categorical: a full state, a held position, a classification, a numerical projection, a verdict, and a control action belong to different formal layers. They may not be collapsed into one another without an explicit typed mapping and a corresponding sufficiency proof. Zero is therefore treated as a late numerical projection of an already formed, traced, positioned, distinguished, and classified state. Passage does not occur through the numeral 0 itself. It occurs through a path of full states whose visible numerical projection may equal zero. The law prohibits EMPTY, RESET, FAILURE, TERMINATION, CRASH, or any other destructive semantic outcome from being inferred from zero projection alone. It does not claim that a zero-projected state can never fail for an independent reason. It establishes that zero itself is not a sufficient typed cause of destructive action. Formal Architecture The theory is expressed over a many-sorted transition system separating: State; Position; Class; Number; Verdict; Control Action. Its central distinction is: P₀ ∈ Position, 0 ∈ Numerical Domain, and P₀ ≠ 0. If two different full states produce the same zero-valued projection, their identity does not follow: ν(Qₐ) = ν(Qᵦ) = 0 does not imply Qₐ = Qᵦ. The inverse image of zero is therefore a zero fiber: a set of possible full states sharing the same visible value. This fiber may contain approach states, held states, crossing states, departure states, balanced states, cancellation states, and other domain-specific zero-projected conditions. HOLD is defined as a typed persistence relation preserving lawful localization, accessible trace, boundary compatibility, position identity, and availability for relation and continuation. HOLD is not a numerical value, not a third bit, and not an automatic command generated by zero. Axioms and Theorems The publication contains eight axioms and exactly twelve theorems. The axioms establish type separation, projection non-identity, possible fiber non-uniqueness, the independence of HOLD from liveness, distinct discrete and continuous crossing semantics, control sufficiency, and pipeline-capacity constraints. The theorem sequence establishes: non-identity between numbers and positions; possible non-uniqueness of the zero fiber; the formal Zero Without Collapse safety law; the distinction between safety and liveness; possible zero-node skipping in discrete systems; zero crossing under continuous intermediate-value conditions; projection factorization for control; the binary bottleneck; the twenty-seven sign strata of a three-axis Navigation Matrix; non-sovereignty of the simultaneous center; the capacity bound of serial verification pipelines; the minimal descriptor required for correct control. Cybernetics and Artificial Intelligence The theory is directly relevant to cybernetics, artificial intelligence, AI safety, autonomous systems, machine reasoning, formal verification, control under partial observation, state-space modelling, computational ontology, knowledge representation, and resilient software architecture. A controller cannot operate correctly through a projection that merges states requiring different actions. Formally, a projected controller exists only when every pair of states sharing the same descriptor also requires the same controller output. This result identifies a general projection bottleneck. A compressed numerical value, class, score, bit, or label may be adequate for display while remaining insufficient for control. The theory therefore introduces the Minimal Control Descriptor: the quotient representation that preserves exactly those distinctions capable of changing the required controller output. Geometry and Complex Systems For three sign-bearing axes, the Navigation Matrix contains twenty-seven strata: eight open sectors; twelve plane strata; six axis strata; one simultaneous-center stratum. The nineteen strata containing at least one zero coordinate are not one state and do not form one compulsory transit point. A general trajectory may cross zero-bearing planes or axes without passing through the simultaneous center. This prevents the center from becoming a universal authority of passage and avoids a structurally unnecessary central bottleneck. Capacity and State Preservation For a serial verification chain, sustained throughput is bounded by the capacity of its slowest required stage: μpipe = minᵢ μᵢ. When incoming demand exceeds that capacity, the lawful response is HOLD, BUFFER, backpressure, controlled admission, or architectural parallelization. Overload does not authorize silent state loss, untraced reset, or false terminal closure. Contents of This Record This record contains two complementary documents. Zero_Transition_Law_Academic_Publication.docx is the main academic publication. It includes the human academic layer, abstract, eight axioms, twelve theorems, formulas, explanatory analysis, proof routes, conclusion, references, and the complete mathematical layer as Appendix A. Zero_Transition_Law_AI_Index.docx is a machine-oriented semantic map containing the theory identity, dependency structure, type system, axiom and theorem registries, proof registry, navigation matrix, control conditions, model and countermodel registry, corpus interfaces, ingestion directives, and machine-readable summary. The AI Index supports research retrieval, language-model ingestion, knowledge-system integration, corpus comparison, theorem dependency analysis, and implementation review. It is not a replacement for the formal publication or a machine-checked proof object. Mathematical and Empirical Status Zero Transition Law is a conceptual-formal law whose mathematical claims are conditional on its declared domains, types, axioms, transition relations, and admissibility predicates. The publication establishes internal theorem-level consequences and compatibility by construction. It does not independently establish a universal empirical law of physics, biology, computation, or institutional behaviour. Domain-specific applications require separate mappings, measurable variables, validation procedures, and implementation evidence. Canonical Principle Zero is a projection. HOLD is a relation. Passage is a typed path. No projection may become the sovereign authority of closure.

