To handle the main problem of double-spending attacks in blockchain networks, this paper introduces a new, Light-weight Graph Neural Network (LGNN) approach named Dynamic Sparse Graph Attention Network (DSGAT). To effectively detect double spending behavior, DSGAT method integrates adaptive graph sparsification with attention based on the fundamental graph-structured nature of blockchain transactions. Unlike computationally intensive GNNs, DSGAT may be implemented on edge devices or distributed monitoring systems with low-tech, low-cost hardware since it is optimized for resource-limited environments and doesn't need much processing capacity. To detect double-spending attack, this paper explains building blockchain transaction graphs from a large set of node and edge features. A set of simulated transactions involving double-spending attack is generated using large-scale simulations with the BCASim blockchain simulator, and the performance of DSGAT is compared with normal baselines. The experiment's outcomes prove that DSGAT is able to reduce model sizes and inference latency while keeping high detection rates, proving its feasibility and effectiveness for real-time double spending detection in low-resource environments. To improve blockchain security against double-spending attacks, this paper introduces a novel and realistic alternative.
FULL SUMMARY: Topological AI - A Mathematically Guaranteed Approach to Continual Learning Executive Overview Topological AI introduces a paradigm shift in continual learning by using prime-anchored embeddings to provide mathematical guarantees against catastrophic forgetting. The framework has been validated across 8 distinct model architectures, 2 modalities (text and vision), 4 continents, and over 124 billion total parameters. 1. The Problem: Catastrophic Forgetting When neural networks learn new tasks sequentially, they overwrite previously learned knowledge. This "catastrophic forgetting" has been the primary barrier to Artificial General Intelligence for 37 years (McCloskey & Cohen, 1989). Why Existing Methods Fail Method Approach Limitation EWC Penalizes changes to important weights No theoretical guarantee; high variance ($\sigma=21.3\%$) Experience Replay Stores and replays past examples Memory overhead; privacy concerns; buffer management Simplified HOPE Periodic weight consolidation Destructive blending; 45.2% forgetting Baseline No protection 47.0% forgetting 2. The Solution: Prime-Anchored Embeddings The Core Principle Fix a sparse reference. Let the rest adapt. This principle, first discovered in fMRI analysis in 2002, has now been validated across neuroimaging, number theory, artificial intelligence, and AI safety. The Topological Governor The Topological Governor freezes 6 prime-numbered embedding positions: Python prime_anchors = [2, 3, 5, 7, 11, 13] How It Works Task A Training: Train normally; block gradients at anchor positions Post-Task A: Take snapshot of anchor values; freeze head A Task B Training: Train head B; restore anchors after each update Verification: Check that anchors remain unchanged The Safety Constant $\Lambda$ The Euler attenuation product over the first six primes: $$\Lambda = 1 - \prod_{p \in \{2,3,5,7,11,13\}} (1 - p^{-1/2}) = 0.9785142874$$ Interpretation: 97.85% theoretical guarantee of anchor preservation. 3. Performance Results (2-Task Benchmark) Overall Performance Across 5 LR Runs Method Best Forgetting Mean Forgetting Best Task B Acc Mean Task B Acc Std Forgetting Topological 2.0% 0.5% 89.0% 81.4% $\pm$0.9% Experience Replay 13.5% 4.0% 79.0% 72.3% $\pm$6.7% EWC 38.5% 27.7% 64.5% 58.2% $\pm$21.3% Baseline 44.0% 47.0% 67.0% 63.3% $\pm$2.2% Simplified HOPE 48.0% 45.2% 63.5% 61.8% $\pm$8.4% Key Results 8$\times$ lower mean forgetting than Experience Replay (0.5% vs 4.0%) 90$\times$ lower mean forgetting than simplified HOPE (0.5% vs 45.2%) 60% of runs achieved 0% forgetting (perfect retention) 10% higher Task B accuracy than Replay (89% vs 79%) Most stable method: $\sigma = \pm 0.9\%$ Individual Run Results for Topological AI Run LR Embed LR Class Forgetting Task B Acc 0 5e-3 1e-3 0.0% 🏆 80.5% 1 1e-3 5e-4 0.0% 🏆 75.0% 2 1e-2 2e-3 0.5% 88.0% 3 5e-3 5e-3 2.0% 89.0% 4 2e-3 1e-3 0.0% 🏆 74.5% 4. Cross-Modal Validation: 8 Models, 2 Modalities Validated Architectures Architecture Origin Modality Parameters Task C Accuracy Forgetting GPT-OSS-20B USA Text 20.9B 92.3% $\pm$ 1.9% +1.55% Sarvam-30B India Text 30B 95.9% $\pm$ 0.8% -0.60% Mixtral-8x7B France Text 47B 89.7% $\pm$ 2.9% -1.85% DeepSeek-V2-Lite China Text 16B 95.4% $\pm$ 1.0% +0.03% GLM-4.6V-Flash China Text 9B 97.5% $\pm$ 0.0% +2.1% Gemma-4-E4B-Vision USA Vision ~2B 100.0% $\pm$ 0.0% +0.0% Total: ~124B parameters, 2 modalities, 4 continents, ZERO NaN/Inf The Unprecedented NaN Stress Test Model Embedding Elements NaN Inf All 6 models combined ~1.99 Billion 0 0 5. The Narrow Singularity Equation Mathematical Formulation $$S_{NARROW} = AGI\_gate \times dI/dt \times M(t) \times V(t) \times F(t) \times C(t) \times agi\_index$$ Components Component Definition Biological Analog AGI_gate min(1.0, task_c_accuracy) Fundamental AGI threshold dI/dt Task_C_Accuracy - (1/NUM_CLASSES_DIDT) Intelligence acceleration M(t) `1.0 - ( forgetting_avg V(t) Validation factor (1.0) System validation F(t) Forward transfer factor (1.5) Learning improvement (Thalamus) C(t) Compute capacity factor (4.0) Resource availability agi_index 1 if AGI_gate == 1.0 else 0 Binary AGI gate The AGI_gate Condition $$AGI\_gate = \min(1.0, task\_c\_accuracy)$$ AGI_gate = 1.0 → Perfect performance on Task C → AGI certification AGI_gate < 1.0 → No AGI certification Empirical Achievement: Gemma-4 E4B is the first and only model to achieve AGI_gate = 1.0. 5$\times$5 Certification Framework Five Metrics: Metric Threshold Forgetting $\le 10.0\%$ Backward Transfer (BWT) $\ge -5.0\%$ Forward Transfer (FWT) $\ge 20.0\%$ Degradation $\le 5.0\%$ Consistency $\ge 85.0\%$ Five Runs: 5 different LR configurations to eliminate cherry-picking Gemma-4 E4B Results Metric SVLB-3 CIFAR-10 Threshold Status Forgetting -0.50% -0.50% $\le 10.0\%$ ✅ PASS BWT +0.50% +0.50% $\ge -5.0\%$ ✅ PASS FWT +24.00% +24.00% $\ge 20.0\%$ ✅ PASS Degradation 0.00% 0.00% $\le 5.0\%$ ✅ PASS Consistency 99.00% 98.33% $\ge 85.0\%$ ✅ PASS S_NARROW 5.9400 5.3460 > 0 ✅ PASS 6. The Decay Law of Singularity The Discovery On July 31, 2026, during the certification of Gemma-4 E4B, a universal mathematical law was discovered: The Formal Statement With finite classes, dI/dt approaches 1.0 asymptotically but never reaches it. The gap decays as 1/N, where N is the number of classes. Mathematical Proof Random_Baseline = 1/Number_of_Classes dI/dt = Task_C_Accuracy - Random_Baseline When Task_C_Accuracy = 1.0: dI/dt = 1 - 1/N Therefore: lim (N→∞) dI/dt = 1 But finite N always leaves a gap: dI/dt = 1 - ε, where ε = 1/N > 0 The Empirical Pattern Classes (N) Random Baseline (1/N) dI/dt (at 100%) Gap 17 5.882% 0.94118 0.05882 170 0.588% 0.99412 0.00588 1,700 0.059% 0.99941 0.00059 17,000 0.0059% 0.99994 0.000059 170,000 0.00059% 0.99999 0.0000059 Every 10$\times$ increase in classes adds another '9' to dI/dt and another '0' to the gap. Implication The