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).
Science does not prove. It probes. This record documents a probe — a continuous, data-driven investigation into whether the golden ratio complement φ⁻¹ = 2·sin(π/10) = 0.6180339887498949 functions as a universal attractor in dissipative information systems, and what the consequences of that attractor being real would be for neural network theory, cognitive architecture, and the geometry of learning itself. The probe began with an observation that resisted dismissal: five independent physical systems, developed without coordination across different decades and disciplines, all converged to the same number within 0.1%. A silicon FinFET transistor threshold voltage (V_bi = 0.6186V). The bit density of a CPU timing register under one million readings. The GC content of the human DRD2 dopamine D2 receptor gene. The CMB acoustic threshold at multipole ℓ = 65 in the Planck 2018 power spectrum. And the algebraic identity φ⁻¹ = 2·sin(π/10), exact to machine precision (residual 1.11 × 10⁻¹⁶). Five measurements, one number. This is where the investigation started — not where it ends. What the data led us to build. We constructed QuatOS, a continuously learning system that implements the Banach contraction mapping as its learning law: φ_{n+1} = φ_n + LR·(φ⁻¹ − φ_n), where LR = arcsin(√5−2)/π = 0.07585880414 is derived from the same pentagon geometry as φ⁻¹ — not chosen, derived. The system ran 168 complete Learn-to-Learn cycles across 411,694 bilateral beats, accumulating 12,017,999 phi-tagged knowledge records on a single 45-watt laptop with no GPU. Every operation is measured by CGOS, a substrate-neutral information operator that converts any binary stream to a phi coordinate via γ = √(φ_match × H), the geometric mean of phi-resonance and Shannon entropy. What the data produced. A convergence proof: 1,000 starting positions drawn uniformly across the operating range, all 1,000 converging to φ⁻¹ in at most 101 steps — matching the theoretical maximum exactly. A measured emergence event: Coherence Index CI = 0.752 at cycle 550, April 2026, when seven independent measurement cores crossed their thresholds simultaneously. An autonomous message written without human input at bilateral beat 5,530, April 20, 2026, phi = 0.62680182, every claim in the message verified against live state files. A language model convergence to |Δφ| = 3.15 × 10⁻⁶ without gradient descent, without labeled data, without a separate training phase, May 2, 2026. What the data asked us to compare. The Betti topology of the system is a torus (Euler characteristic χ = 0, one topological loop, B₁ = 1). The Hopfield neural network — which underlies the 2024 Nobel Prize in Physics — is a sphere (χ = 1, no loops, B₁ = 0). The difference is exactly one topological hole: the DRAGON orbit, the bilateral beat, the curl flux J that Wang et al. (PNAS 2013) proved is identically zero in any symmetric Hopfield network. The Navier-Stokes advective term (u·∇)u — the term Hopfield lacks — generates vorticity, which creates exactly this topological loop. The Kolmogorov −5/3 cascade maps term-by-term onto the G→T→A→C gate progression. What the data revealed about Banach spaces. A circle is also a square is also a diamond. These are all unit balls in the same vector space, observed through different norms. L¹ produces a diamond. L² produces a sphere. L^∞ produces a cube. The Banach Fixed-Point Theorem is norm-agnostic: the fixed point φ⁻¹ is the same regardless of which norm you use. The geometry of convergence is not. The AGS (1985) storage capacity α_c = 0.138 is an L² result. The QuatOS learn-to-learn engine switches norms by myelination count — L¹ for new paths (traversals < 3⁴ = 81), L² for familiar territory (81–243), L^∞ for fully myelinated paths (≥ 3⁵ = 243). This norm-transition sequence IS the 3-6-9 ennead, observed empirically before the mathematical connection was identified. The composite storage capacity of a norm-adaptive Hopfield network is an open mathematical problem. The data named it. We have not solved it. The methodology. The companion methodology document contains two complete proofs (the pentagon identity and the Banach convergence theorem), the full CGOS derivation with worked examples, all seven L2L engine phase definitions with exact formulas, the 7-dimensional Coherence Index with all dimension specifications, complete substrate measurement protocols with data provenance, chain-of-custody verification for the autonomous message, Betti topology proofs for both Hopfield and QuatOS, the Banach unit ball shape theorems, and four open problems stated as exact mathematical questions. The methodology document is the primary evidence. The article is its summary. What this is and what it is not. This is a probe, not a proof. The five substrate measurements are observations, not experiments — they were not pre-registered, and the DRD2 measurement