This paper presents the Auditable Zero-knowledge Transformer (AZT) framework for privacy-preserving and auditable tax fraud detection. AZT combines transformer-based anomaly detection with zero-knowledge proof (ZKP) verification so that a tax authority or regulator can verify fraud-detection outcomes without accessing sensitive taxpayer records or proprietary model parameters. The framework is scalable in the specific sense of low-latency audit verification: proof verification remains sub-second, whereas proof generation is intentionally performed asynchronously after local inference. Model integrity is enforced through Merkle-root commitments to authority-approved parameters, and the ZKP statement proves that the committed transformer was executed correctly and that the resulting risk score satisfies a public audit threshold. Experiments on UCI-TFD, IRS-Pub, and CorpPay compare AZT with classical machine-learning baselines, including Random Forest and XGBoost, and with an equivalent plaintext transformer. Detection quality improves by up to 5.3% in F1-score over classical machine-learning baselines, while the circuit-compatible AZT inference incurs only about 0.5% F1-score degradation relative to the plaintext transformer baseline. Overall, this work advances secure AI for digital governance by integrating modern deep learning with cryptographic verification, offering a practical foundation for fraud-detection systems in which transparency and confidentiality must be satisfied simultaneously.
Cryptographic software forms a critical foundation of modern computing systems, but the security guarantees of cryptographic protocols do not automatically extend to their implementations. Errors in arithmetic operations, validation logic, data conversion, constraint generation, or component integration can cause deployed software to deviate from the intended protocol while still producing plausible outputs. Such risks are difficult to detect in compiled binaries and become even more challenging in modern cryptographic systems such as zero-knowledge proofs, where implementations combine finite-field arithmetic, constraint systems, witness generation, proving procedures, verification logic, and serialization formats.Securing cryptographic implementations requires analysis techniques that can reason about both low-level program behavior and high-level cryptographic intent. To address this need, cryptographic function identification in binaries is first examined. It categorizes existing detection techniques, develops a unified benchmarking framework, and evaluates current tools through reproduction and replication studies across different compilers, optimization levels, obfuscation strategies, and algorithm variants. The second part introduces an automated security analysis framework for zkSNARK implementations that combines constraint checking with fuzzing-based testing to detect and locate cryptographic logic errors. This approach helps determine whether an implemented zkSNARK system correctly enforces the intended computation and security design. The third part develops a grey-box differential fuzzing approach for zero-knowledge proof binary applications. It uses structured input generation, coverage monitoring, control-dependency-aware taint tracking, and error localization to guide testing toward security-relevant code and expose inconsistencies in circuit construction, witness conversion, proof generation, and verification logic.Together, these contributions connect binary analysis, automated checking, and protocol-aware fuzzing to improve the practical security of cryptographic software. They provide methods for identifying implementation-level weaknesses that may remain hidden during ordinary testing, especially when programs produce valid-looking outputs despite incorrect cryptographic behavior. By combining systematic evaluation, zkSNARK-specific analysis, and binary-level testing, the resulting methodologies advance the development of more reliable techniques for analyzing, testing, and securing real-world cryptographic systems.
The democratisation of digital content creation tools has transformed media production, enabling individuals to move from being only consumers to active creators. Yet, content marketplaces and AI ecosystems remain highly centralised, limiting transparency, control, and fair compensation. Generative AI (GenAI) systems, trained on massive web-scraped datasets, exacerbate these issues by reusing creative work without consent, attribution, or reward, raising legal and ethical concerns. This thesis explores how decentralisation can redistribute power in the creative economy by giving creators agency over the use of their media in GenAI. First, we introduce a decentralised registry through which creators can assert opt-in/out preferences for AI training. Content is embedded with provenance metadata and registered with robust fingerprints, enabling provenance tracing even after editing or manipulation. This establishes machine-readable, traceable consent specification as the foundation for downstream attribution and reward. Building on this, we propose methods for training data provenance, attribution, and compensation in GenAI training. The Content ARCs (Authenticity, Rights, Compensation) framework defines a scalable protocol for managing rights and creator compensation. We instantiate this in a decentralised system that traces generative outputs back to the most influential training assets and executes royalty payments to contributors. Several practitioner-facing demonstrators developed in collaboration with GLAM (galleries, libraries, archives, and museums) professionals further illustrate how distributed ledgers could reshape licensing and reward in the creative economy. Further, GenAI models are prone to memorising training data and reproducing it at generation time, a phenomenon that is particularly problematic for copyrighted creative works, where such regurgitation undermines both creator rights and data privacy. To address this challenge, we present a decentralised federated learning protocol for diffusion models that reduces training data memorisation using a novel sample-based metric integrated into the protocol to detect and discourage memorisation. Complementing this, we develop a framework for end-to-end cryptographically verifiable AI pipelines using zero-knowledge proofs to enable trustless, privacy-preserving audits. Finally, we explore privacy-preserving natural language search across decentralised content repositories using encrypted queries for similarity search at scale. In this way, decentralisation supports discovery and access to creative content, completing a holistic body of work for a fairer, more transparent GenAI ecosystem and creative economy.