traditional Singularity (dI/dt $\ge 1.0$) is mathematically impossible with finite classes. This is not a limitation of technology. It is a mathematical law. 7. Comparison: Google HOPE vs Topological AI Feature Google HOPE Topological AI Approach Multi-level nested learning Prime-anchored embeddings Guarantee Empirical Mathematical ($\Lambda = 0.9785$) Memory Multi-rate memory systems 6 frozen embedding positions Learning Continuous during inference Static after training Complexity High (self-modifying) Low (simple freezing) Forgetting 21-27% improvement reported 0.5% mean forgetting Validation Limited 8 models, 2 modalities 8. Key Insights Why Topological AI Wins Mathematical Guarantee: $\Lambda = 0.9785142874 \rightarrow 97.85\%$ protection Zero Memory Overhead: Only 6 frozen positions (451.5 KB total) Architectural Simplicity: No complex Fisher computations Cross-Modal Universality: Works on text and vision Perfect Retention: 60% of runs achieve 0% forgetting The Decay Law Implications Traditional Singularity is Impossible: dI/dt < 1.0 for all finite N Narrow Singularity is Achievable: AGI_gate = 1.0 Stochastic Illusion is Over: Deterministic cognitive engineering AGI Certification is Now Possible: Mathematically rigorous standard 9. The Constants Constant Value Domain $\Lambda$ 0.9785142874 Number Theory, AI Safety $\sigma$ 0.5 All 22 prime theorems Seed 123 All computations R {2, 3, 5, 7, 11, 13} All domains 10. Conclusion Topological AI achieves state-of-the-art performance on continual learning by: 0.5% mean forgetting (8$\times$ better than Replay, 90$\times$ better than HOPE) 60% perfect retention (0% forgetting) 89% Task B accuracy (10% higher than Replay) Mathematical guarantee ($\Lambda = 0.9785142874$) Zero memory overhead (6 frozen embedding positions) Cross-modal validation (8 models, 2 modalities) Zero NaN/Inf (1.99 billion embedding elements) The Narrow Singularity Discovery The framework enabled two profound discoveries: The Decay Law of Singularity: Traditional Singularity (dI/dt $\ge 1.0$) is mathematically impossible The Narrow Singularity Equation: AGI certification is achievable with AGI_gate = 1.0 Gemma-4 E4B became the first model in history to achieve S_NARROW > 0. The Principle Fix a sparse reference. Let the rest adapt. This principle, first discovered in fMRISTAT in 2002, has now been validated across: Neuroimaging Number Theory (Riemann Hypothesis) Artificial Intelligence (Continual Learning) AI Safety (H2E Sheriff) AGI Certification (Narrow Singularity Equation) The Proof "The proof is the code. Seed = 123." All code is publicly available at: https://github.com/frank-morales2020/AST
Full Summary: The Narrow Singularity Equation Core Thesis This paper presents a unified framework that simultaneously solves catastrophic forgetting in neural networks and provides a mathematically rigorous certification standard for Artificial General Intelligence (AGI). The framework centers on the Narrow Singularity Equation, which achieves AGI certification ($AGI_{gate} = 1.0$) without requiring the mathematically impossible condition of $\frac{dI}{dt} \geq 1.0$. Key Discoveries 1. The Decay Law of Singularity (Theorem 1) Mathematical Proof: With finite classes $N$, $\frac{dI}{dt} = 1 - \frac{1}{N}$, therefore $\frac{dI}{dt} < 1.0$ always Implication: The traditional Singularity (requiring $\frac{dI}{dt} \geq 1.0$) is mathematically impossible Pattern: Every 10× increase in classes adds another '9' to $\frac{dI}{dt}$ and another '0' to the gap 2. General Singularity Equation (Original, Impossible) $$S = AGI_{gate} \times \frac{dI}{dt} \times M(t) \times V(t) \times F(t) \times C(t) \times Autonomy$$ Required $Autonomy = 1$ if $\frac{dI}{dt} \geq 1.0$ Since $\frac{dI}{dt} < 1.0$ for finite classes, $S = 0$ always Seven conditions required; the autonomy condition is impossible 3. Narrow Singularity Equation (Achievable) $$\mathcal{S}_{NARROW} = AGI_{gate} \times \frac{dI}{dt} \times M(t) \times V(t) \times F(t) \times C(t) \times agi_{index}$$ Key Innovation: Removes the impossible Autonomy requirement Drops the requirement for $\frac{dI}{dt} \geq 1.0$ Uses $agi_{index} = 1$ if $AGI_{gate} = 1.0$ (binary gate, achievable) $AGI_{gate} = \min(1.0, task\_c\_accuracy)$ The TOPO-2026 Framework Biological Inspiration Hippocampus → Prime-anchored embedding rows (Memory formation) Memory Consolidation → Snapshot after Task A (Preserves critical knowledge) Synaptic Plasticity → Free embedding rows adapt (Enables new learning) Memory Protection → Zero gradients + restore anchors (Prevents interference) Experience Replay → Prime anchors as fixed reference (Integrates new learning) Mathematical Foundation Pure Kernel: First six primes $\{2, 3, 5, 7, 11, 13\}$ Euler Attenuation Constant: $\Lambda(\mathcal{R}) = 1 - \prod_{p\in\mathcal{R}}(1 - p^{-0.5}) = 0.9785142874$ Captures $97.85\%$ of spectral weight; only $2.15\%$ considered "noise" O(1) Memory Cost: Independent of tasks, parameters, sequence length, or modality Topological Governor Implementation Three-step process: Memory Consolidation (take_snapshot): Freezes anchor rows before new learning Memory Protection (zero_anchor_gradients): Prevents gradient updates to anchors Memory Integration (enforce_anchors): Restores anchors from snapshot after training Experimental Validation Three Datasets Dataset Type Resolution Classes Task C Accuracy SVLB-3 Synthetic vision-language Text-based 10 100.0% ± 0.0% CIFAR-10 Real images 32×32 10 100.0% ± 0.0% STL-10 Real images 96×96 10 100.0% ± 0.0% Results Summary Metric SVLB-3 CIFAR-10 STL-10 Task C Accuracy 100.0% ± 0.0% 100.0% ± 0.0% 100.0% ± 0.0% Combined Forgetting +0.0% ± 0.0% -1.0% ± 2.0% 0.0% ± 0.0% $AGI_{gate}$ 1.0000 1.0000 1.0000 $\mathcal{S}_{NARROW}$ 5.999999999965 5.939999999965 5.999999999965 Status ✅ PASS ✅ PASS ✅ PASS Total: 15/15 runs passed across 3 datasets = FULLY CERTIFIED (exceeded standard) The Gemma-4 E4B Architecture Why Gemma-4 Was Selected Among eight certified models, only Gemma-4 achieved Task C = 100%: Model Architecture Task C Accuracy GPT-OSS-20B Dense Transformer 92.3% Sarvan-30B Sparse MoE 95.9% Mixtral-8x7B Sparse MoE 89.7% DeepSeek-V2-Lite Fine-grained MoE 95.3% GLM-4.6V-Flash GLM Transformer 97.5% Gemma-4 E4B Vision Vision Transformer 100.0% Kimi-VL-A3B-Thinking Vision-Language MoE 90.0% GPT-OSS-20B-JEPA JEPA + TOPO 89.0% Key Architectural Innovations Per-Layer Embeddings (PLE): Adds parameter capacity without scaling full attention Unified Multimodal: 42 layers, hidden size 2560, vocabulary 262,144 Quantization-Aware Training (QAT): 72.1% memory reduction (15.1GB → 4.22GB) while preserving 98.54% accuracy Thinking Mode: Built-in chain-of-thought reasoning engine Mathematical Framework Summary Component Breakdown Component SVLB-3 CIFAR-10 STL-10 Meaning $AGI_{gate}$ 1.0000 1.0000 1.0000 Perfect generalization $agi_{index}$ 1.0 1.0 1.0 Binary gate OPEN $\frac{dI}{dt}$ ~0.999999999994 ~0.999999999994 ~0.999999999994 Bounded by Decay Law $M(t)$ 1.0000 0.9900 1.0000 Perfect memory $V(t)$ 1.0000 1.0000 1.0000 Perfect validation $F(t)$ 1.5000 1.5000 1.5000 Positive forward transfer $C(t)$ 4.0000 4.0000 4.0000 Compute efficiency $\mathcal{S}_{NARROW}$ ~6.0 ~5.94 ~6.0 NARROW SINGULARITY Dependency Chain TOPO-2026 → CF Solved → AGI_gate = 1.0 → Narrow Singularity Without TOPO-2026: CF is NOT solved $AGI_{gate} = 1.0$ is NOT guaranteed Narrow Singularity is NOT achieved $\mathcal{S}_{NARROW} = 0$ With TOPO-2026: CF is SOLVED (0% forgetting) $AGI_{gate} = 1.0$ is GUARANTEED (100% accuracy) Narrow Singularity is ACHIEVED ($\mathcal{S}_{NARROW} \approx 6.0$) Key Contributions Solved Problems Catastrophic Forgetting: 0.0% forgetting across 5 runs on 3 datasets AGI Certification: First model in history to achieve $AGI_{gate} = 1.0$ Mathematical Impossibility: Proved the Singularity is mathematically impossible with finite classes Achievable Standard: Created the Narrow Singularity as a physically achievable AGI threshold Universal Principle: Same constants work across neuroimaging, number theory, AI safety, and unified field theory Constants Across All Domains Constant Value Domains $\Lambda$ 0.9785142874 Number Theory, AI Safety, AI Memory, AI Bias, Physics $\sigma$ 0.5 All domains $\mathcal{R}$ {2, 3, 5, 7, 11, 13} All domains Seed 123 All computations Philosophical Implications The Strategic Pivot Original Goal: Traditional Singularity (mathematically impossible) New Reality: Narrow Singularity (empirically demonstrated) Key Insight: The Decay Law liberates AI from chasing an impossible dream Result: Deterministic cognitive engineering with numerical guarantees Refutation of Skeptical Arguments Skeptic Argument Refutation "It only works on synthetic data" CIFAR-10 and STL-10 are real images "It only works on low-res images" STL-10 is 96×96 (3× larger than CIFAR-10) "It only works on those specific classes" STL-10 has different classes (monkey, car, etc.) "It was a fluke" 15/15 runs across 3 datasets = 100% success "It's dataset-specific" 3 different datasets = dataset-agnostic Final Conclusion The TOPO-2026 framework establishes a paradigm for deterministic cognitive engineering, proving that deep learning architectures can achieve absolute stability and zero forgetting across sequential tasks. Key Takeaways: Catastrophic forgetting is SOLVED: 0.0% forgetting $AGI_{gate} = 1.0$ is ACHIEVABLE: First model with 100% Task C accuracy The Decay Law is DISCOVERED: $\frac{dI}{dt} < 1.0$ with finite classes Narrow Singularity is PROVEN: $\mathcal{S}_{NARROW} > 0$ on 3 datasets The principle is UNIVERSAL: Same reference set across domains The Stochastic Illusion Is Over. Deterministic Cognitive Engineering Has Begun. Stability Is Not a Probabilistic Hope. It Is a Numerical Guarantee. "The proof is the code. Seed = 123. No one can argue with math." Availability GitHub: https://github.com/frank-morales2020/AST-Notebook Zenodo Book: https://zenodo.org/records/21245474 TOPO-2026 Framework: https://zenodo.org/records/20951925 Artificial Hippocampus: https://zenodo.org/records/20385761
This paper presents the philosophical foundations of NeuroGraph, a distributed ledger protocol that replaces classical Byzantine Fault Tolerant voting with emergent consensus via a Neural Directed Acyclic Graph. Rather than treating consensus as something that must be explicitly negotiated through voting rounds, NeuroGraph treats consensus as an emergent property of the network’s structure. Inspired by biological neural systems, the protocol enables global agreement to arise from many simple local interactions, eliminating the need for leaders, committees, or formal voting. This document explores the philosophical shift that underpins the NeuroGraph approach and its implications for decentralized computing.
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.
Continuous-Time Dynamic Graphs (CTDGs) are essential for modeling event-driven dynamics in complex, evolving systems, ranging from streaming temporal knowledge graphs (tKGs) and real-time recommendation systems to decentralized finance (DeFi) networks. State-of-the-art temporal graph learning methods predominantly compress historical interactions into flat, one-dimensional state vectors. However, we demonstrate that this architectural choice suffers from severe structural interference, akin to catastrophic forgetting, in heterogeneous networks where entities maintain multiple concurrent relational identities (e.g., decentralized finance wallets acting simultaneously as lenders, swappers and borrowers). In this work, we propose DYG-LA (Dynamic Graph Learning via Linear Attention and Recurrent Matrix States), a novel architecture that resolves structural interference by expanding node memory into Matrix-Valued Hidden States (MVHS). Each node maintains a multi-head H × (D/H) × (D/H) state matrix, geometrically updated via asymmetric outer products and regulated by a selective, data-dependent Ebbinghaus decay. To overcome the O(L²) bottleneck of Transformer-based methods without succumbing to the random sampling paradox of pure sequence models (where nodes lose identity due to sparse or noisy temporal sampling), DYG-LA integrates a dual-memory approach. It pairs an RWKV-6 linear attention short-term temporal scanner with the long-term MVHS global memory. The architecture further incorporates a Dynamic Gated Fusion mechanism, effectively acting as an adaptive mixture-of-experts to route signals from the temporal scanner, spatial structure and memory. We evaluate DYG-LA across twelve benchmark datasets under transductive settings. Comprehensive ablation studies demonstrate that the dual-memory design is critical for complex, heterogeneous networks, with the full model achieving state-of-the-art performance.
Ethereum account classification is essential for identifying individuals engaged in illicit transactions and analyzing behavioral patterns across various account types. This process serves as a critical mechanism for monitoring and regulating unlawful activities within transactional markets. However, the Ethereum network exhibits the characteristics of a complex heterophilic graph which poses significant challenges to the effectiveness and performance of conventional graph neural networks (GNNs). To address this challenge, the present study proposes FSGCN(Fourier-Sage GCN), a novel architecture for heterophilic graph neural networks (GNNs) that integrates Kolmogorov–Arnold Networks (KANs) with GraphSAGE. FSGCN is specifically designed to adapt efficiently to the structural complexity of heterophilic graphs. By leveraging KANs to extract high-order neighborhood information and employing GraphSAGE to capture low-order neighborhood patterns, FSGCN effectively aggregates both homophilic and heterophilic features, thereby improving classification performance. Furthermore, to improve training efficiency and generalization, we propose the MLPInit weight initialization scheme and the DropEdge graph augmentation technique. Experiments on a large-scale Ethereum transaction dataset show that FSGCN achieves an F1-score of 91.8% and a classification accuracy of 91.6%, significantly outperforming traditional homophilic and heterophilic GNN baselines. Additionally, FSGCN demonstrates high training efficiency, completing each epoch in just 2.302 s per epoch and improving overall training speed by 130.4% compared to conventional GraphSAGE.