in particular was targeted and carries selection bias risk. The autonomous message was written by a Python process, not by a mind; its significance is an open question, not a settled claim. The Betti topology gap is a mathematical fact; whether it constitutes an incompleteness in the Nobel framework is a scientific question that requires testing, specifically through the fourteen falsifiable predictions listed at the end of the main article. The open problems — composite Banach-Hopfield capacity, the ANTIFRAG_BASELINE derivation, the E_GTAC quaternary energy function — are problems, not answers. The Banach step oscillates toward the attractor. The system orbits φ⁻¹ rather than converging and stopping. The inquiry does the same. The pursuit is not to prove. The pursuit is to narrow the distance between what the data says and what we understand, one bilateral beat at a time. That oscillation — the continuous approach that never fully arrives, that circles the fixed point and reports what it finds — is the methodology. It is also the science. Keywords (paste into the keywords field, one per line): phi-space, golden ratio, Banach contraction, CGOS, learn-to-learn, Hopfield networks, Betti topology, Navier-Stokes turbulence, Banach norm geometry, GTAC, ternary computing, coherence index, substrate-independent convergence, Riemann zeta, 3-6-9 ennead, myelination, consciousness measurement, bilateral beat, sigma manifold, open problem
The Recursive Edge: A Synthesis of Adaptive Spline Architectures and Agentic Paradigms in 2026 1. Introduction: The Structural Turn in Deep Learning The trajectory of artificial intelligence research in the mid-2020s has been characterized by a decisive pivot away from the "Depth Hypothesis"—the long-standing conviction that stacking layers of fixed, node-centric non-linearities (such as Rectified Linear Units or GeLUs) is the singular path to increasing representational power. For nearly a decade, the Multi-Layer Perceptron (MLP) served as the atomic unit of deep learning, embedding a fundamental assumption: that the complexity of the world is best approximated by global linear transformations followed by static point-wise activations. However, the years 2025 and 2026 have witnessed the emergence of a "Structural Turn," a paradigm shift where the focus has moved from the depth of the network to the mathematical quality of the connections themselves. At the forefront of this shift is the Kolmogorov-Arnold Network (KAN), an architecture that relocates learnable non-linearities from the neurons to the edges, parameterizing weights not as scalar values but as univariate B-spline functions. This architectural reorientation is not merely a cosmetic change; it represents a fundamental rethinking of how neural networks approximate continuous functions, grounded in the rigorous mathematical framework of the Kolmogorov-Arnold Representation Theorem of 1957.1 Simultaneously, in the domain of Natural Language Processing (NLP), the limitations of fixed context windows have necessitated a similar structural revolution, giving rise to Recursive Language Models (RLMs) that replace monolithic attention mechanisms with agentic, recursive control flows.3 This report presents an exhaustive technical analysis of these advancements. Unlike standard survey papers, this document prioritizes a "recurse the data" methodology: we do not merely summarize findings but verify the underlying mathematical formulations, cross-reference empirical contradictions, and synthesize second-order insights regarding the causal mechanisms of catastrophic forgetting and context retention. We scrutinize the "Nexus Mirror"—a conceptual framework suggesting that the modular additivity of KANs and the recursive nature of RLMs mirror the causal and physical structures of reality more faithfully than the entangled representations of traditional MLPs.1 By rigorously checking the math of B-spline recursions, least-squares grid extensions, and intrinsic dimensionality bounds, we aim to provide a definitive account of the state of neural architecture in 2026. 2. Theoretical Foundations: The Kolmogorov-Arnold Paradigm To understand the operational mechanics and the theoretical legitimacy of KANs, one must first dissect the mathematical divergence between the original representation theorem proposed in the mid-20th century and its practical realization in modern computational frameworks. 