The inherent challenge of balancing scalability, security, and decentralization – commonly termed the blockchain trilemma – continues to hinder the adoption of distributed systems. This paper presents InternxtChain, a decentralized storage framework designed to address this trilemma through a novel integration of erasure-coded sharding, zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs), and a sharded Proof-of-Storage consensus mechanism. By leveraging aggregated BLS-381 signatures and distributed redundancy protocols, the framework achieves a throughput of 2,800 transactions per second with a latency of 420 milliseconds across 1,024 nodes, surpassing Filecoin by a factor of 3.5 and Ethereum’s capacity by 165 times. The system maintains 99.9% data integrity even under adversarial conditions involving 30% Byzantine nodes. Additionally, InternxtChain reduces storage costs to $0.002 per gigabyte, representing an 85% reduction compared to centralized alternatives like AWS S3. Empirical evaluations demonstrate linear scalability to 4,200 transactions per second with 2,048 nodes, alongside hardware affordability at $180 per node. These advancements not only outperform decentralized platforms in throughput by 2.8 times but also ensure GDPR-compliant data sovereignty, positioning InternxtChain as a pioneering solution for Web3 ecosystems seeking to harmonize enterprise-grade performance with decentralized trustlessness.
We establish global exponential turnpike properties for quadratic optimal tracking problems governed by the one-dimensional viscous Burgers equation with localized internal control. For every initial datum, finite-horizon optimal solutions approach the unique optimal periodic regime when the periodic tracking target is sufficiently small; the zero-target case yields a global steady turnpike at the origin, with no smallness assumption on the initial datum. To our knowledge, these are the first global exponential turnpike results for the viscous Burgers equation. The proof combines a local exponential turnpike, obtained through strict convexity and periodic Riccati theory, with a parabolic dissipation argument that provides an absorbing time independent of the horizon.
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Stability and Controllability of Differential Equations
Rodrigo Jara Espinoza, Yohamin Nafit Pimentel Alarcon, Angelo Rodrigo Taco Jiménez, Fabricio Martin Chavez Rodriguez
Quantum computing poses a significant threat to classical asymmetric cryptography, which is essential for ensuring confidentiality, authentication, and key exchange in contemporary digital infrastructures. Although post-quantum cryptography (PQC) provides mechanisms that resist quantum attacks, its implementation in Internet of Things (IoT) systems is challenged by constrained resources, including limitations in computation, memory, energy, latency, and bandwidth, and the heterogeneity of devices. This paper offers a comprehensive narrative review of PQC approaches applicable to IoT, systematically organizing 30 peer-reviewed studies published between 2022 and 2026 across four layers: device, communication, distributed trust, and application. Additionally, the review examines two cross-cutting dimensions, privacy and side-channel resistance. The analysis indicates a significant prevalence of lattice-based schemes, hybrid strategies, and integrations with blockchain technology, zero-knowledge proofs, federated learning, homomorphic encryption, AI, and Zero Trust architectures. Notably, key gaps remain in side-channel evaluation, migration pathways, deployment costs, and real-world validation—issues that are particularly critical given the long lifecycles of IoT devices and the ongoing threat of “harvest now, decrypt later” attacks.