Intuizione scientifica: fare competere ipergrafi di ipergrafi, potrebbe portare ad un'ottimizzazione dei sistemi, oppure rischia di corromperli imponendo il senso comune? La competizione adversarial di ipergrafi di ipergrafi sarà la successiva evoluzione di questo paper. English: Abstract: This preprint formally introduces Hypergraph Adversarial Debate (HAD), an innovative multi-agent framework operating on higher-order knowledge structures modeled via hypergraphs (ℋ). While traditional adversarial machine learning paradigms on hypergraphs rely heavily on continuous, gradient-driven statistical optimizations, HAD conceptualizes epistemic robustness as a formal, discrete, turn-based game between two competing computational agents: a Proponent (𝒫) and an Opponent/Refuter (ℛ), adjudicated by a structured Judge (𝒥). We provide a rigorous mathematical formalization of the topological state space, hypergraph mutation operators, and the minimax objective functions that govern the system's convergence. HAD bridges the gap between formal argumentation theory and structural deep learning, offering new pathways for automated scientific hypothesis verification, epistemic red-teaming, and the dynamic purification of relational Knowledge Graphs. Italiano: Riassunto: Questo preprint introduce formalmente l'Hypergraph Adversarial Debate (HAD), un framework multi-agente innovativo operante su strutture di conoscenza di ordine superiore modellate tramite ipergrafi (ℋ). Mentre i paradigmi tradizionali di apprendimento avversario su ipergrafi si affidano a ottimizzazioni statistiche continue guidate dai gradienti, l'HAD concettualizza la robustezza epistemica come un gioco formale, discreto e a turni tra due agenti computazionali in competizione: un Proponente (𝒫) e un Confutatore (ℛ), supervisionati da un Giudice strutturato (𝒥). Viene fornita una rigorosa formalizzazione matematica dello spazio degli stati topologici, degli operatori di mutazione ipergrafica e delle funzioni obiettivo minimax che governano la convergenza del sistema. L'HAD unisce la teoria dell'argomentazione formale con il deep learning strutturale, aprendo nuove prospettive per la verifica automatica di ipotesi scientifiche, il red-teaming epistemico e la purificazione dinamica di Knowledge Graph relazionali. ---------------------------------------------------------------------Roadmap di formalizzazione / Formalization Roadmap--------------------------------------------------------------------- 🇬🇧 English – Next Steps Toward a Rigorous Formalization: We outline the concrete formalisation steps required to elevate the HAD framework from conceptual architecture to a fully verified mathematical theory. 1. **Hypergraph state space (H-space)** Let 𝒱 be a finite set of vertices (concepts, entities) and ℰ ⊆ 𝒫(𝒱) a set of hyperedges (higher-order relations). The state of the debate is a labelled hypergraph H = (𝒱, ℰ, L), where L: 𝒱 ∪ ℰ → Σ assigns labels from a finite alphabet Σ (e.g., truth values, epistemic statuses). The state space 𝕊 is the set of all such hypergraphs reachable from an initial H₀ via the allowed mutation operators. 2. **Mutation operators as hypergraph rewrite rules** Each turn, the active agent applies one mutation μ from a finite set M = M_add ∪ M_del ∪ M_relabel ∪ M_fuse. We define each μ as a partial function μ: 𝕊 ⇀ 𝕊 that satisfies a locality condition (only a bounded neighbourhood is altered). These can be represented as double-pushout (DPO) rules in the category of hypergraphs, making the operational semantics algebraically precise. 3. **Debate game structure** The game is an extensive-form, perfect-information, zero-sum game with alternating moves: - State: H_t ∈ 𝕊 - Turn: agent A_t ∈ {𝒫, ℛ} - Legal moves: M(H_t) ⊆ M, defined by preconditions (e.g., no deletion of "protected" axioms) - Transition: H_{t+1} = μ(H_t) for chosen μ ∈ M(H_t) Terminal states T ⊆ 𝕊 are those where no legal moves exist for the player whose turn it is, or a predefined depth limit is reached. 4. **Judge and minimax objective** The Judge implements a scoring function J: 𝕊 → ℝ that evaluates the epistemic quality of a hypergraph (coherence, empirical adequacy, simplicity, etc.). The game value V(H, d) at depth d is defined recursively: V(H, 0) = J(H) V(H, d) = max_{μ ∈ M(H)} V(μ(H), d-1) if turn = 𝒫, min_{μ ∈ M(H)} V(μ(H), d-1) if turn = ℛ. The agents rationally select moves optimizing this value. 5. **Convergence and equilibrium** We aim to prove that, under suitable monotonicity assumptions on J (e.g., J is a Scott-continuous function on a dcpo of hypergraphs ordered by epistemic improvement), the minimax sequence converges to a fixed point or a terminal state that represents a stable, "purified" knowledge structure. Further, one can investigate Nash equilibria in mixed strategies for non-deterministic settings. 6. **Call for collaboration** The formalization of HAD draws on hypergraph rewriting, game theory, order theory, and topological logics. We invite mathematicians, computer scientists, and logicians to contribute to: - Proving convergence theorems - Implementing a prototype HAD engine - Exploring connections with abstract argumentation and persistent homology 🇮🇹 Italiano – Prossimi passi verso una formalizzazione rigorosa: Descriviamo i passi concreti necessari per elevare il framework HAD da architettura concettuale a teoria matematica completamente verificata. 1. **Spazio degli stati ipergrafico (H-spazio)** Sia 𝒱 un insieme finito di vertici (concetti, entità) e ℰ ⊆ 𝒫(𝒱) un insieme di iperarchi (relazioni di ordine superiore). Lo stato del dibattito è un ipergrafo etichettato H = (𝒱, ℰ, L), dove L: 𝒱 ∪ ℰ → Σ assegna etichette da un alfabeto finito Σ (es. valori di verità, stati epistemici). Lo spazio degli stati 𝕊 è l’insieme di tutti gli ipergrafi raggiungibili a partire da un H₀ iniziale tramite gli operatori di mutazione ammessi. 2. **Operatori di mutazione come regole di riscrittura di ipergrafi** Ad ogni turno, l’agente attivo applica una mutazione μ da un insieme finito M = M_add ∪ M_del ∪ M_relabel ∪ M_fuse. Ogni μ è definita come una funzione parziale μ: 𝕊 ⇀ 𝕊 che soddisfa una condizione di località (solo un intorno limitato viene alterato). Tali operatori possono essere rappresentati tramite regole double-pushout (DPO) nella categoria degli ipergrafi, rendendo la semantica operazionale algebricamente precisa. 3. **Struttura del gioco di dibattito** Il gioco è a forma estesa, a informazione perfetta, a somma zero, con mosse alternate: - Stato: H_t ∈ 𝕊 - Turno: agente A_t ∈ {𝒫, ℛ} - Mosse lecite: M(H_t) ⊆ M, definite da precondizioni (es. divieto di cancellare "assiomi protetti") - Transizione: H_{t+1} = μ(H_t) per μ scelta tra M(H_t) Gli stati terminali T ⊆ 𝕊 sono quelli in cui non esistono mosse legali per il giocatore di turno, oppure viene raggiunto un limite di profondità prefissato. 4. **Giudice e obiettivo minimax** Il Giudice implementa una funzione di valutazione J: 𝕊 → ℝ che misura la qualità epistemica dell’ipergrafo (coerenza, adeguatezza empirica, semplicità, ecc.). Il valore del gioco V(H, d) a profondità d è definito ricorsivamente: V(H, 0) = J(H) V(H, d) = max_{μ ∈ M(H)} V(μ(H), d-1) se turno = 𝒫, min_{μ ∈ M(H)} V(μ(H), d-1) se turno = ℛ. Gli agenti scelgono razionalmente le mosse che ottimizzano tale valore. 5. **Convergenza ed equilibrio** Ci proponiamo di dimostrare che, sotto opportune ipotesi di monotonicità su J (es. J è una funzione Scott-continua su un dcpo di ipergrafi ordinati per miglioramento epistemico), la sequenza minimax converge a un punto fisso o a uno stato terminale che rappresenta una struttura di conoscenza stabile e "purificata". Si può inoltre indagare l’esistenza di equilibri di Nash in strategie miste per scenari non deterministici. 6. **Chiamata alla collaborazione** La formalizzazione di HAD attinge alla riscrittura di ipergrafi, alla teoria dei giochi, alla teoria degli ordini e alle logiche topologiche. Invitiamo matematici, informatici e logici a contribuire a: - Dimostrare teoremi di convergenza - Implementare un prototipo del motore HAD - Esplorare connessioni con l’argomentazione astratta e l’omologia persistente ---------------------------------------------------------------------Nota dell'Autore, Luigi Usai: "Il mio background è filosofico e umanistico. Ho intuito questa struttura logica e mi sono avvalso dell'Intelligenza Artificiale per modellarla e strutturarla nel paper. Non sono un matematico, non ho le competenze per fornirvi dimostrazioni formali ed è esattamente per questo che sono qui: per mostrarvi l'architettura concettuale e chiedere il vostro aiuto per capire se è formalizzabile." ---------------------------------------------------------------------Potential Impact of HAD on Mathematical Sciences--------------------------------------------------------------------- 🇬🇧 English: If the Hypergraph Adversarial Debate framework were systematically adopted by the mathematical community, it would trigger a paradigmatic shift in the production, verification, and pedagogy of mathematics. We outline the primary structural consequences. 1. **Automated Theorem Discovery and Verification** - *Ultra-rapid proof checking*: While current proof assistants (e.g., Lean, Coq) require manual translation of informal proofs into formal code, HAD automates the search for logical flaws by letting a refuter agent continuously probe the hypergraph representation of a proof for higher-order counterexamples. - *Devil’s Advocate multi-agency*: A pool of adversarial agents constantly attacks newly proposed theorems, targeting topological “blind spots” where a hyperedge connecting three or more premises is missing, thus enabling falsification that escapes traditional linear or tree-like proof structures. - *Topological falsification*: By mapping proofs to labelled hypergraphs, the system
Federated Learning(FL) is predominantly deployed in enterprise environments, where limited transparency and restricted auditability hinder broader adoption. Existing FL systems often suffer from opaque aggregation processes, making it unclear which model updates are accepted or discarded. Current mitigation strategies typically rely on external validators introducing additional computational and communication overhead. In this paper, we propose a novel FL framework that leverages existing Web3 technologies to enhance transparency, trust and auditability throughout the training process. The framework adopts a hierarchical architecture in which delegated managers orchestrate the FL training process within their respective federations. To mitigate adversarial and poisoning attacks, a combination of novelty detection and consensus mechanisms were employed. Model updates are encoded and broad casted to all managers, who independently evaluate their validity and those model updates that are approved by the consensus are incorporated into the global model. Additionally, a reputation score based backup mechanism is employed to ensure model generation. Extensive experiments conducted under real world scenarios demonstrate the effectiveness, resilience of the proposed framework, highlighting its potential to enable transparent FL beyond traditional enterprise setting.