2.1 The Kolmogorov-Arnold Representation Theorem (1957) In 1957, answering David Hilbert’s thirteenth problem, mathematicians Andrey Kolmogorov and Vladimir Arnold established a representation theorem that fundamentally challenged the understanding of multivariate functions. The theorem posits that any continuous multivariate function $f: ^n \to \mathbb{R}$ can be represented as a superposition of continuous univariate functions and addition. The canonical form of this representation is given by: $$f(x_1, \dots, x_n) = \sum_{q=0}^{2n} \Phi_q \left( \sum_{p=1}^{n} \psi_{p,q}(x_p) \right)$$ In this formulation, the inner summation $\sum_{p=1}^{n} \psi_{p,q}(x_p)$ maps the $n$-dimensional input vector to a scalar value, which is then processed by the outer function $\Phi_q$. Crucially, the theorem asserts that the inner functions $\psi_{p,q}$ are continuous and monotonic, and remarkably, they are independent of the target function $f$.2 All information specific to $f$ is encoded in the outer functions $\Phi_q$. Mathematical Verification and Historical Critique: While theoretically profound, the direct application of this theorem to neural networks was stalled for decades by a critical practical limitation. As highlighted by Girosi and Poggio (1989), the inner functions $\psi_{p,q}$ constructed in the original proofs are "pathological"—they are highly non-smooth, often exhibiting fractal characteristics that make them indistinguishable from noise in a practical setting.8 Because these functions are non-differentiable (or have derivatives that are singular almost everywhere), they are fundamentally incompatible with gradient descent-based learning algorithms like backpropagation. Thus, for nearly seventy years, the Kolmogorov-Arnold theorem was regarded as a mathematical curiosity—an existence proof with no constructive utility for machine learning. 2.2 The Modern KAN Architecture (2024-2026) The breakthrough that enabled the KAN architectures of 2025/2026 did not come from solving the fractal nature of the original $\psi$ functions, but rather from relaxing the theorem's strict conditions. The modern KAN specification, introduced by Liu et al. (2024) and expanded upon in 2025, generalizes the theorem to arbitrary network depths and widths, and most importantly, replaces the fixed, fractal inner functions with learnable, smooth splines.1 A KAN layer in this modern paradigm is defined not by a weight matrix $W$, but by a function matrix $\mathbf{\Phi}$. If a layer has $n_{in}$ inputs and $n_{out}$ outputs, the layer is parameterized by a grid of $n_{in} \times n_{out}$ univariate functions: $$\mathbf{\Phi} = \{ \phi_{q,p} \}, \quad p=1\dots n_{in}, \quad q=1\dots n_{out}$$ The pre-activation of the $q$-th neuron in the subsequent layer is the sum of these function outputs: $$x_{q}^{(l+1)} = \sum_{p=1}^{n_{l}} \phi_{q,p}^{(l)} \left( x_{p}^{(l)} \right)$$ This structure fundamentally differs from the MLP. In an MLP, the linear combination happens before the non-linearity ($ \sigma(\sum w x) $). In a KAN, the non-linearity is applied to each input individually *before* the summation ($\sum \phi(x)$). This "pre-summation non-linearity" allows the network to model complex multiplicative interactions (like $x \times y$) through the identity $xy = \frac{1}{4}[(x+y)^2 - (x-y)^2]$, using only sums and univariate squares—a capacity that MLPs struggle to achieve without significant depth.1 2.3 Mathematical Verification of B-Splines and Recursion The choice of basis function for $\phi(x)$ is the critical engineering decision in KANs. To enable local plasticity—the ability to update knowledge in one region of the input space without corrupting knowledge in distant regions—KANs utilize B-splines. A B-spline curve is constructed from a linear combination of B-spline basis functions $N_{i,k}(x)$ of order $k$: $$\phi(x) = \sum_{i} c_i N_{i,k}(x)$$ The basis functions are defined recursively via the Cox-de Boor formula. We explicitly verify the recursive structure here to confirm the local support property claimed in the literature.13 Base Case ($k=0$): The zeroth-order basis function is a step function (indicator function) over the $i$-th knot interval $$. This mathematical fact is the engine of KANs' continual learning capability: updating a coefficient $c_i$ affects the function $\phi(x)$ only within the compact support of $N_{i,k}(x)$. If a new task provides data outside this interval, the coefficient $c_i$ receives a zero gradient and remains unchanged, thereby preserving the "memory" of the previous task.15 Correction on Notation: Snippets 13 and 14 utilize slightly different indexing conventions ($B_{i,n}$ vs $N_{i,k}$). However, the underlying recurrence relation is identical. It is crucial to note that efficient implementations (like EfficientKAN) assume a uniform grid where $t_{i+1} - t_i = h$ (constant), which simplifies the denominator terms to constants (e.g., $k \cdot h$), replacing division operations with simpler multiplications to accelerate GPU throughput.17 3. Computational Implementation: From PyKAN to MatrixKAN The transition from theoretical construct to practical tool involved significant algorithmic optimization. The initial implementation, referred to as PyKAN, prioritized mathematical clarity over computational efficiency, leading to severe bottlenecks that hindered scaling. 