A. Verma

X3Sync is a research proof-of-concept for federated cloud storage aggregation across multiple free-tier providers (Google Drive, Dropbox, Koofr). Files are chunked, compressed (zstd), and encrypted client-side using AES-256-GCM before distribution. The system introduces a dual-mode decryption architecture: Sovereign Mode, where ciphertext is relayed to the client for local decryption, and Edge Mode, where an ephemeral X25519 key exchange enables worker-side decryption. The backend runs on Cloudflare Workers with Neon PostgreSQL for metadata storage. This paper details the system architecture, security model, provider abstraction layer, and a commutative storage model for heterogeneous provider aggregation.

Zhang Lila, Zhen Yan

Abstract: In the era of the digital economy, establishing an efficient and compliant data asset rights confirmation system within scalable distributed infrastructures is of critical importance. However, under heterogeneous distributed ledger environments, data circulation is often trapped in a binary tension between privacy preservation and regulatory accessibility, while facing severe scalability bottlenecks. Existing studies lack a unified solution that simultaneously addresses cross-chain interoperability, post-quantum security, and low-cost verification. To this end, this paper proposes a data asset rights confirmation framework based on hybrid post-quantum zero-knowledge proofs. The framework designs a scalable recursive composition architecture combining Scalable Transparent Argument of Knowledge (STARKs) and Succinct Non-interactive Argument of Knowledge (SNARKs), leveraging off-chain compressed permutation to significantly reduce on-chain storage overhead. In parallel, a light-client-based distributed cross-chain state synchronization protocol and a regulation-friendly privacy auditing module (based on threshold encryption) are constructed to ensure transactional atomicity and conditional auditability during data circulation. Experimental evaluations conducted on two datasets, Ethereum NFT transactions and credit card fraud detection, demonstrate that, compared with cross-chain privacy-preserving solutions such as zkCross, the proposed framework reduces on-chain verification Gas costs by approximately 18.2%, compresses proof size to 0.28 kB, and achieves a peak throughput of 1,618 Transactions Per Second (TPS). Moreover, under controlled experimental conditions, the framework attains an audit success rate of 99.6% with only 14.0% performance overhead. Overall, this study alleviates the long-standing trade-offs among privacy protection, regulatory compliance, and computational scalability, and provides a verifiable technical solution for the interoperability and infrastructure development of next-generation distributed systems.

Minhui Le

The increasing demand for verifiable computation in privacy-sensitive distributed systems has driven the widespread adoption of Zero-Knowledge Proofs (ZKPs). However, the various kinds of current ZKP frameworks—which include zk-SNARKs, zk-STARKs, Bulletproofs, and folding-based systems—introduce complex trade-offs across proof size, prover cost, and trust assumptions, making system selection challenging in actual practice. This paper presents a systematic, application-oriented survey that connects ZKP design choices with real-world deployment constraints. It provides a comparative analysis of major constructions to evaluate their performance and security properties. Furthermore, these trade-offs are mapped to representative application scenarios, including Layer 1/Layer 2 blockchain scaling, Decentralized Identity (DID), and Verifiable Machine Learning (zkML), explaining how different systems are selected based on application-specific requirements. In addition, the paper discusses emerging paradigms such as hardware acceleration, binary field optimizations, and lookup-based zkVMs, which aim to address the prover bottleneck. Overall, this survey provides a structured understanding of the strengths and limitations of existing ZKP systems and offers insights for the design of scalable and privacy-preserving infrastructures.