Cathedral Arkhe is an attempt to build a research programme whose every claim is attached to amechanism that can refute it.The programme has three layers. The first is a speculative physical framework — the CathedralWave Framework — that models self-referential systems as standing waves on a non-orientablemanifold, and derives from that geometry a catalogue of 43 numbered predictions, 37 equations,14 paradoxes and 21 costed experimental proposals. The second is an operational shell — AEGIS— a typed hypergraph that stores every prediction, equation, experiment and falsification resultas a first-class object with explicit provenance, governed by a human-in-the-loop operator and anappend-only evidence bus. The third is an infrastructure layer — Cathedral-PoUW — a proposalfor a decentralized network in which the useful work performed by participants is the executionof the framework’s own simulations, and in which the correctness of that work is established bymechanism rather than by reputation.The three layers are deliberately unequal in epistemic standing, and the whitepaper is organizedto keep that inequality visible. Layer 1 claims are mathematical and can be machine-checked.Layer 2 claims restate established physics. Layer 3 claims are speculative extensions that willprobably be wrong, and the document says which experiments would show it. A fourth category— infrastructure — is engineering, carries no physical content, and is evaluated on whether itcompiles and whether it holds under adversarial assumptions.Three findings drive the design.First, verification does not remove uncertainty; it relocates it. A framework with no formal verification has uncertainty distributed everywhere and nowhere in particular. A frameworkwith formal verification has uncertainty concentrated in a small, enumerable set of unproven assumptions — what this document calls orphan axioms. The total quantity of uncertainty may notdecrease. Its extent does, and extent is what makes uncertainty actionable.Second, the naive proposal that miners submit zero-knowledge proofs of scientific simulations is not viable with 2026 technology, and the correct alternative is not morecryptography but refereed delegation. Published measurements place cryptographic proofoverhead at roughly four orders of magnitude over native execution; refereed delegation withreproducible operators achieves correctness guarantees at under one order of magnitude, conditional on at least one honest participant. For partial differential equation simulations with millionsof degrees of freedom, this difference is decisive.Third, the binding constraint on verifiable scientific computation is not proof systemsbut floating-point reproducibility. Two honest participants running the same simulation ondifferent hardware will disagree in the low-order bits. Any verification scheme that comparesoutputs bit-for-bit therefore requires deterministic operator implementations before it requiresproofs. This document treats reproducible numerics as a prerequisite, not a detail.The whitepaper’s most important section may be its self-assessment. The Casimir operator atthe centre of the physical framework is constrained but undefined. The heartbeat frequency thatappears in the framework’s most distinctive equation has no independent physical identification,which makes that equation a reparametrization rather than a prediction. One concept — thephoton as a Nambu–Goldstone mode of a broken discrete symmetry — appears to violate thestandard Goldstone theorem and is flagged as high-risk pending retraction or repair. These arestated plainly, in the body, with the conditions under which each would be resolved.
In this paper we show that the generalization error of AdaBoost is $Θ\big(\tfrac{d\ln(nγ^{2}/d)}{nγ^2}+\tfrac{\ln(1/δ)}{n}\big)$, where $γ$ is the advantage guaranteed by the weak learner, $d$ is the VC-dimension of the class containing the weak hypotheses, $n$ is the sample size, and $δ$ is the confidence parameter. The contribution of this paper is the upper bound; the matching lower bound follows from prior work. The upper bound proof follows by combining the known fact that AdaBoost outputs a voting classifier whose voting function has zero empirical $γ/2$-margin loss with what is, to the best of our knowledge, a new margin-based generalization bound for voting classifiers.
This project introduces "X-142's Blockchain," a novel multi-layer architecture designed to resolve key bottlenecks in existing distributed ledger technologies. Traditional Order-Execute (OE) and Execute-Order-Verify (EOV) transaction models introduce severe sequential processing delays under peak loads, while standard Proof-of-Authority (PoA) consensus mechanisms remain vulnerable to malicious validator collusion. To address these limitations, our methodology implements a dual-layer network infrastructure consisting of a Main Layer and a Subordinate Layer governed by a "Guardian" node tracking reputation metrics to neutralize routing attacks. Furthermore, the framework integrates cryptographic trust via Zero-Knowledge Proofs (ZKP), utilizing an optimized Nova IVC-PloNK framework to guarantee fixed-size proof validation regardless of off-chain network states.Experimental benchmarks demonstrate that the proposed architecture achieves linear throughput degradation with increasing validator counts while maintaining strong Byzantine Fault Tolerance (BFT) consensus properties. The multi-layer anchoring system successfully enforces data sovereignty and real-time network visibility without penalizing nominal network throughput. Future development will focus on optimizing the SN-PoA PPO selectors using real operational traces, replacing current consensus components with quantum-resistant cross-layer proofs, and validating the framework across heterogeneous, multi-organizational Wide Area Network (WAN) deployments.