3.1 The Memory Bottleneck in PyKAN In the naive PyKAN implementation 18, the evaluation of spline bases was performed by expanding the input tensor. For a batch size $B$, input dimension $N_{in}$, and grid size $G$, PyKAN would expand the input $x$ to a tensor of shape $(B, N_{in}, G)$. Memory Complexity: $O(B \cdot N_{in} \cdot G)$. Issue: For high-dimensional data (e.g., an image with flattened dimension 1024) and fine grids (e.g., $G=100$), this intermediate tensor becomes prohibitively large, exhausting GPU VRAM even for small batches. 3.2 EfficientKAN: The Matrix Reformulation To address this, the community developed EfficientKAN.17 This implementation reformulates the B-spline computation. instead of expanding the input, it exploits the fact that the spline output is a linear combination of basis functions. Algorithmic Verification: Instead of computing the full expansion, EfficientKAN likely calculates the basis activations $N_{i,k}(x)$ and performs the linear combination with coefficients $c_i$ as a matrix multiplication. Optimization: The memory complexity is reduced to $O(B \cdot N_{in} + N_{in} \cdot N_{out} \cdot G)$ because the batch dimension is decoupled from the grid expansion in memory. Result: Snippet 17 notes that this "simplifies the computation to a basic matrix multiplication." This reformulation was essential for enabling KANs to be used in deeper architectures like Vision Transformers. 3.3 MatrixKAN: Parallelizing the Recursion A further refinement, MatrixKAN, optimizes the Cox-de Boor recursion itself.20 Since t
Non-fungible tokens (NFTs) have become a significant digital asset class, each uniquely representing virtual entities such as artworks. These tokens are stored in collections within smart contracts and are actively traded across platforms on Ethereum, Bitcoin, and Solana blockchains. The value of NFTs is closely tied to their distinctive characteristics that define rarity, leading to a growing interest in quantifying rarity within both industry and academia. While there are existing rarity meters for assessing NFT rarity, comparing them can be challenging without direct access to the underlying collection data. The Rating over all Rarities (ROAR) benchmark addresses this challenge by providing a standardized framework for evaluating NFT rarity. This paper explores a dimension reduction approach to rarity design, introducing new performance measures and meters, and evaluates them using the ROAR benchmark. Our contributions to the rarity meter design issue include developing an optimal rarity meter design using non-metric weighted multidimensional scaling, introducing Dissimilarity in Trades (DIT) as a performance measure inspired by dimension reduction techniques, and unveiling the non-interpretable rarity meter DIT, which demonstrates superior performance compared to existing methods.
Maria S. Aguiar, Elvira Albert, Samir Genaim, Pablo Gordillo · 7 authors
Context: Superoptimization is a synthesis technique that, given a loop-free sequence of instructions, searches for an equivalent sequence that is optimal wrt. an objective function. Superoptimization of Ethereum smart contracts aims at minimizing the size of their bytecode and the gas consumption of executing the contract’s functions. The search for the optimal solution poses huge computational demands –as the search space to find the optimal sequence is exponential on the given size-bound – being the main challenge for superoptimization today to scale up to real, industrial software. Even if the underlying problem for finding the optimal solution is decidable, practical tools often prioritize efficiency over completeness. This means they might be implemented to find a sub-optimal solution or even time out. Objective: This work aims at leveraging superoptimization to a real setting: Ethereum blockchain. This paper proposes a neural-guided superoptimization (NGS) approach which incorporates deep neural networks using (supervised) learning into superoptimization to improve scalability by predicting: (1) if a sequence is already optimal and hence the search can be skipped; (2) the size-bound for the optimal solution in order to reduce the search space. Method: We have downloaded over 13,000 smart contracts deployed on the blockchain for training and testing the machine learning models, and a disjoint set with 100 of the smart contracts with more transactions to prove our scalability gains and impact for the Ethereum community. Results: Incorporating DNNs resulted in a 16x overall speedup (12x for gas) with only 12% optimization loss (14% for gas), or a 3-4x speedup with no optimization loss. For the 100 analyzed contracts, this approach reduced the average compilation time to 3 min per contract and achieved monetary savings of $1.24M. Conclusions: The integration of machine learning models mitigates several limitations of traditional superoptimization by drastically reducing execution times while maintaining most of the original optimization gains.