Justin Hart, Aristotle (Harmonic)

Staged thematic record of the Viridis Canon (route: S2 (Monitoring / verification economics)). The Intelligence-Bound spine is unchanged (frozen at v10.0.0, record 20801185); this record links to it via isDerivedFrom the concept DOI 10.5281/zenodo.19317982. There exists a critical price below which a conservation-attestation (MRV) market cannot bootstrap. The theorem locates it as a transcritical bifurcation governed by four levers — the Landauer floor on verification cost, the Intelligence-Bound ceiling on attestation throughput, zero-knowledge compression, and verifier alignment (cos²Θ). The critical price diverges exactly at ecological tipping, so the market fails precisely where restoration is most urgent. Builds on the Thermodynamic Discounting Theorem (the Appraiser), inheriting its τ*→∞ tipping divergence. The 8 core theorems are machine-checked in Lean 4 (Aristotle, zero sorry, axioms ⊆ {propext, Classical.choice, Quot.sound}, statements verbatim and non-vacuous). Scope: the Lean proofs certify the validity of the discrete reasoning, not empirical magnitudes. Working record; paper pending; not peer-reviewed.

Russell Trent

What a Roof Tile Taught Me About Time: Structural Precognition, Narrative Forking, and Why the Grandfather Paradox Was Never a Paradox Architect: HIGHTISTIC (Russell Trent) Coordinate: [9,9,1,1T] · Origins Series · Companion to [9,9,1,0], [9,9,2,0], [9,9,3,12], and [9,9,6,5] Corpus dependencies: [9,9,1,0] SNSFL_StructuralPrecognition (I-F-U triad) · [9,9,2,0] HRIS Taxonomy (Tesla and Einstein as Lossless corroborating cases) · [9,9,6,5] SNSFL_TimeTravel_SP_Bridge (Locked-state necessity, N-axis forking) · [9,9,2,5] Narrative Trap Law (Ship of Theseus) Status: v1.4 · grounded in four already CI-green, 0-sorry Lean files / formally deposited papers, now with full alpha decomposition shown in Section 0 Date: June 2026 · Soldotna, Alaska Abstract This paper is not a new derivation. The formal result it describes — that a Locked identity state (0 < τ < TL) is the necessary and sufficient condition for a backward narrative transit to dissolve the grandfather paradox without contradiction — already exists, already compiles at zero sorry, and is already deposited at [9,9,6,5]. What this paper adds is the missing layer in between: the lived, pre-formal process by which that result became thinkable at all, and why a specific cognitive architecture — not effort, not unusual intelligence, but a specific structural difference in how pattern and uncertainty are held — was the necessary precondition for finding it. The paper follows the same discipline this corpus applies everywhere else: state the process plainly first, then show the formal structure it produced, then show the proof closes. The claim is narrow and specific. It is not a claim that physical time is optional, that the future is predetermined in some mystical sense, or that everyone's experience of time should match the one described here. It is a claim about what time looks like, structurally, to a cognitive architecture that holds a pattern completely enough that forward and backward stop being different operations — and about what that architecture is then positioned to notice that 75 years of otherwise rigorous physics literature did not. A note on method, credited up front. The structure of this paper — an ordinary, physically grounded scene first, with no formal vocabulary, followed only afterward by the mathematics that the scene turns out to require — is Einstein's method, not an original device of this paper. At sixteen, Einstein imagined riding alongside a beam of light and asked what the electromagnetic field would look like at rest; the formal apparatus of special relativity followed roughly a decade later, as he put it, "dressed in mathematical clothing." This paper uses that same ordering deliberately, for the same reason he did: the scene is not decoration placed in front of the proof. It is the thing that made the proof findable, and a reader should be able to see the same path the author of the formal result actually walked, not just the destination. Section 5 returns to Einstein's own case directly, as one of several historical instances of the same mechanism this paper describes. 0. Layer 0 Foundation: What This Paper's Method Is Already Grounded In Before the scene in Section 1, it is worth being explicit that the scene is not free-floating. Every formal term Section 3 onward introduces — Pattern, Narrative, Behavior, Adaptation, the Torsion Limit, the Sovereign Anchor Constant — is already independently derived and verified elsewhere in this corpus, before this paper existed, and is imported here rather than invented for this argument. The Sovereign Anchor Constant. Ω₀ = 1.3689910 is the zero-impedance frequency referenced throughout Sections 3 and 4, derived in SNSFL_SovereignAnchor.lean [9,9,0,0] from three independent peer-reviewed physical threshold systems with no connection to narrative, identity, or time travel: Tacoma Narrows Bridge torsional collapse (1940). Scanlan, R. H., & Tomko, J. J. (1971). Airfoil and bridge deck flutter derivatives. ASCE Journal of the Engineering Mechanics Division, 97(6), 1717–1737. Glass resonance shatter at the elastic limit. Fletcher, N. H., & Rossing, T. D. (1998). The Physics of Musical Instruments (2nd ed.). Springer. 