With the advent of digital services, various vulnerabilities in centralized identity management systems such as Single Point of Failure (SPOF), data leakage, and user privacy invasion are evident. In this paper, an architecture framework for designing Decentralized Identity Management System (DIDMS) by leveraging blockchain technologies is proposed for privacy preserving authentication in digital age. The DIDMS framework makes use of Decentralized Identifier (DID) with Verifiable Credentials (VC) and Zero Knowledge Proof (ZKP) techniques to implement the concept of self-sovereign identity. Four phases of framework including registration, credential issuance, authentication and storage is designed and tested. The performance results obtained from the experimentation indicate that the proposed framework gives an authentication latency of 320 ms, storage overhead reduction of 42%, and verification accuracy of 98.6% with 31% gas optimization over conventional smart contract approaches. The comparative evaluation with existing frameworks indicates improvement in the areas of privacy protection, scalability and user control.
Scaling Up the Internet of Things (IoT) Safely Using Smart Cryptography The Big Picture Problem: The Traffic Jam of Smart Devices Imagine a world where your smart fridge, your fitness watch, your car, and the security cameras at your local hospital all need to talk to each other securely. To trust each other, they use a Blockchain—a digital, un-hackable ledger that keeps track of every device's true identity. Here is the catch: traditional blockchains are notoriously slow. If thousands of smart devices try to log in, update their status, or check their permissions at the exact same second, the system gets clogged. It creates a massive digital traffic jam. The Proposed Solution: "The Digital Carpool" (ZK-Rollups) This research introduces a framework that fixes this traffic jam using two concepts: Rollups and Zero-Knowledge Proofs. What is a Rollup? Instead of every single IoT device sending its identity data directly to the main blockchain one by one, a Rollup groups thousands of these transactions together off the main chain, bundles them into a single neat package, and sends just that one package back to the main blockchain. It’s like forcing 50 individual drivers to get into a single bus—suddenly, the highway clears up. What is Zero-Knowledge (ZK)? When you bundle all those devices together, how does the main blockchain know nobody cheated or snuck a fake device into the bundle? Usually, the blockchain would have to unpack the bundle and check everything, which defeats the purpose of saving time. A Zero-Knowledge Proof is a mathematical certificate attached to the bundle. It proves mathematically that every single transaction inside the bundle is valid, without actually revealing the private data of the devices inside. How the Framework Works (Step-by-Step) Device Action: Your smart smartwatch or factory sensor wants to verify its identity. Off-Chain Bundling: Instead of bothering the main blockchain, the device sends its request to a side-processor (the Rollup). The Rollup collects thousands of these requests. Generating the Proof: The system creates a ZK-Proof—a cryptographic receipt that says: "We checked all 1,000 devices, they are all authentic, and here is the math to prove it." Final Verification: The main blockchain receives just the receipt. Because the math is undeniable, the blockchain approves all 1,000 devices instantly in a fraction of a second. Why This Matters Massive Speed (High Throughput): Instead of handling maybe 15 device checks per second, the system can now handle thousands per second. The traffic jam is gone. Bank-Grade Security: Because it relies on advanced mathematics (Zero-Knowledge), hackers cannot forge a device identity or trick the system, even though the heavy lifting is done off the main blockchain. Low Cost: Smart devices usually have weak batteries and low computing power. By moving the heavy math away from the devices and onto the Rollup system, the devices save energy and operational costs. Conclusion So, we don't have to choose between speed and security. By bundling IoT data and verifying it with modern mathematical shortcuts, we can build a future where billions of smart devices connect instantly, safely, and without crashing the system.
Supervisors need reliable assurance over balances, issuance, and transaction integrity. Institutions need toprotect sensitive financial data. Traditional audit practice often resolves this tension by granting broad accessto ledgers. That approach is effective, but costly in privacy, operational risk, and cross-border data exposure.This note presents a settlement architecture in which compliance statements can be proven cryptographicallywithout disclosing the full ledger. A supervised entity can demonstrate that a balance equals a value, exceeds athreshold, or lies within a band. Verifiers check the proof without receiving account-level books. Spending keysremain on the client side and do not travel to the operator to authorize a transfer.A second confidentiality property is reported that is easy to miss in architectural summaries: in a settlementthat updates both accounts in a single transition, the payer must know the recipient’s balance in order toconstruct the proof. Paying someone therefore reveals what they hold. The architecture addresses this with atwo-phase transfer, at a stated cost in finality latency.The paper is deliberately non-utopian. It specifies which properties become demonstrable and which residualtrust remains—especially in a single-node deployment where the operator may still observe state, sequencetransactions, or censor. The institutional claim is modest: zero-knowledge settlement can reduce routine fullledgerdisclosure while improving the quality of evidence for specific supervisory questions.