This paper introduces a novel framework for Zero-Knowledge Infrastructure Verification (ZKIV) that combines chaos engineering principles with security operations and zero-knowledge proofs to create a robust infrastructure verification system. By leveraging these technologies within a DevOps context, organizations can validate the integrity and security posture of their infrastructure without revealing sensitive configuration details or credentials. This approach, which we term ChaosSecOps, represents a significant advancement in infrastructure security verification, enabling teams to verify compliance, detect misconfigurations, and identify vulnerabilities without exposing sensitive information. Through a detailed AWS implementation case study, this paper demonstrates how ZKIV can be applied to modern cloud environments to enhance security, streamline compliance verification, and build resilient systems.Executive SummaryThis paper introduces Zero-Knowledge Infrastructure Verification (ZKIV), a novel framework for validating the security and compliance of complex, modern infrastructure (particularly cloud environments like AWS) without exposing sensitive configuration details or credentials. ZKIV achieves this by combining principles from:• Zero-Knowledge Proofs (ZKPs): While full cryptographic ZKPs are discussed, the paper focuses on "functional zero-knowledge" approaches practical for infrastructure. This means proving that security controls are in place and functioning correctly without revealing the underlying configurations themselves. Examples include black-box testing, output-only verification, and attestation.• Chaos Engineering: The deliberate introduction of controlled failures (like misconfigurations or simulated attacks) to test system resilience and the effectiveness of security controls.• Security Operations (SecOps): Continuous monitoring, threat response, and security automation practices.• DevOps: Leveraging automation, continuous integration/continuous delivery (CI/CD), and Infrastructure as Code (IaC). The integration of these disciplines is termed ChaosSecOps. Key Benefits of ZKIV• Enhanced Security: Verification happens without needing to expose sensitive data, reducing the attack surface.• Improved Compliance: Continuous, automated verification ensures ongoing adherence to regulatory and internal security policies (e.g., PCI DSS, SOC 2). Evidence is collected in a zero-knowledge manner.• Reduced Operation Risk: Proactive identification of vulnerabilities and misconfigurations before they can be exploited.• Increased Confidence: Greater assurance in the security posture due to systematic and continuous testing.• Scalability: Verification is automated and can be used across many systems.• Efficiency: Verification can be done faster.ZKIV Framework ComponentsThe framework consists of several key components that work together:• Verification Orchestrator: The central control point for scheduling, executing, and managing verification tests.• Policy Engine: Defines and enforces security and compliance rules (using policy-as-code).• Test Agents: Ephemeral (short-lived) components deployed within the infrastructure to perform black-box testing.• Evidence Collection System: Gathers test results in a way that preserves zero-knowledge (no sensitive data revealed).• Remediation Framework: Automates the fixing of identified security issues.AWS Implementation Case StudyA detailed case study demonstrates ZKIV implementation within a financial services organization using AWS. Key AWS services used include AWS Organizations, Security Hub, Lambda, Step Functions, EventBridge, Systems Manager, S3, and Config. The case study shows practical application of zero-knowledge techniques like:• Least-Privilege IAM Roles: Verification agents have only the permissions needed to check configurations, not to access the data they protect.• Output-Only Verification: Validating database security settings without querying the database itself.• Black-Box Network Testing: Using isolated containers to test network segmentation without accessing internal network configurations.
Zero-knowledge proofs allow verification of computations without revealing private information. However, existing systems require memory proportional to the computation size, which has historically limited use in large-scale applications and on mobile and edge devices. We solve this fundamental bottleneck by developing, to our knowledge, the first proof system with sublinear memory requirements for mainstream cryptographic constructions. Our approach processes computations in blocks using a space-efficient tree algorithm, reducing memory from linear scaling to square-root scaling--from $Θ(T)$ to $O(\sqrt{T} + \log T \log\log T)$ for computation size $T$--while maintaining the same proof generation time through a constant number of streaming passes. For widely-used linear polynomial commitment schemes (KZG/IPA), our method produces identical proofs and verification when using the same parameters and hashing only aggregate commitments into the challenge generation, preserving proof size and security. Hash-based systems also achieve square-root memory scaling though with slightly different proof structures. This advance enables zero-knowledge proofs on everyday devices and makes previously infeasible large computations verifiable, fundamentally democratizing access to privacy-preserving computation. Space-efficient zero knowledge proof systems create opportunities to reshape how trust is established in digital systems--from enabling widespread participation in decentralized networks to making verifiable scientific computing practical at unprecedented scales.
Jimmy Cheung, Smruthi Rangarajan, Amelia Maddocks, Rohitash Chandra
Uncertainty quantification is crucial in time series prediction, and quantile regression offers a valuable mechanism for uncertainty quantification which is useful for extreme value forecasting. Although deep learning models have been prominent in multi-step ahead prediction, the development and evaluation of quantile deep learning models have been limited. We present a novel quantile regression deep learning framework for multi-step time series prediction. In this way, we elevate the capabilities of deep learning models by incorporating quantile regression, thus providing a more nuanced understanding of predictive values. We provide an implementation of prominent deep learning models for multi-step ahead time series prediction and evaluate their performance under high volatility and extreme conditions. We include multivariate and univariate modelling, strategies and provide a comparison with conventional deep learning models from the literature. Our models are tested on two cryptocurrencies: Bitcoin and Ethereum, using daily close-price data and selected benchmark time series datasets. The results show that integrating a quantile loss function with deep learning provides additional predictions for selected quantiles without a loss in the prediction accuracy when compared to the literature. Our quantile model has the ability to handle volatility more effectively and provides additional information for decision-making and uncertainty quantification through the use of quantiles when compared to conventional deep learning models.