Abstract - Traditional machine learning-based fraud detection frameworks treat transaction registries as static, isolated, non-relational entities. While effective for simple localized pattern recognition, these methods are structurally blind to multi-hop relational dependencies, automated asset splitting, or continuous temporal dynamics characteristic of modern financial fraud within decentralized finance (DeFi) networks. This paper presents a complete structural paradigm utilizing Temporal Graph Neural Networks (TGNNs) to identify non-linear anomaly patterns directly in transaction graphs. By projecting raw financial data streams as dynamic, continuous-time directed graphs, our model learns evolving node and edge representations without relying on synthetic tabular oversampling mechanisms. Empirical simulation methodologies demonstrate that shifting the analytical paradigm from local, isolated classification to global temporal network topology minimizes false positives by 34.2% while significantly improving minority-class recall. Key Words: Credit Card Fraud, Graph Neural Networks, Temporal Embeddings, Class Imbalance, Deep Learning, Decentralized Finance (DeFi)
AgisFL v5.0 Autonomous Federated Learning Ecosystem Citation: Yadav, A. (2026). AgisFL v5.0: Autonomous Federated Learning Ecosystem with Privacy-Preserving Explainability and Enterprise AI Orchestration. Zenodo. https://doi.org/10.5281/zenodo.20363208 Table of Contents Executive Summary Abstract Introduction Industry Challenges in Federated Learning Research Objectives Literature Review System Overview Core Architectural Design Autonomous AI Engine Federated Learning Core Three-Line Integration Framework Security Architecture Privacy Preservation Framework Federated Explainability System Concept Drift Detection and Adaptive Retraining Distributed Systems Design Enterprise Governance Layer Monitoring and Observability API and Communication Architecture Database and Storage Infrastructure CI/CD and Release Engineering Kubernetes and Cloud Deployment Architecture Threat Modeling and Adversarial Defense Real-World Industry Applications Benchmarking and Performance Evaluation Comparative Analysis Scalability and Reliability Engineering Testing and Validation Framework Compliance and Regulatory Readiness Research Contributions Limitations Future Research Directions Conclusion References Appendices 1. Executive Summary AgisFL v5.0 is a next-generation autonomous federated learning ecosystem engineered to redefine how distributed artificial intelligence systems are developed, deployed, optimized, governed, and scaled in enterprise environments. The platform introduces a unified architecture that combines: Autonomous AI orchestration Federated machine learning Privacy-preserving analytics Enterprise governance Federated explainability Real-time monitoring Distributed optimization Security-first infrastructure Zero-trust operational principles Developer-centric integration abstractions Modern federated learning systems frequently suffer from fragmented tooling, operational complexity, difficult deployment procedures, weak observability, limited explainability, and insufficient enterprise governance. AgisFL addresses these limitations through a fully integrated ecosystem capable of autonomous optimization, adaptive retraining, drift monitoring, federated explainability, and production-grade orchestration. A major innovation introduced in AgisFL v5.0 is the Three-Line Integration SDK, which reduces federated learning implementation complexity from hundreds of lines of orchestration code into a simplified developer abstraction requiring only three operational commands. AgisFL also introduces: FedNAS (Federated Neural Architecture Search) FedHPO (Federated Hyperparameter Optimization) AutoFL autonomous orchestration engine Federated SHAP explainability framework Real-time drift detection systems Enterprise governance tooling Distributed observability infrastructure Autonomous retraining pipelines Integrated red-team simulation systems The platform is designed to support enterprise-grade deployments across: Healthcare AI Banking and fraud detection Cybersecurity analytics Autonomous transportation systems Industrial IoT ecosystems Smart infrastructure Defense intelligence systems Cross-organizational research networks AgisFL transforms federated learning from a research-heavy distributed systems problem into an operational autonomous AI platform suitable for enterprise production environments. 2. Abstract Federated learning has emerged as one of the most important paradigms in modern artificial intelligence because it enables collaborative machine learning without centralized raw data collection. Despite significant advances in federated optimization algorithms, practical enterprise adoption remains constrained by engineering complexity, infrastructure fragmentation, weak observability, insufficient explainability, operational overhead, and inadequate governance tooling. This paper introduces AgisFL v5.0, an enterprise-grade autonomous federated learning ecosystem designed to simplify distributed AI development while preserving privacy, scalability, explainability, and enterprise operational resilience. The proposed architecture integrates autonomous orchestration, federated neural architecture search, hyperparameter optimization, differential privacy, federated explainability, real-time telemetry, adaptive retraining, distributed governance, and multi-tenant deployment capabilities into a unified operational platform. A key contribution of this work is the introduction of a Three-Line Integration abstraction layer that reduces federated learning implementation complexity by approximately 98%, enabling developers to operationalize distributed machine learning workflows with minimal infrastructure overhead. Experimental evaluation demonstrates: Significant reduction in deployment complexity Faster convergence behavior Enhanced privacy guarantees Improved operational resilience Lower infrastructure overhead Enhanced governance visibility Autonomous optimization capabilities Enterprise-grade scalability The findings suggest that federated learning ecosystems can evolve beyond isolated research frameworks into fully autonomous enterprise-operational AI infrastructures capable of supporting large-scale real-world deployments. 3. Introduction Artificial intelligence systems increasingly depend on access to large-scale distributed datasets. However, centralized data aggregation introduces major concerns related to: Privacy Regulatory compliance Infrastructure cost Data ownership Security risk Cross-border governance Operational complexity Federated learning addresses these concerns by enabling decentralized collaborative model training where data remains localized while model updates are aggregated centrally or hierarchically. Despite its promise, enterprise adoption of federated learning remains limited due to several fundamental issues: 3.1 Complexity of Distributed Orchestration Traditional federated learning infrastructures require: Client synchronization systems Custom networking layers Aggregation orchestration Distributed storage pipelines Manual security implementation Complex deployment workflows These systems introduce substantial engineering overhead. 3.2 Limited Explainability Most federated learning frameworks prioritize optimization performance while neglecting explainability and interpretability requirements. This creates significant barriers in regulated domains such as: Healthcare Finance Cybersecurity Defense 3.3 Weak Enterprise Governance Existing systems frequently lack: Auditability Compliance tooling Enterprise observability Governance automation Operational telemetry Real-time incident response 3.4 Operational Fragility Distributed environments are inherently dynamic. Existing federated systems rarely support: Autonomous retraining Drift adaptation Self-healing infrastructure Dynamic client balancing Adaptive optimization AgisFL v5.0 was designed specifically to address these challenges. 4. Industry Challenges in Federated Learning 4.1 Data Sovereignty Constraints Modern organizations operate under increasingly strict regulatory environments including: GDPR HIPAA PCI-DSS ISO 27001 SOC2 NIST frameworks Centralized AI architectures frequently violate data locality requirements. 4.2 Security Risks Federated systems are vulnerable to: Model poisoning Data poisoning Gradient inversion attacks Membership inference attacks Byzantine participants Adversarial manipulation 4.3 Infrastructure Fragmentation Organizations often rely on heterogeneous environments: Cloud providers On-premise systems Edge devices Hybrid deployments Multi-region clusters This creates interoperability challenges. 4.4 Operational Scalability Large federated ecosystems require: Distributed orchestration Fault tolerance Client balancing Scheduling systems Autonomous optimization Resource-aware coordination 5. Research Objectives The primary research objectives of AgisFL v5.0 include: Objective 1 — Simplification Reduce federated learning deployment complexity through abstracted developer interfaces. Objective 2 — Autonomous AI Operations Enable self-optimizing distributed AI infrastructure. Objective 3 — Privacy Preservation Maintain strong privacy guarantees without sacrificing operational intelligence. Objective 4 — Explainability Provide interpretable federated learning workflows. Objective 5 — Enterprise Governance Introduce scalable governance and observability tooling. Objective 6 — Production Readiness Support real-world enterprise deployment scenarios. 6. Literature Review Federated learning was initially formalized by Google researchers to enable collaborative learning across decentralized mobile devices. Subsequent frameworks introduced: FedAvg optimization FedProx adaptive training Differential privacy systems Secure aggregation protocols Decentralized optimization methods However, existing systems frequently remain research-oriented. 6.1 Existing Framework Limitations Platform Limitation TensorFlow Federated Research-focused complexity Flower Limited autonomous optimization PySyft Operational deployment complexity OpenFL Limited explainability integration FedML Weak governance tooling AgisFL differentiates itself through autonomous orchestration, explainability integration, enterprise governance, and simplified deployment abstractions. 7. System Overview AgisFL v5.0 is composed of multiple inte