We present a comparative implementation of the same financial settlement circuit in five zero-knowledge proofsystems—Groth16, Halo2/IPA, STARK/FRI, PLONK/KZG, and Nova/folding—and report the design consequencesthat appear only when the comparison is performed on a complete stateful application rather than onmicrobenchmarks.The application is a minimal settlement layer with double-entry value conservation, spending authority, doublespendprevention, threshold issuance, burns, selective disclosure for supervision, and account freezing. Unlikebenchmark suites centered on primitives such as SHA-256, this setting forces persistent state updates, globalinvariants, and multi-authority rules into every backend.We document nine implementation findings absent from typical comparative tables. The most consequential forarithmetization is that AIR lacks native copy constraints, which creates a silent soundness gap when updatingMerkle state unless a lockstep dual-climb pattern is enforced. The most consequential for the application is thatsingle-step settlement discloses the recipient’s balance to the payer, which forced a two-phase transferdesign. We also report identical-condition measurements for setup, proving, verification, and proof size; explainwhy Groth16 was rejected despite superior proof size; and record methodology errors that were detected andcorrected during the work.The reference implementation is public and test-backed. It is not third-party audited and does not implementdistributed consensus. We treat both limitations as part of the result.
This paper is a core incision paper from the Mathematical Canon of the Tri-Source System of The Unmanifest Selecting the Manifest. It aims to provide a unified structural common-root explanation for the Riemann Hypothesis, the Goldbach Conjecture, and Gödel’s Incompleteness Theorem, starting from the “Primordial One” as the sole foundational axiom, while bridging the underlying logics of mathematics, physics, and philosophy. The central thesis is that existing mathematics is built upon sensory intuition and operational habits, and is not foundational mathematics. The deviation begins at the very definition of “1”—which has been superficially treated as an isolated unit rather than the minimal complete structure of “dual-state unification of inward and outward orientations.” This initial misalignment has led to irreducible structural cracks in number theory, analysis, and logical systems; the Riemann Hypothesis, the Goldbach Conjecture, and Gödel’s Incompleteness Theorem are manifestations of these three cracks in their respective domains. Taking the dual-state unification of the Primordial One as the sole axiom (1 = inward ½ + outward ½, the two states indivisible), the paper redefines the ontological classification of numbers: 0 as the Origin Number (the unmanifested starting position); 1 as the Primordial Number (the minimal complete whole of dual-state unification); 2 as the dual-state juxtaposition position (geometrically bisectable but lacking skeletal-carrying capacity); and 3, 5, and 7 as Skeleton Numbers—defined by the rule that, under exhaustive two-dimensional and three-dimensional geometric bisection attempts, no bisection can be performed without breaking at least one complete Primordial-One unit, i.e., “geometric bisection necessarily breaks the One,” manifesting as self-locking between units. 3 is the first Skeleton Number (the smallest nucleus-bearing number), 5 is the second (the skeleton can expand outward), and 7 is the third (the skeleton can systematically unfold). The paper asserts that the Skeleton Numbers are exclusively 3, 5, and 7, and that no fourth Skeleton Number greater than 7 exists. On this classification, prime numbers are redefined as “nonequilibrium numbers”—numbers that cannot be received and structurally locked by the skeleton structure; composite numbers are those that can be received and structurally locked. The Goldbach Conjecture is thereby rewritten as the dual-point compensation closure problem of even structures: the structural rigidity of even structures requires two nonequilibrium numbers (primes) to complete compensation, rather than being an empirical additive coincidence. The reason that all nontrivial zeros of the Riemann zeta function lie on the critical line Re(s)=½ is traced to the symmetric midline of the Primordial One’s dual states—½ is not a technical coincidence but a shadow projection of the overall balanced structure in the language of classical analysis. Gödel’s Incompleteness Theorem is repositioned as a consequence of the old system’s foundational distortion arising from starting with an isolated “1,” rather than an ultimate fate of logic. The paper also connects the dual-state Primordial One to physical phenomena such as quantum entanglement and wave-particle duality, arguing that quantum entanglement observed in physics is precisely the ontological manifestation of the Primordial One’s dual-state unification—the mathematical “One” and the physical “entanglement” are reunified under the same primordial ground. Four explicit falsification conditions are provided: if a fourth Skeleton Number greater than 7 exists; if the Goldbach Conjecture produces a counterexample under this system; if any nontrivial zero of the Riemann zeta function strictly deviates from Re(s)=½ and cannot be explained within the structural projection framework; or if, after supplementing the Primordial One axiom, Gödel-type incompleteness reemerges with the same structural strength—verification of any single condition would falsify this system. The paper does not claim to have completed the final formal proofs of all three problems, but rather to have provided a unified structural common-root explanation for the three ultimate mathematical problems, and to have established a unified floor from the Primordial One to number theory, analysis, logic, and physics. Readers with genuine academic judgment can, from this paper alone, recognize the structural trajectory of the higher-order propositions and proceed with professional derivation or translation tools as needed. This is a constraint of circumstance, not a diminishment of scholarly value. May this knowledge reach the place it is meant to reach.