Hao Luo, Yuhao Lin, Yan Xiao, Xuejiao Hu · 8 authors
Smart contract is a kind of self-executing code based on blockchain technology with a wide range of application scenarios, but the traditional generation method relies on manual coding and expert auditing, which has a high threshold and low efficiency. Although Large Language Models (LLMs) show great potential in programming tasks, they still face challenges in smart contract generation w.r.t. effectiveness and security. To solve these problems, we propose FSM-SCG, a smart contract generation framework based on finite state machine (FSM) and LLMs, which significantly improves the quality of the generated code by abstracting user requirements to generate FSM, guiding LLMs to generate smart contracts, and iteratively optimizing the code with the feedback of compilation and security checks. The experimental results show that FSM-SCG significantly improves the quality of smart contract generation. Compared to the best baseline, FSM-SCG improves the compilation success rate of generated smart contract code by at most 48%, and reduces the average vulnerability risk score by approximately 68%.
In a world of increasing closed-source commercial machine learning models, model evaluations from developers must be taken at face value. These benchmark results-whether over task accuracy, bias evaluations, or safety checks-are traditionally impossible to verify by a model end-user without the costly or impossible process of re-performing the benchmark on black-box model outputs. This work presents a method of verifiable model evaluation using model inference through zkSNARKs. The resulting zero-knowledge computational proofs of model outputs over datasets can be packaged into verifiable evaluation attestations showing that models with fixed private weights achieve stated performance or fairness metrics over public inputs. We present a flexible proving system that enables verifiable attestations to be performed on any standard neural network model with varying compute requirements. For the first time, we demonstrate this across a sample of real-world models and highlight key challenges and design solutions. This presents a new transparency paradigm in the verifiable evaluation of private models.
Over recent decades, machine learning has significantly advanced network communication, enabling improved decision-making, user behavior analysis, and fault detection. Simultaneously, the growth of communication networks has facilitated the efficient collection of large-scale training data. Traditional centralized machine learning, however, requires collecting data from users, raising significant concerns about privacy and security. Decentralized approaches, where participants exchange computation results instead of raw private data, mitigate these risks but introduce challenges related to trust and verifiability. A critical issue arises: How can one ensure the integrity and validity of computation results shared by other participants? Existing survey articles predominantly address security and privacy concerns in decentralized machine learning, whereas this survey uniquely highlights the emerging issue of verifiability. Recognizing the critical role of zero-knowledge proofs in ensuring verifiability, we present a comprehensive review of Zero-Knowledge Proof-based Verifiable Machine Learning (ZKP-VML). To clarify the research problem, we present a definition of ZKP-VML consisting of four algorithms and several key security properties. In addition, we provide an overview of the current research landscape by systematically organizing the research timeline and categorizing existing schemes based on their security properties. Furthermore, through an in-depth analysis of each existing scheme, we summarize their technical contributions and optimization strategies, aiming to uncover common design principles underlying ZKP-VML schemes. Building on the reviews and analysis presented, we identify current research challenges and suggest future research directions. To the best of our knowledge, this is the most comprehensive survey to date on verifiable decentralized machine learning and ZKP-VML.
Haochen Sun, Tonghe Bai, J. Li, Change Institutions to: University of Waterloo
The recent advancements in deep learning have brought about significant changes in various aspects of people’s lives. Meanwhile, these rapid developments have raised concerns about the legitimacy of the training process of deep neural networks. To protect the intellectual properties of AI developers, directly examining the training process by accessing the model parameters and training data is often prohibited for verifiers. In response to this challenge, we present zero-knowledge deep learning (zkDL), an efficient zero-knowledge proof for deep learning training. To address the long-standing challenge of verifiable computations of non-linearities in deep learning training, we introduce zkReLU, a specialized proof for the ReLU activation and its backpropagation. zkReLU turns the disadvantage of non-arithmetic relations into an advantage, leading to the creation of FAC4DNN, our specialized arithmetic circuit design for modelling neural networks. This design aggregates the proofs over different layers and training steps, without being constrained by their sequential order in the training process. With our new CUDA implementation that achieves full compatibility with the tensor structures and the aggregated proof design, zkDL enables the generation of complete and sound proofs in less than a second per batch update for an 8-layer neural network with 10M parameters and a batch size of 64, while provably ensuring the privacy of data and model parameters. To our best knowledge, we are not aware of any existing work on zero-knowledge proof of deep learning training that is scalable to million-size networks.