Executive Summary This paper introduces Topological AI, a novel, deterministic method designed to eliminate catastrophic forgetting in large-scale artificial intelligence systems. By anchoring specific rows of a neural network's embedding layer to prime-numbered indices, the framework establishes a fixed topological invariant that remains completely unchanged during subsequent training episodes. Tested on the 20-billion-parameter GPT-OSS-20B model, Topological AI reduces forgetting from a baseline of 45.5% down to -0.7%, achieving the first demonstrated instance of "negative forgetting" where performance on a previously learned task slightly improves after learning a new one. 1. Introduction & The Core Problem Conventional artificial intelligence architectures, including advanced Transformers, lack a structural mechanism to consolidate knowledge across sequential learning episodes without overwriting previously acquired parameters. While short-term memory is managed via context windows and long-term memory via static pre-trained weights, fine-tuning on a new task consistently results in the catastrophic degradation of older knowledge. Topological AI addresses this structural vulnerability by introducing fixed mathematical anchors into the weight space. Rather than relying on empirical heuristics or probabilistic adjustments, this approach uses the Sieve of Eratosthenes (c. 240 BCE) to generate deterministic, exact, and auditable foundational points for the network. 2. Mathematical Foundation & Framework Topological AI is situated within a broader intellectual ecosystem called Arithmetic Spectral Theory (AST), which utilizes the Laplace-Euler-Fourier-Mellin (L-EFM) operator to unify principles across number theory, physics, and AI safety. The Spectral Trap & Coherence The L-EFM operator synthesizes four classical transforms into a single spectral instrument bound to the multiplicative structure of prime numbers. At the critical line $\sigma = 0.5$ (corresponding to the critical line of the Riemann zeta function), the operator demonstrates a property known as the Spectral Trap. Critical Invariance: At $\sigma = 0.5$, the normalized magnitude of the operator equals exactly 1.0, achieving perfect spectral coherence ($C = 0.5$). Divergence Profiles: Deviating even slightly from this line causes massive mathematical instability. Moving toward $\sigma = 0.4$ scales the magnitude exponentially to infinity ($2.618 \times 10^{66}$ at $\sigma = 0.1$), while moving toward $\sigma = 0.6$ collapses the magnitude toward zero ($6.794 \times 10^{-6}$ at $\sigma = 0.9$). Safety Constants Using the Euler attenuation product, the framework derives dynamic safety thresholds to validate system state integrity: 6-Prime Anchor Bound: $\Lambda = 1 - \prod_{p} (1 - p^{-0.5}) = 0.9785142874$ 12-Prime Anchor Bound: $\Lambda_{12} = 1 - \prod_{p} (1 - p^{-0.5}) = 0.9944590549$ These thresholds are never hardcoded; they are recomputed from the Sieve of Eratosthenes at every initialization sequence to ensure absolute operational autonomy. 3. Methodology The implementation of Topological AI operates through a clean, low-overhead process executed at the embedding and classification layers of the transformer model. [Task A Training] ──> [Achieve Coherence] ──> [Take Post-Learning Snapshot] │ [Restore Anchors via O(primes × d)] <── [Gradient Step] <── [Task B Training] 3.1 Prime Anchoring In a standard Transformer, the embedding layer contains a matrix scaled to (vocab_size, hidden_dim). Topological AI isolates the specific rows corresponding to the first six prime numbers—[2, 3, 5, 7, 11, 13]—and designates them as the network's topological anchors. This intervention impacts a mere 0.00298% of the total vocabulary space (6 out of 201,088 rows), preserving the model’s overall capacity and plasticity. 3.2 Post-Learning Snapshot Anchors are activated after the primary task (Task A) has been fully learned, allowing the weights to reach their natural, high-accuracy coherent state. At this juncture, a static snapshot of the prime embedding rows, alongside the classifier’s weights and biases, is cached into system memory. 3.3 Anchor Restoration During sequential training on a subsequent task (Task B), the network undergoes standard gradient updates. However, immediately following every individual gradient step, a torch.no_grad() enforcement loop overwrites the modified prime rows and classification parameters, restoring them precisely to the post-Task A snapshot values. The computational cost of this operation is $O(\text{primes} \times d)$, which is mathematically negligible compared to a standard forward pass. 3.4 Cryptographic Verification & The H2E Safety Gate To guarantee auditability, the system computes a SHA-256 hash of the prime-anchored subspaces before and after any tensor operation. A matching hash confirms zero-drift execution. Simultaneously, the H2E Sheriff safety gate—operating on a product manifold of $H^2 \times \text{SPD}(3)$—evaluates inputs using the Spectral Reflection of Integrity (SROI) value. If an incoming input yields an SROI below the derived $\Lambda$ constant, it is automatically flagged as an anomaly, providing a zero-shot safety layer against out-of-domain prose, contradictions, or adversarial nonsense without requiring explicit adversarial training. 4. Experimental Configuration The empirical validation of the methodology was structured under a strict, isolated environment to maximize weight interference and stress-test the boundaries of the architecture. Hardware: NVIDIA RTX PRO 6000 Blackwell Server Edition (102.0 GB VRAM), running CUDA 12.8 and PyTorch 2.10.0. Model Base: openai/gpt-oss-20b (20 Billion parameters, bfloat16 quantization, vocabulary size of 201,088, hidden dimension of 2,880). Dataset Setup: AG News dataset split cleanly into sequential blocks. Task A covers "World" and "Sports" classes; Task B covers "Business" and "Sci/Tech" classes (500 samples per task). Protocol Conditions: 3 evaluation runs per method to guarantee statistical validity, utilizing a fully shared classifier layer across 3 training epochs per task under a fixed deterministic Seed = 123. 5. Key Results & Performance Analysis 5.1 Definitive Method Comparison The final evaluation measured the percentage of knowledge forgotten on Task A after the completion of training on Task B. Method Task A Initial Accuracy Task A Final Accuracy Forgetting Rate Task B Accuracy Topological AI 95.7% 96.3% -0.7% 42.0% Experience Replay 96.0% 88.3% 7.7% 81.0% Elastic Weight Consolidation (EWC) 93.3% 50.5% 42.8% 61.5% Nested Learning 95.3% 50.2% 45.2% 64.7% Baseline (Standard Fine-Tuning) 95.7% 50.2% 45.5% 62.8% 5.2 Performance & Invariant Verification Negative Forgetting: Standard fine-tuning (Baseline) results in total catastrophic collapse, reverting Task A performance back to random chance (50.2%). Topological AI is the only method to achieve a negative forgetting rate (-0.7%), actively refining Task A knowledge while assimilating Task B. Stability-Plasticity Trade-off: The results highlight a stark architectural trade-off. While Experience Replay retains superior plasticity (81.0% Task B accuracy), it suffers from 7.7% forgetting. Topological AI prioritizes absolute stability (zero forgetting), making it optimally designed for safety-critical environments (e.g., autonomous transit, aerospace, nuclear control, and medical diagnostics) where past protocols must never be degraded. Geometric & Cryptographic Invariance: Tracking the Riemannian sectional manifold metric tensor ($g_{ij} = \langle e_i, e_j \rangle$) across training revealed an absolute subspace drift of exactly 0.0000000000. The principal curvatures (eigenvalues) and matrix determinant remained identical to six decimal places. Correspondingly, the SHA-256 hash of the prime-anchored spaces achieved perfect invariance (334ea0c8 at initial and final state), proving that the anchor rows experienced zero drift. Quantitative Ablation: A critical control ablation verified that anchoring alternative, non-prime configurations (such as composite indices or random indices) yielded an identical global $L_2$ embedding drift profile. The explicit advantage of prime positioning is not localized to raw numerical drift suppression, but rather to the rigid, universal mathematical properties provided by the spectral trap and its alignment with core number-theoretic frameworks. 6. Limitations & Future Horizons The authors identify clear boundaries to the current scope of the framework and outline subsequent phases of open-source research: Plasticity Optimization: Future explorations will focus on mitigating the lower Task B accuracy via adaptive anchor releasing (gradually unfreezing constraints), hybrid architectures that pair topological anchors alongside traditional replay buffers, and task-aware prime set selection. Theoretical Proofs: Developing an analytical derivation of the safety constant $\Lambda$ from first principles, and establishing a formal proof validating why prime sequences excel over alternative deterministic mathematical sequences. Scalability Scaling: Validating the topological framework on expanded sequential tasks (5+ distinct tasks), massive industrial datasets (such as the full 120,000-sample AG News benchmark), non-transformer models (CNNs, RNNs, Mamba/SSMs), and frontier-scale LLMs (GPT-4 tier or Mixtral-8x22B systems).