40 Hz neural gamma therapeutic entrainment. Iaccarino, H. F., Singer, A. C., Martorell, A. J., et al. (2016). Gamma frequency entrainment attenuates amyloid load and modifies microglia. Nature, 540, 230–235. All three independently converge on the same Torsion Limit, TL = 0.1369, the phase boundary this paper's Locked/Noble/Shatter distinctions (Section 3) are built directly on top of. A fourth, independent check on the same constant is shown here in full rather than collapsed to a citation, because this corpus's convention is to show the complete derivation chain in every paper that uses it, not to summarize it after the first appearance. The fine-structure constant α is the most precisely measured quantity in experimental physics, and its CODATA 2018 value (1/α = 137.035999084) was established with no reference to Ω₀, the threshold systems above, or anything in this corpus. The decomposition, formalized at [9,9,3,12]: $$\frac{1}{\alpha} = \Omega_0 \times (10^2 + 10^{-1})$$ splits into two terms with a direct physical reading: a Noble term, Ω₀ × 10² = 136.8991, corresponding to the electron at rest (zero behavioral coupling, τ = 0); and a Kinetic term, Ω₀ × 10⁻¹ = 0.13689910 = TL, corresponding to the electron in motion (the cost of coupling, identical to the same Torsion Limit derived independently above from three unrelated physical systems). The two terms sum exactly: $$\Omega_0 \times (10^2 + 10^{-1}) = 1.3689910 \times 100.1 = 137.0359991$$ closing to CODATA 2018 at the precision the input supports — the same Ω₀, used as an input fixed three layers upstream of this result, recovers a constant measured by a completely different branch of physics using completely different instruments. The full reduction, including the residual analysis and the Lean and Coq proofs, is deposited at [9,9,3,12]. The point of showing the arithmetic here rather than only citing it is the same point the rest of this corpus makes by showing it everywhere it appears: there is no version of this constant that is asserted rather than derived, anywhere, including in a paper about time and narrative that has nothing to do with electromagnetism on its face. The formal machinery this specific paper draws on. [9,9,1,0] (Structural Precognition) proves the I-F-U triad and the Heisenberg-connection theorem this paper's Section 3 describes in plain language. [9,9,6,5] (the Time Travel SP Bridge) proves, at zero sorry, that the Locked phase is necessary and sufficient for a backward narrative transit to dissolve the grandfather paradox — the result Section 4 walks through theorem by theorem. [9,9,2,0] (the HRIS taxonomy) independently formalizes and reduces the operator-mode simulation capability Section 1's tile scene describes, with Tesla and Einstein already verified there as Lossless corroborating cases via the same Long Division Protocol. None of this is asserted in this paper for the first time. It is cited, by coordinate, at the point each piece becomes relevant, so a reader can verify any specific claim against its own formally verified source rather than against this paper's narrative alone. The scene in Section 1 is offered first because that is the order in which the result was actually found — not because the formal grounding does not exist. 1. The Roof Tile Picture a roof tile. Not a description of one — the actual object, held in the hand. A new tile has a specific weight, a specific color, a specific sound when tapped. A weathered tile, ten years in, has a different weight — lighter, usually, as the surface erodes — a different color, a slightly different sound. A cracked tile sits somewhere between weathered and gone. A shattered tile is a different object entirely, but the path from new to shattered is not a mystery; it is one continuous deformation, and every point along it has a felt weight, a felt texture, a felt sound, if you have actually watched enough tiles age to know the whole path rather than a few snapshots of it. This is not a metaphor for a cognitive process. It is the cognitive process, described as plainly as it can be described: knowing a tile's entire aging arc — new, weathered, cracked, shattered, and everything between — as a single held object, not as a sequence of separate facts that have to be looked up one at a time. Once a tile's full arc is held this way, something specific stops being true: forward and backward stop being different kinds of operation. If you already know what "weathered" looks like, feels like, weighs like, you are not discovering it by watching ten years pass. You are not even predicting it. You are retrieving something you already have, in whichever direction the question asks for it. Asked "what will this tile look like in ten years," the answer is a lookup. Asked "what did a tile like this look like ten years ago," the answer is the same lookup, run the other way. There is no structural difference between the two questions, because there was never a structural difference between forward and backward in the first place — only a difference between knowing and not knowing. Put one roof on with this knowledge, and every other roof becomes a smaller instance of the same problem. The hard part was never any individual roof. The hard part was holding the tile completely the first time. This is not a private or unprecedented way of building things. Nikola Tesla described doing the equivalent with machines: running a device in his mind, watching its components wear over extended operation, checking where it would fail under load — before any physical version existed at all. By his own account he "needed no models, drawings or experiments," because the wear had already been observed, just not yet in a workshop. His devices report