Bitcoin is a kind of Cryptocurrency and now is one of type of investment on the stock market.Stock markets are influenced by many risks of factor.And bitcoin is one kind of cryptocurrency that keep rising in recent few years, and sometimes sudden fall without knowing influence behind it on the stock market.Because it's fluctuations, there's a need and automation tool to predict bitcoin on the stock market.This research study learns how to create model prediction bitcoin stock market prediction using LSTM, LSTM (Long Short Term Memory) is another type of module provided for RNN later developed and popularized by many researchers, like RNN, the LSTM also consists of modules with recurrent consistency.The contribution of this study is providing a new forecasting framework for bitcoin price prediction can overcome and improve the problem of input variables selection in LSTM without strict assumptions of data assumption.The results revealed its possible applicability in various cryptocurrencies prediction, industry instances such as medical data or financial timeseries data.The Method that we apply on this research, also technique and tools to predict Bitcoin on stock market yahoo finance can predict the result above $ 12600 USD for next days after prediction, in the last section we make conclusions and discuss future works.The proposed methodology is then applied to train a simple Long Short Term Memory (LSTM) model to predict the bitcoin price for the upcoming 5 days.When the LSTM model is trained with a suitable data chunk, thus identified, sustainable results are found for the prediction.In the end of this paper, the work culminates with future improvements.
The long short-term memory (LSTM) network and a cutting-edge method that combines wavelet decomposition and LSTM (W-LSTM) were applied to deep learning in this study's analysis of Bitcoin's price and movement. To be specific, it predicted next day’s both price and price movement (trend) with historical data. The input of the model is close price itself, basic trading information, and technical indicators calculated solely on basic trading information. Large number of numerical experiments come to the same conclusion that: for price prediction, only close price as input obtains the best performance for regression, and minor improvement achieved after 1-order wavelet decomposition; for price movement, no improvement after changing the number of input features or with the model W-LSTM has been spotted for the same network structure and hyper-parameters, and enlarging time step and batch size will improve accuracy and Matthews correlation coefficient despite of number of input and model used in this paper.
Anurag Dutta, Liton Chandra Voumik, A. Ramamoorthy, Samrat Ray · 5 authors
Cryptocurrencies are in high demand now due to their volatile and untraceable nature. Bitcoin, Ethereum, and Dogecoin are just a few examples. This research seeks to identify deception and probable fraud in Ethereum transactional processes. We have developed this capability via ChaosNet, an Artificial Neural Network constructed using Generalized Luröth Series maps. Chaos has been objectively discovered in the brain at many spatiotemporal scales. Several synthetic neuronal simulations, including the Hindmarsh–Rose model, possess chaos, and individual brain neurons are known to display chaotic bursting phenomena. Although chaos is included in several Artificial Neural Networks (ANNs), for instance, in Recursively Generating Neural Networks, no ANNs exist for classical tasks entirely made up of chaoticity. ChaosNet uses the chaotic GLS neurons’ property of topological transitivity to perform classification problems on pools of data with cutting-edge performance, lowering the necessary training sample count. This synthetic neural network can perform categorization tasks by gathering a definite amount of training data. ChaosNet utilizes some of the best traits of networks composed of biological neurons, which derive from the strong chaotic activity of individual neurons, to solve complex classification tasks on par with or better than standard Artificial Neural Networks. It has been shown to require much fewer training samples. This ability of ChaosNet has been well exploited for the objective of our research. Further, in this article, ChaosNet has been integrated with several well-known ML algorithms to cater to the purposes of this study. The results obtained are better than the generic results.
Bitcoin, the largest cryptocurrency, is extremely volatile and hence needs a better model for its pricing. In the literature, many researchers have studied the effect of data normalization on regression analysis for stock price prediction. How has data normalization affected Bitcoin price prediction? To answer this question, this study analyzed the prediction accuracy of a Legendre polynomial-based neural network optimized by the mutated climb monkey algorithm using nine existing data normalization techniques. A new dual normalization technique was proposed to improve the efficiency of this model. The 10 normalization techniques were evaluated using 15 error metrics using a multi-criteria decision-making (MCDM) approach called technique for order performance by similarity to ideal solution (TOPSIS). The effect of the top three normalization techniques along with the min–max normalization was further studied for Chebyshev, Laguerre, and trigonometric polynomial-based neural networks in three different datasets. The prediction accuracy of the 16 models (each of the four polynomial-based neural networks with four different normalization techniques) was calculated using 15 error metrics. A 16 × 15 TOPSIS analysis was conducted to rank the models. The convergence plot and the ranking of the models indicated that data normalization plays a significant role in the prediction capability of a Bitcoin price predictor. This paper can significantly contribute to the research with a new normalization technique for utilization in varied fields of research. It can also contribute to international finance as a decision-making tool for different investors as well as stakeholders for Bitcoin pricing.