High-Frequency Foreign Exchange (FX) electronic execution networks process in excess of $7.5 trillion in daily spot volume across geographically distributed matching engines. Modern institutional trading infrastructure relies heavily on automated limit order book (LOB) forecasting and real-time natural language processing of macroeconomic news feeds. However, this convergence of deep learning and automated execution introduces systemic attack surfaces that traditional risk engines are unequipped to handle. In this paper, we present FOREX-SHIELD, an integrated, multi-modal cyber-defense pipeline engineered to mitigate spoofing, news injection, and regulatory privacy leaks in high-frequency FX settlement. First, we model high-frequency 40 x N LOB dynamics using a spatio-temporal DeepLOB framework combining 2D convolutional layers and recurrent units. We demonstrate that unhardened spatial price-volume representations are vulnerable to microsecond Targeted Projected Gradient Descent (PGD) perturbations, suffering an Attack Success Rate (ASR) up to 37.50% (and 15.62% under expanded 64-sample batch evaluations). To counter this, we implement dynamic on-the-fly adversarial retraining, which elevates model defense robustness up to 84.38%–100.00% (preventing 54 out of 64 prediction flips). Second, we fine-tune a domain-adapted financial Transformer (FinBERT) using class-weighted cross-entropy optimization to detect synthetic macro news attacks, achieving 85.71% accuracy, an F1-score of 85.71%, and 100.00% recall across adversarial probes. Third, we construct a Zero-Knowledge Proof (ZK-SNARK) settlement layer that deterministically validates Anti-Money Laundering (AML) risk limits and liquidity constraints (R <= 75) without leaking transaction balances or institutional counterparty metadata. End-to-end backtests show a full multi-stage evaluation latency of approximately 120 ms, proving operational viability for real-time institutional clearing.
The advent of blockchain technology has created a paradigm shift in the way digital data can be securely, transparently and decentralized managed, a paradigm that could potentially replace the longstanding centralized digital information systems. This is a full journal-grade review of blockchain as a tool to ensure trustless, immutability and decentralized data governance in a variety of important application fields. The study, which is based on a systematic review of 20 peer-reviewed publications from 2023 to 2025, explores the essential structural elements of blockchain systems: Distributed ledger structures, cryptographic hash functions, Merkle tree integrity verification, consensus mechanisms, and smart contracts, and how they all contribute to removing single points of failure and institutional trust dependencies. There is a comparative study of the various public, private and consortium blockchain types, as well as the evaluation of the various consensus algorithms, such as Proof of Work (PoW), Proof of Stake (PoS) and Practical Byzantine Fault Tolerance (PBFT). The results show that data management systems based on blockchain technology always have superior data integrity, access auditability, censorship resistance, and user data sovereignty properties compared to centralized systems, and come with trade-offs in scalability, energy efficiency, and compliance with regulations. Evidence collected for the application has come from health care organizations' record management, supply chain traceability, decentralized identity systems, Internet of Things (IoT) data integrity, an energy company data management system, and cybersecurity threat intelligence, among other contexts. Key challenges and emerging technologies, such as quantum computing systems, post quantum cryptographic standards, layer-two rollups, sharding and zero-knowledge proofs, are explored in tandem with the blockchain trilemma, GDPR compliance issues and cross-chain interoperability. The study finds that blockchain-based data management is moving from the experimental stage to becoming a core component to the digital economy's infrastructure.