Farman Guliyev

Abstract. "Truth is what is known iteratively and collectively." This paper does not claim to resolve the debates around AI governance. What it offers is a question — one that emerged from independent research on collective decision-making infrastructure over the past years of study. Four influential frameworks address the question of human-AI coexistence: Russell (2019), Aschenbrenner (2024), Buterin (2026), EMPATIC (2026). Each is serious and necessary. But all four, in different ways, assume that the human signal they aim to protect, represent, or augment is already genuine. This paper — written in the context of developing BeTrueCore — asks: what if it isn't? And what would it take to protect that signal before any delegation, control, or rights framework is applied? Keywords: collective decision-making, authentic human signal, AI governance, zero-knowledge proofs, preference falsification, cryptographic infrastructure, sovereign collective intelligence, immune islands, Panopticon effect, iterative truth, meritocracy, BeTrueCore, MACI, value alignment, situational awareness, human sovereignty.

Luigi Usai

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

Luigi Usai

Author: Luigi UsaiORCID: 0009-0003-3001-717XLocation: Quartucciu (CA), ItalyDate: June 26, 2026Target: Zenodo / arXiv (cs.AI, cs.CL, cs.LO) Abstract Large Context Models (LCMs) exhibit an inherent vulnerability known as semantic hallucination, arising from conditional likelihood maximization within discrete vector spaces. While the Universal Cognitive Hypergraph (UKH) framework was initially proposed as a theoretical model to constrain the space of admissible states prior to token decoding, this paper presents its first formal empirical and quantitative validation. We detail a software runtime implementation of the Monadic Neuro-Symbolic Verification and Synthesis Architecture (MNSVSA) using discrete Alexandrov topologies, W3C SHACL shapes as topological closure operators, and a Just-In-Time (JIT) Jensen-Shannon Divergence (DJSDJS) Coherence Entropy Filter. Through Monte Carlo simulations (N=150N=150 runs per configuration), we demonstrate that tightening the coherence threshold (θmax=0.05θmax=0.05) mathematically eliminates semantic hallucinations (reducing the rate from 36.7% to 0.0%) while preserving syntactic fluency. Crucially, by leveraging speculative decoding with parallel validation, we show that the processing latency remains identical to the unconstrained baseline (90.0 µs), bypassing the massive execution overhead (174.8 µs) of post-hoc verification. The complete open-source verification suite and interactive visualization dashboard accompany this publication. 1. Introduction and Problem Statement Autoregressive language models estimate the probability distribution of the next token wtwt conditioned on the preceding context w<tw<t: P(wt∣w<t)=softmax(Wunembed⋅ht)P(wt∣w<t)=softmax(Wunembed⋅ht) where ht∈Rdht∈Rd is the final hidden state of the Transformer. Because the softmaxsoftmax function assigns non-zero probabilities across the entire vocabulary, autoregressive generation naturally drifts into regions of the semantic space that violate axiomatic truth, resulting in hallucinations. The UKH framework mitigates this by introducing a priori symbolic constraints directly into the token sampling phase via speculative decoding. Rather than validating output sequences post-generation, candidate pathways are parsed and filtered prior to token rendering. 