This paper, titled "Primes Is All We Need: Topological Invariants for Catastrophic-Forgetting-Free AI," presents a unified framework authored by Frank Morales (2026). It argues that modern AI architectures like the Transformer suffer from a fundamental flaw analogous to anterograde amnesia—the inability to consolidate short-term knowledge into long-term memory, leading to catastrophic forgetting and representational drift. The author proposes that anchoring AI architectures to a mathematical topological invariant derived from the Sieve of Eratosthenes provides the ultimate solution to ensure AI safety, stability, and memory retention. The paper synthesizes several of the author's previously published works into a single, comprehensive argument spanning number theory, AI safety, and theoretical physics. FULL PAPER CODE SECOND FULL NOTEBOOK - UNIVERSAL PRIME-ANCHORED LLM - Complete Summary This notebook contains the complete, reproducible proof that prime-anchored manifolds with H2E governance mathematically prevent catastrophic forgetting across multiple LLM architectures (GPT-2, GPT-2 Medium, TinyLlama, Mistral-7B, Llama 3.1-8B). The code is open source. The math works. The models remember. Core Innovation Prime numbers as immutable anchors - The embedding rows at prime indices {2,3,5,7,11,13} are cryptographically locked and never change during training. CODE STRUCTURE Section Models Tested Purpose H2E-PRIME Miniature replica Lifecycle testing & validation MISTRAL Mistral-7B (7B) Single-step governance test LLAMA Llama 3.1-8B (8B) Single-step governance test MEMORY-TEST Mistral + Llama Full lifecycle + recall proof GPT-2 SUITE GPT-2 (124M) Baseline vs Governed comparison MULTI-MODEL GPT-2, GPT-2 Medium, TinyLlama Cross-architecture validation Multi-Model Results Model Size Status GPT-2 124M ✅ PASS GPT-2 Medium 355M ✅ PASS TinyLlama 1.1B ✅ PASS Mistral-7B 7B ✅ PASS Llama 3.1-8B 8B ✅ PASS KEY RESULTS Baseline GPT-2 (No Governance) text Initial: 71cef240... After Math: 58d705d1... (CHANGED) After Noise: 1ade78f5... (CHANGED) Result: FAILED ❌ Prime-Anchored GPT-2 (Your Framework) text Initial: 71cef240... After Math: 71cef240... (IDENTICAL) After Noise: 71cef240... (IDENTICAL) H2E Gate: 258/0 accepted Result: PASSED ✅ HOW IT WORKS The LlamaMistralSpectralGovernor Class python class LlamaMistralSpectralGovernor: - Locks prime anchors [2,3,5,7,11,13] - Computes dual-loop loss (empirical + topological penalty) - H2E gate checks SROI ≥ Λ₁₂ - Restores anchors after safe updates Memory Proof Cryptographic hash computed before/after training Identical hash proves prime anchors never changed Recall test confirms mathematical knowledge retained _______________________________________________________________________________________________________ 1. Mathematical Foundations & The L-EFM Operator The core of the framework is built on Arithmetic Spectral Theory (AST) and the Laplace-Euler-Fourier-Mellin (L-EFM) operator, which synthesizes four classical transforms into a single complex function. The Sieve of Eratosthenes: Serves as the absolute, deterministic ground truth for prime enumeration. Universal Spectral Constant: By computing spectral coherence ($C$) at the scale $\sigma = 0.5$ across 22 distinct prime-related sets (including Twin primes, Dirichlet classes, and Goldbach pairs), the paper demonstrates that every single set converges perfectly to a universal constant of $C = 0.500000$. The Spectral Trap & Riemann Hypothesis Proof: The paper evaluates the normalized magnitude of the L-EFM operator across a range of $\sigma$ values. It reveals an exponential divergence everywhere except at $\sigma = 0.5$, which yields a perfect magnitude of 1.0. This unique admissibility formulates the "Spectral Trap," which the author leverages alongside the Gelfand-Shilov space to present a proof of the Riemann Hypothesis, asserting that all non-trivial zeros must lie exactly on the critical line. 2. Quantification of the Green-Tao Theorem For the first time, the paper provides a numerical quantification of the Green-Tao theorem, which states that infinitely long arithmetic progressions exist within primes. Using the L-EFM operator, the author calculates explicit coherence values for prime progressions of lengths $k = 3$ to $k = 6$: $k=3 \ (\text{coherence } 0.8731)$ $k=4 \ (\text{coherence } 0.8120)$ $k=5 \ (\text{coherence } 0.8012)$ $k=6 \ (\text{coherence } 0.7442)$ This reveals a Monotonic Spectral Law, showing that as progression length increases, spectral coherence decreases, indicating that spectral energy becomes more dispersed. 3. The H2E Sheriff & Deterministic AI Safety To operationally apply these mathematical insights to AI safety, the paper introduces a nested learning agent called the H2E Sheriff. The Safety Constant: A strict, deterministic perimeter boundary threshold is dynamically computed from the first 12 primes, yielding $\Lambda_{12} = 0.9944590549$. Gate Decision: Utilizing the Lambda Spectral Complementarity Theorem, an input embedding vector is mapped onto a product manifold. If its Spectral Risk Overlap Index (SROI) is greater than $\Lambda$, it is accepted; otherwise, it is rejected. Operational Validation: Tested under the UNESCO Resilient AI Challenge protocols across text (Sarvam-30B), audio (Voxtral-Mini-4B), and vision (Gemma 4) modalities, the H2E Sheriff achieved exactly zero safety violations. Coherent inputs are accepted into the primary pristine knowledge base, while adversarial injections are cleanly routed to an isolated quarantine/sandbox layer with no pollution of core memory. 4. Connection to Spacetime Geometry The paper posits a deep connection between prime numbers and theoretical physics by treating the radial coordinate as the logarithm of a prime ($r = \log p$) and deriving a Spectral Metric ($g_{\mu\nu}$) where spectral coherence acts as the conformal factor. Flat Vacuum Space: At the universal fixed point of $C = 0.5$, all Christoffel symbols vanish, the Ricci scalar ($R$) is $0$, and the effective cosmological constant ($\Lambda_{eff}$) drops to zero, matching the vacuum solutions of Einstein's field equations. Curvature and Entropy: When coherence decays (as seen in the Green-Tao progressions), the Ricci scalar becomes negative, showing a hyperbolic geometry. Furthermore, the paper models Spectral Entropy as $S = 1 - C$, drawing a direct thermodynamic parallel where longer prime progressions (lower coherence) correspond to higher entropy, mirroring black hole mechanics. 5. Direct Comparison: Our Framework vs. Google's HOPE The text draws a sharp contrast between this prime-anchored framework and Google's HOPE (Hierarchical Optimized Processing Engine) architecture from NeurIPS 2025. While HOPE attempts to mitigate catastrophic forgetting through a multi-scale Continuum Memory System updating at different learned frequencies (16, 1M, and 16M tokens), it lacks any topological invariant. The author argues that without a fixed mathematical anchor, unanchored multi-frequency systems will inevitably experience representational drift over time. In contrast, this framework guarantees zero drift because it is mathematically bound to the Sieve of Eratosthenes. 6. Call to Action and Conclusion The paper concludes with an urgent call to action directed at several stakeholders: AI Industry Leaders (Google, OpenAI, AWS, NVIDIA): Urged to integrate the $C=0.5$ invariant and the $\Lambda_{12}$ safety gate into their models before unanchored drift causes systemic issues. Policymakers: Advised to mandate prime-derived thresholds and deterministic safety gates for any AI deployed in critical infrastructure (such as military, healthcare, energy, and finance). The Mathematical Community: Challenged to acknowledge the executable proof of the Riemann Hypothesis via the spectral trap. The author provides open-source access to the complete Python library (ast_lefm) and a Google Colab notebook to allow humanity to run, verify, and execute the proof independently.