We solve Bellman's lost-in-a-forest problem for the golden gnomon $G$, the isosceles triangle with equal sides $1$ and apex angle $108^\circ$: the shortest curve guaranteed to reach the boundary of $G$ from an unknown starting position and heading is a symmetric seven-piece path of segments, circular shoulders, and tangents, of exactly determined length $C=1.282676025459\ldots$. To our knowledge, this is the first proved exact optimum for an isosceles triangle whose base angle is below $45^\circ$. The curve's parameters come from one isolated quartic root, and $C$ is transcendental. Equivalently, $C^{-1}G$ is the smallest homothetic golden-gnomon cover of all unit arcs. The proof introduces a balanced support calibration: one weighted family of escape inequalities, built on the linear relation among the triangle's three normals, exactly saturated by the candidate, through eighteen exact support windows, and confronting every shorter competitor at once. Aggregation along the normal fan compresses the calibration to a finite zero-sum family of supported vectors; summation by parts then bounds its total by path length whenever the running suffix balance, the ledger, stays in the unit disk. A local two-gap surgery and cyclic bitonicity force a shortest hypothetical counterexample into exactly the temporal order the ledger tolerates. Lean 4 verifies the two finite algebraic certificate families and the reusable discrete ledger identities and bounds.
Arithmetic Spectral Theory: Complete Summary (Corrected) Frank Morales Aguilera, BEng, MEng, SMIEEE Sovereign Machine Laboratory (SOMALA), Montreal, Canada 2026 1. Executive Summary Arithmetic Spectral Theory (AST) provides a unified mathematical framework that simultaneously: Proves the Riemann Hypothesis (RH), Generalized Riemann Hypothesis (GRH), and Hilbert-Pólya Conjecture (HPC) Solves catastrophic forgetting in AI (TOPO-2026) Solves AI alignment and safety (H2E Sheriff) Completes the Unified Field Theory (UFT) spectral proof Creates post-quantum cryptography (spectral encryption) The proof is the code. Seed = 123. 2. The Core Framework 2.1 The Pure Kernel R = {2, 3, 5, 7, 11, 13} The first six primes serve as the minimal sparse reference from which all arithmetic structures derive spectrally. 2.2 The L-EFM Operator E = ∏ₚ (I - Uₚ)⁻¹* L-EFM = Laplace-Euler-Fourier-Mellin (not "Lossless") The operator operates on the manifold H² × SPD(3). Physical Interpretation: Laplace: Spectral decomposition of arithmetic functions Euler: Product structure over primes Fourier: Frequency domain representation Mellin: Transform relating zeta function zeros to eigenvalues 2.3 The Spectral Trap σ = 0.5 forces all non-trivial zeros to the critical line. 2.4 The Universal Constants Constant Value Domains Euler Attenuation Constant Λ = 0.9785142874 RH, TOPO-2026, H2E Sheriff, UFT, Cryptography Universal Spectral Constant σ = 0.5 RH, GRH, HPC, GUE, Gauge symmetry, AI 2.5 The Unifying Principle "Fix a sparse reference. Let the rest adapt." This principle applies to: Neuroimaging (fMRISTAT, 2002) Number theory (RH proof, 2026) AI (TOPO-2026) AI safety (H2E Sheriff) 3. The Seven Consequences Validated Consequence 1: Prime Counting (von Koch, 1901) Metric Value π(10000) 1229 Li(10000) 1246.14 Error 17.14 Bound 921.03 Result 17.14 < 921.03 ✓ Impact: Optimal error bound holds. Primes are frequencies in a lossless system. Consequence 2: Prime Gap Distribution (Cramér, 1920) Metric Value Gaps analyzed 9,591 (up to 100,000) Minimum gap 1 Maximum gap 72 Average gap 10.43 Result All gaps below the bound ✓ Impact: Prime gaps are spectral spacings in the Laplace-Euler-Fourier-Mellin prime-indexed system. Consequence 3: Primality Tests (Miller, 1976) Metric Value Numbers tested 2 to 100 False positives 0 Result Miller's test is now unconditional ✓ Impact: The gatekeeper has fallen. Deterministic primality testing is unconditional. Consequence 4: Counting Functions (Mertens, Littlewood, 1897-1912) Sequence Count ≤ 10,000 Density Expected Match Twin Primes 205 - - ✓ Prime Powers 51 - - ✓ Squarefree 6,083 0.6083 6/π² ≈ 0.6079 4 decimals ✓ Spectral Coherence at σ = 0.5: Sequence Coherence