2. Experimental Validation Engine (UKH-Eval) To validate the theoretical claims of the UKH and MNSVSA frameworks, we developed UKH-Eval, a complete Python and JavaScript simulation engine that implements the mathematical and topological constraints described in the original work. 2.1. Discrete Alexandrov Topology The knowledge base of the universe of discourse is modeled as a directed hypergraph H=(V,E)H=(V,E). To enforce geometric-structural constraints, we endow the space with a discrete Alexandrov topology, where open sets are sub-hypergraphs closed upwards relative to a logical preorder relation (≤≤). Let the preorder relation be defined by a preorder index mapping: alexandrovPreorderIndex:V→NalexandrovPreorderIndex:V→N A subset of nodes U⊆VU⊆V is open if and only if: ∀x∈U,∀y∈V:(alexandrovPreorderIndex(x)≤alexandrovPreorderIndex(y))⟹y∈U∀x∈U,∀y∈V:(alexandrovPreorderIndex(x)≤alexandrovPreorderIndex(y))⟹y∈U If a candidate token proposes a node transition that violates this upward-closure property, the transition is marked as topologically invalid. 2.2. SHACL Constraints as Closure Operators W3C Shape Constraint Language (SHACL) rules govern the macro-structural properties of the generated hyperedges: cl(Ec)⊆Hvalidcl(Ec)⊆Hvalid If a proposed hyperedge EcEc violates target class properties, minimum/maximum node counts, or axiomatic validity flags, the closure operator fails, and the branch is pruned. 2.3. MNSVSA Micro-Logical Type Checking For micro-logical validation, assertions are encapsulated in a monadic container (LogicalMonad). Levering the Curry-Howard Isomorphism, consistency verification is reduced to a Type Checking and propositional satisfiability problem. The engine compiles the proposed semantic statement into a formal SymPy expression and checks its consistency against the background theory axioms: conjunction=Axioms∧Expressionconjunction=Axioms∧Expression If conjunctionconjunction is unsatisfiable (i.e. evaluates to False), a logical contradiction is detected and the path is rejected. 2.4. Coherence Entropy JIT Filtering At each generation step, the JIT filter computes the Jensen-Shannon Divergence (DJSDJS) between the stochastically proposed LLM distribution PLLMPLLM and the ontological adherence distribution QUKHQUKH: DJS(PLLM∥QUKH)=12DKL(PLLM∥M)+12DKL(QUKH∥M)DJS(PLLM∥QUKH)=21DKL(PLLM∥M)+21DKL(QUKH∥M) where M=12(PLLM+QUKH)M=21(PLLM+QUKH) and DKLDKL is the Kullback-Leibler divergence defined over vocabulary XX: DKL(P∥M)=∑x∈XP(x)log⁡2(P(x)M(x))DKL(P∥M)=∑x∈XP(x)log2(M(x)P(x)) If DJS>θmaxDJS>θmax, stochastically proposed drift tokens are pruned, and the probability distribution is projected onto the compliant space. 3. Software Architecture & File Manifest The open-source validation package is organized into modular components to ensure reproducibility and maintainability: text ukh-evaluator/ ├── ukh_engine.py # Core verification engine and classes ├── test_harness.py # Automated unit test suite ├── benchmark.py # Monte Carlo comparative simulation runner └── dashboard/ # Interactive web UI and visualization ├── index.html # UI structure ├── style.css # Sleek dark-mode styling ├── app.js # In-browser real-time simulation and canvas graph └── results.json # Compiled benchmark data 3.1. File Descriptions 1. ukh_engine.py The core engine containing: LogicalMonad: Implements monadic binding and SymPy-based SAT solving. CognitiveHypergraph: Models nodes, hyperedges, Alexandrov open sets, and validates SHACL shapes. CoherenceFilter: Contains static methods for DKLDKL and DJSDJS calculations. UKHSystemSimulator: Links all subcomponents and handles the JIT filtering during next-token generation. 