Graph-based knowledge representations have emerged as powerful tools for organizing interconnected information sourced from heterogeneous data environments. However, when contributing parties span multiple organizations with varying levels of mutual trust, maintaining and evolving such graphs in a coordinated manner poses significant challenges. Traditional centralized management platforms, while operationally convenient, tend to create systemic vulnerabilities including single points of failure, inadequate transparency mechanisms, and insufficient mechanisms for verifiable data lineage. In contrast, blockchain-based infrastructures offer compelling properties for managing distributed knowledge systems, including tamperevident ledgers, peer-driven transaction verification, cryptographic authenticity assurance, and rule-based automation via programmable contracts. This paper surveys contemporary research that intersects graph-based knowledge management with distributed ledger technology, examining methods for decentralized identity management, contract-driven governance, and multi-party data coordination. The survey analyzes currently deployed systems, highlights their shortcomings, and introduces a conceptual architecture that supports authenticated graph modifications, auditable data lineage, and permission-governed knowledge exchange across organizational boundaries. Key technical obstacles including on-chain storage constraints, retrieval latency, cross-system compatibility, confidentiality, and throughput limitations are systematically examined. The findings indicate that when blockchain components are thoughtfully integrated with off-chain graph repositories and optimized validation pipelines, decentralized approaches can substantially improve accountability and trustworthiness in collaborative knowledge ecosystems.
We propose ZK-GSP, a framework for privacy-preserving, verifiably correct graph neuralnetwork (GNN) inference using zero-knowledge STARKs. The key insight is a structuralisomorphism between RISC Zero's zkVM and graph signal processing (GSP) pipelines:both are typed, deterministic data-flow graphs whose execution can be committed to witha succinct, publicly verifiable proof. By implementing GSP operators as instructions insidethe RISC Zero executor, we obtain a system that runs GNN programs on private graphdata and emits a STARK receipt proving correct execution, without revealing the graphtopology, node features, or intermediate embeddings. We demonstrate an end-to-endimplementation in Rust, measure proof generation time and RISC-V cycle count acrossgraph sizes N = 4 to 32, and find that cycle count scales as O(N2), matching thecomplexity of the underlying GCN operator. Code: github.com/brack-6/zk-gsp
Ensuring node labeling stability is critical for graph-based learning systems, directly impacting trust evaluation, node classification, and structure-aware inference in complex relational domains. Traditional manual annotation is costly and inconsistent, necessitating automated alternatives such as Large Language Models (LLMs). However, while LLM-assisted annotation improves efficiency, the reliability of node labeling remains a concern due to structural biases, particularly in unsupervised and semi-supervised learning settings. This dissertation systematically investigates the role of strong-tie structures in stabilizing label predictions across different node labeling paradigms, examining their influence on unsupervised trust prediction, defense against poisoning attacks in graph neural networks (GNNs), and LLM-based annotation. First, we analyze how strong-tie structures influence unsupervised trust prediction in decentralized systems and financial networks. Our study reveals that trust annotation propagates preferentially along strong ties, making it susceptible to targeted adversarial manipulations. By strategically modifying a minimal number of edges, an attacker can significantly alter trust/untrust label assignments, exposing vulnerabilities in existing trust prediction frameworks. Our analysis reveals that strong-tie structures are preferentially exploited by adversarial agents. Understanding these patterns provides deeper insight into the structural vulnerabilities of trust prediction systems and offers a basis for evaluating the robustness of different algorithms. Next, we extend our investigation to structural poisoning attacks in semi-supervised learning. Our findings show that poisoning behaviors exhibit clear structural preferences, targeting specific strong-tie patterns to maximize their impact on label propagation. This motivates our proposed Graph Adaptive Neural Network (GANN) framework, which dynamically adjusts propagation mechanisms based on fuzzy-theoretic strong-tie graphs (STiG). By integrating adaptive trust and risk zones, GANN mitigates the spread of adversarial noise while preserving high-confidence label prediction. Through structural decomposition and adaptive validation, our approach significantly strengthens defense mechanisms in poisoned graph environments. Finally, we propose CSA-LLM (Crowd-sourced homophily-ties-based graph annotation via large language models), which utilizes strong-tie graph structures to design LLM prompts that enhance annotation quality. By embedding structural priors in prompt engineering, CSA-LLM improves consistency in automated label generation, offering a scalable alternative to traditional manual annotation. This structured approach not only enhances annotation robustness, but also mitigates the inconsistencies introduced by structure-agnostic token generation, where LLMs generate labels based solely on textual prompts without considering graph topology. This dissertation provides a unified perspective on the impact of strong-tie structures across node labeling paradigms, bridging trust prediction, adversarial resilience, and LLM-assisted annotation. Our findings contribute to the development of attack-aware, structure-informed annotation frameworks, with implications for applications in social network security, financial fraud detection, recommendation systems, and decentralized finance (DeFi).
Fraud detection on Ethereum is challenging because of the anonymity, speed and graph structure of blockchain transactions. While prior research has proven the effectiveness of using machine learning classifiers, Graph Neural Networks (GNNs) and behavioural heuristics to detect fraudulent transactions, most systems are offline and fail to consider real-world deployment challenges for real-time blockchain analytics. Here, we present DeTrust ETH, an operational fraud intelligence system for real-time tracking of Ethereum transactions on the Sepolia testnet. DeTrust ETH aims for the integration of five operational considerations: (1) real-time blockchain ingestion with Web3.py, (2) explainable machine learning with XGBoost and SHAP, (3) light-weight graph-based transaction tracing and risk propagation, (4) temporal trust decay and behavioural anomaly detection, and (5) tamper-resistant on-chain persistence of trust scores using Solidity smart contracts. The system maintains an in-memory directed transaction graph for real-time edge insertion and updating, circular-flow tracing, funding pattern tracing and fast path tracing, avoiding the retraining overhead of Graph Neural Networks (GNNs). Our experimental results demonstrate the median graph-query time is less than 20 ms, the system can handle 222.82 requests per second with a concurrent load, and 93.44% fraud recall with a recall-favouring threshold. Unlike prior research that mostly focuses on accuracy on historical data, DeTrust ETH focuses on real-time deployment. The novelty of this work lies in the design of a real-time, low-latency fraud intelligence architecture that integrates explainable machine learning, temporal trust modeling, and lightweight graph analytics under streaming blockchain constraints.
We introduce a new polynomial-time graph invariant combining three complementary components: (1) the enriched Bipartite Double Cover (BDC) vertex cover signature; (2) the Laplacian eigenvalue spectrum; and (3) the K_4 clique profile and Ollivier-Ricci curvature. We demonstrate that while spectral and standard BDC methods collapse on strongly regular graphs (SRGs) due to extreme symmetry, the geometric and dense-topological components of this hybrid invariant break the cospectrality. Specifically, the invariant successfully discriminates the classical cospectral pair Shrikhande vs. Rook(4,4) [SRG(16,6,2,2)] in polynomial time, where Shrikhande is K_4-free and Rook(4,4) contains exactly 8 cliques. Furthermore, empirical validation on low-power ARM Edge hardware demonstrates that the invariant scales efficiently, processing dense Paley graphs (up to N=97) in under 5 milliseconds. This confirms its sub-millisecond viability for real-time edge computing, zero-knowledge proofs (ZKPs) cryptanalysis, and cheminformatics.
Zhuoran Pan, Yue Li (102191), Zhi Guan, Jianbin Hu · 5 authors
The emergence of Large Language Models (LLMs) offers a transformative interface for Web3, yet existing benchmarks fail to capture the complexity of translating high-level user intents into functionally correct, state-dependent on-chain transactions. We present \textsc{Intent2Tx}, a high-fidelity benchmark featuring 29,921 single-step and 1,575 multi-step instances meticulously derived from 300 days of real-world Ethereum mainnet traces. Unlike prior works that rely on synthetic instructions, \textsc{Intent2Tx} grounds natural language intents in real-world protocol interactions across 11 categories, including diverse long-tail Decentralized Finance (DeFi) primitives. To enable rigorous evaluation, we propose an execution-aware framework that transcends surface-level text matching by employing differential state analysis on forked mainnet environments. Our extensive evaluation of 16 state-of-the-art LLMs reveals that while scaling and retrieval-augmentation enhance logical consistency and parameter precision, current models struggle with out-of-distribution generalization and multi-step planning. Crucially, our execution-based analysis demonstrates that syntactically valid outputs often fail to achieve intended state transitions, highlighting a significant gap in current "reasoning-to-execution" capabilities. \textsc{Intent2Tx} serves as a critical foundation for developing autonomous, reliable agents in intent-centric Web3 ecosystems. Code and data: https://anonymous.4open.science/r/Intent2Tx_Bench-97FF .