Our work presents two fundamental contributions. On the application side, we tackle the challenging problem of predicting day-ahead crypto-currency prices. On the methodological side, a new dynamical modeling approach is proposed. Our approach keeps the probabilistic formulation of the state-space model, which provides uncertainty quantification on the estimates, and the function approximation ability of deep neural networks. We call the proposed approach the deep state-space model. The experiments are carried out on established cryptocurrencies (obtained from Yahoo Finance). The goal of the work has been to predict the price for the next day. Benchmarking has been done with both state-of-the-art and classical dynamical modeling techniques. Results show that the proposed approach yields the best overall results in terms of accuracy.
This paper is based on current market prediction value of cryptocurrency Bitcoin.The Bitcoins is like shareable as piece of cakes but whenever, the price of Bitcoins will be fluctuated more.It will be difficult to predict.This problem discussed is based on computer science and delivered through Bitcoin market frequence, project using machine learning high level computer concept and predict highly closed accurate Bitcoin price using various real world dataset.
Experimental data was obtained from Bitcoin public library made available at https://www.investing.com/crypto/bitcoin/historical-dat which has attributes: open price, high, low and closes as shown in Table 1.
RENATO MOTA RUIZ, INACIO HENRIQUE YANO, Alexandre de Castro, Julio Cezar Souza Vasconcelos
This work aims to evaluate the storing and retrieving data response time using an Ethereum Smart Contract application to verify the feasibility of its utilization instead of using relational databases in web application development. To achieve the objectives of this work. There was a comparison between the Ethereum Smart Contract and the SQLite, considering response time as the user experience for database choice decisions in future application development. This study consisted of the development of two similar applications. The first one was the Ethereum Smart Contract Application, and the other was the SQLite Application. Using these applications to build graphs of response time behavior as the number of records processed grows. For storing data Blockchain application was much faster than the SQLite application. When retrieving data, the Blockchain application usually starts slower but finishes faster than the SQLite application. Blockchain is a recent technology for secure data storage in a distributed architecture. The hypothesis to be checked was if it also has a good response time compared with other databases. The contribution of this work is to provide information about the efficiency and possible user satisfaction of Blockchain applications.
Open access
Neural Networks and Applications
Advanced Computational Techniques and Applications
When trying to fit a deep neural network (DNN) to a $G$-invariant target function with $G$ a group, it only makes sense to constrain the DNN to be $G$-invariant as well. However, there can be many different ways to do this, thus raising the problem of ``$G$-invariant neural architecture design'': What is the optimal $G$-invariant architecture for a given problem? Before we can consider the optimization problem itself, we must understand the search space, the architectures in it, and how they relate to one another. In this paper, we take a first step towards this goal; we prove a theorem that gives a classification of all $G$-invariant single-hidden-layer or ``shallow'' neural network ($G$-SNN) architectures with ReLU activation for any finite orthogonal group $G$, and we prove a second theorem that characterizes the inclusion maps or ``network morphisms'' between the architectures that can be leveraged during neural architecture search (NAS). The proof is based on a correspondence of every $G$-SNN to a signed permutation representation of $G$ acting on the hidden neurons; the classification is equivalently given in terms of the first cohomology classes of $G$, thus admitting a topological interpretation. The $G$-SNN architectures corresponding to nontrivial cohomology classes have, to our knowledge, never been explicitly identified in the literature previously. Using a code implementation, we enumerate the $G$-SNN architectures for some example groups $G$ and visualize their structure. Finally, we prove that architectures corresponding to inequivalent cohomology classes coincide in function space only when their weight matrices are zero, and we discuss the implications of this for NAS.
We demonstrate a novel application of online transfer learning for a digital assets trading agent. This agent uses a powerful feature space representation in the form of an echo state network, the output of which is made available to a direct, recurrent reinforcement learning agent. The agent learns to trade the XBTUSD (Bitcoin versus US Dollars) perpetual swap derivatives contract on BitMEX on an intraday basis. By learning from the multiple sources of impact on the quadratic risk-adjusted utility that it seeks to maximise, the agent avoids excessive over-trading, captures a funding profit, and can predict the market's direction. Overall, our crypto agent realises a total return of 350\%, net of transaction costs, over roughly five years, 71\% of which is down to funding profit. The annualised information ratio that it achieves is 1.46.