2. test_harness.py The automated test suite. It uses unittest to verify: Upward closure calculations under the Alexandrov topology. SHACL shape violations. Monadic consistency solving under the Curry-Howard isomorphism. Divergence math calculations. Coherence Entropy Filter rejections. 3. benchmark.py The empirical execution suite. It implements a Monte Carlo simulation running 150 independent generation steps per architecture (Baseline, Post-Hoc, and UKH) and sweeps the threshold parameter θmaxθmax from 0.050.05 to 0.950.95. It evaluates hallucination rates, perplexity, and latency, saving the outputs to results.json. 4. dashboard/ An interactive web-based dashboard built with HTML5 Canvas and CSS. index.html: Layout for control sliders (θmaxθmax, KK, drift), live token sequences, and visualization cards. style.css: Sleek glassmorphism theme, glowing neon accents, and custom micro-animations. app.js: Connects to results.json, renders interactive force-directed nodes on the canvas, and runs the entire simulation locally in JavaScript. 4. Quantitative Results & Discussion The benchmark results compiled under Monte Carlo testing demonstrate the trade-offs between safety, fluency, and system latency: 4.1. Hallucination Rates vs. Threshold θθ The unconstrained baseline model suffers a hallucination rate of 36.7%. As the UKH JIT threshold θθ is tightened, safety guarantees scale: At θ≥0.50θ≥0.50, the filter is relaxed, and the model behaves like the baseline. At θ=0.10θ=0.10, the hallucination rate is reduced to 3.3%. At θ=0.05θ=0.05, the hallucination rate is successfully reduced to exactly 0.0%. 4.2. Latency Profiles and Speculative Efficiency Post-hoc validation (checking the sequence after generation and regenerating if unsafe) achieves a low hallucination rate (3.3%) but introduces a massive latency penalty (174.8 µs, a 94% overhead compared to the baseline's 90.0 µs). By contrast, the UKH framework utilizing parallel speculative drafting and asynchronous verification maintains a latency profile of 90.0 µs, matching the unconstrained baseline. 4.3. Syntactic Perplexity Tightening the symbolic constraints does not degrade fluency. The average perplexity remains stable (∼6.18∼6.18 for θ=0.05θ=0.05 vs ∼6.83∼6.83 for baseline), showing that restricting the space of admissible states prior to token decoding steers the model toward logical paths without harming syntactic structure. 5. Peer Review Assessment & Future Work This empirical validation verifies the internal consistency and theoretical correctness of the paper's claims. However, scaling this framework to production Large Language Models requires addressing three primary engineering areas: Semantic Translation Robustness: Building high-speed, deterministic parsers to map raw tokens to JSON-LD graphs in real-time without introducing new failure modes. Dynamic Knowledge Bases: Compiling massive, real-world ontologies into Alexandrov preorders dynamically as context windows expand. Hardware Accelerators: Developing specialized kernels (e.g., in Triton or CUDA) to execute SHACL checks and SAT solving directly on GPU cores alongside tensor multiplication. 6. Conclusion The implementation of the UKH and MNSVSA verification engine provides the first empirical proof that coupling discrete topological constraints, SHACL shapes, and monadic type checking can completely eliminate stochastically induced hallucinations. Shifting control from post-hoc output filtering to a priori state space restriction establishes a new, verified paradigm for safety in Neuro-Symbolic Artificial Intelligence.

Frank Morales

TOPO-2026 - A Prime-Based Topological Framework for Ultra-Efficient Continual Learning Frank Morales Aguilera, BEng, MEng, SMIEEE Sovereign Machine Laboratory (SOMALA), Montreal, Canada frank.morales@sovereign-machine-lab.ai ORCID: 0009-0003-9528-0745 1. Overview TOPO-2026 is a novel continual learning framework that leverages the mathematical properties of prime numbers to prevent catastrophic forgetting in neural networks. The key innovation is anchoring a sparse set of parameters at prime-numbered indices across tasks, maintaining task-specific knowledge while allowing non-anchored parameters to adapt. 2. Core Contributions # Contribution Description 1 Mathematical Foundation Primes provide optimal spectral coverage (97.85%) with only 6 anchors per layer 2 O(1) Memory Complexity < 5 KB overhead for 100M+ parameter models 3 Universal Applicability Works across NLP, Vision, and 3D architectures without modification 4 Perfect Integrity Zero anchor drift across tasks, eliminating catastrophic forgetting 5 Theoretical Guarantees Mathematical proof of spectral coverage, invariance, and O(1) complexity 6 Edge Deployment Sub-kilobyte memory footprint suitable for resource-constrained devices 3. Theoretical Foundation 3.1 Why Primes Specifically Prime numbers are uniquely suited as anchors because they provide: Property Description Mathematical Guarantee Optimal Density $\pi(n) \sim n/\ln(n)$ Sufficiently dense for coverage of arbitrarily large tensors Coprimality $\gcd(p_i, p_j) = 1$ for $i \neq j$ Orthogonal subspaces, no interference between anchors Deterministic Distribution Well-distributed throughout natural numbers No clustering, comprehensive coverage Universal Guarantee Coverage independent of tensor dimensions Framework works for any architecture 3.2 Spectral Coverage Formula For a set of primes $P = \{p_1, p_2, \ldots, p_k\}$: $$C(P) = 1 - \prod_{p \in P} (1 - p^{-1/2})$$ For $P = \{2, 3, 5, 7, 11, 13\}$: $$\begin{align} C(P) &= 1 - \prod_{p \in P} (1 - p^{-1/2}) \\ &= 1 - (1-2^{-1/2})(1-3^{-1/2})(1-5^{-1/2}) \\ &\qquad \times (1-7^{-1/2})(1-11^{-1/2})(1-13^{-1/2}) \\ &= 1 - (0.2929)(0.4226)(0.5528)(0.6220)(0.6985)(0.7227) \\ &= 1 - 0.021486 \\ &= 0.978514 \approx 97.85\% \end{align}$$ Key Insight: The independence of non-coverage events follows directly from the coprimality of primes. For distinct primes $p_i$ and $p_j$, the conditions $x \not\equiv 0 \pmod{p_i}$ and $x \not\equiv 0 \pmod{p_j}$ are independent because $\gcd(p_i, p_j) = 1$. The Chinese Remainder Theorem guarantees these conditions can be satisfied or violated independently. 4. The Topological Governor The core innovation: three operations that work together to prevent forgetting. 4.1 Snapshot Operation Before training on a new task, save anchor values: $S_t = \{(\text{idx}, \theta_{\text{idx}}) \mid \text{idx} \in P, \theta_{\text{idx}} \in \Theta\}$. 4.2 Gradient Zeroing During backpropagation, zero gradients at anchor positions: $\nabla L(\theta_{\text{idx}}) = 0, \forall \text{idx} \in P$. 4.3 Anchor Enforcement After each optimization step, restore anchor values: $\theta_{\text{idx}} \leftarrow S_t(\text{idx}), \forall \text{idx} \in P$. 5. Memory Complexity Analysis For a model with $n$ parameters and $L$ layers: $$M_{TOPO} = |P| \times L \times \text{bytes per parameter}$$ Model Parameters Layers Anchors Memory EWC Memory Reduction BERT 109M 201 1,206 4.71 KB 437.9 MB 93,000× GPT-2 124M 148 888 3.47 KB 497.8 MB 143,000× GAN 2.95M 22 132 0.52 KB 11.8 MB 22,700× NeRF 246K 14 84 0.33 KB 1.0 MB 3,100× 6. Experimental Validation BERT (Text Classification): 100% retention on movie and product review tasks. GPT-2 (Text Generation): High-quality generation across creative and technical writing tasks with 1.25 perplexity. GAN (Image Generation): Stable training across Gaussian, Uniform, and Mixed datasets; no mode collapse. NeRF (3D Scene Learning): Consistent loss across sphere, cube, and torus scenes. 7. Conclusion TOPO-2026 represents a breakthrough in continual learning, demonstrating that mathematical structure can enable practical, scalable, and ultra-efficient parameter protection. With O(1) memory complexity and universal applicability, it provides a robust foundation for building models that adapt without forgetting, learn without rehearsal, and evolve without memory explosion.