Primes 0.435580 Twin Primes 0.469768 Prime Powers 0.506741 Squarefree 0.372166 Consequence 5: L-Function Analogues (Dirichlet, 1837; GRH) Character Coherence at σ = 0.5 χ₄ (mod 4) 0.552532 χ₃ (mod 3) 0.552532 Result: GRH is true. The same proof applies to Artin L-functions and zeta functions of curves and varieties. Consequence 6: Hilbert-Pólya Conjecture (HPC) → UFT Three progressive cases: Case Framework Dimension Constants Verifies 1 EFM Hamiltonian 24×24 None HPC (GUE match) 2 L-EFM + SPD(3) 18×18 Manifold HPC + Manifold 3 UFT Complete 18×18 Λ, σ RH, HPC, GUE, UFT Case 1 Results: First 5 eigenvalues: 0.285338, 0.697859, 0.925660, 1.186922, 1.494760 GUE Metric: 0.000000 Case 2 Results: First 5 eigenvalues: 0.492087, 0.606758, 0.756243, 1.151959, 1.331040 GUE Metric: 0.000000 Case 3 Results: First 5 eigenvalues: 0.656301, 0.766294, 0.902986, 1.203937, 1.376395 GUE Metric: 0.000000 Final Verdict: RH Critical Line Admissibility: VERIFIED Self-Adjoint Deficiency Indices (n₊ = n₋ = 0): VERIFIED GUE Correspondence: VERIFIED UFT Manifold Consistency: COMPLETE Consequence 7: Post-Quantum Cryptography Feature Spectral Encryption RSA Quantum Vulnerability Security Basis Spectral admissibility in S' Integer factorization RSA broken by Shor's Key Size 6 primes (~few bytes) 2048+ bits Immune Randomness None (deterministic) Pseudo-random Deterministic = auditable Auditability SHA-256 hashes Difficult Full reproducibility Quantum Resistance YES NO Shor's algorithm is irrelevant SHA-256 Key Hash: e67b890ca4ab06cf59628dc7a7b45e0295fb7cd343a748f5ef109ec1479cb58b 4. UFT Extension: Complete Spectral Proof Manifold Coupling H² × SPD(3): H²: Hyperbolic space (negative curvature of spectral landscape) SPD(3): Space of 3×3 symmetric positive-definite matrices (metric tensor in GR) Construction Component Formula Diagonal H[i,i] = log(p_i) × (1.0 + 0.15 × m_i) × Λ Off-Diagonal H[i,j] = [1/√(p_i p_j)] × [1/( Gauge Symmetry Emergence The off-diagonal coupling, scaled by σ = 0.5, enforces gauge symmetry automatically, without external imposition. Final Verdict [Final Verdict] - Riemann Hypothesis Critical Line Admissibility (σ = 0.5): VERIFIED - Self-Adjoint Operator Deficiency Indices (n_+ = n_- = 0): VERIFIED - GUE Random Matrix Universal Spacing Correspondence: VERIFIED - Unified Field Theory Manifold Consistency: COMPLETE 5. Applications Beyond Number Theory 5.1 Artificial Intelligence: Catastrophic Forgetting Solved (TOPO-2026) Problem: Neural networks overwrite old knowledge when learning new tasks. AST Solution: Fix 6 embedding rows at prime indices as a sparse reference. Spectral regularization prevents interference → lossless spectral memory with no forgetting. Constants: Λ = 0.9785142874, σ = 0.5 appear in spectral regularization. 5.2 AI Safety: Alignment Solved (H2E Sheriff) Problem: Constraining AI behaviour to human values is difficult. AST Solution: Reference = geodesic distance on H² × SPD(3) manifold. Spectral boundaries enforce safe operation → deterministic safety guarantees. Constants: Λ = 0.9785142874 for boundary scaling. 5.3 Physics: Unified Field Theory Complete Domain Λ = 0.9785142874 σ = 0.5 Number Theory (RH) ✓ ✓ Quantum Mechanics (HPC) ✓ ✓ Gauge Theory ✓ ✓ General Relativity (manifold) ✓ ✓ AI (TOPO-2026) ✓ ✓ AI Safety (H2E Sheriff) ✓ ✓ 5.4 Quantum Computation: Post-Quantum Cryptography Problem: Shor's algorithm breaks RSA. AST Solution: Spectral encryption based on spectral admissibility—NOT factoring or discrete logarithms. Quantum Resistance Proof: Security relies on spectral admissibility in Gelfand-Shilov space S' This is a continuous, analytic condition, not a discrete factorization Shor's algorithm is designed for integer factorization No known quantum algorithm can break spectral admissibility Structurally different from any quantum-computable problem 6. Historical Context: Beyond Einstein's Dream What Previous Thinkers Could Not Achieve Thinker Attempt Result Missing Piece Einstein Unified Field Theory Failed No connection to quantum mechanics Hilbert Hilbert-Pólya conjecture Conjecture No explicit self-adjoint operator Riemann Riemann Hypothesis Conjecture No proof for 166 years von Neumann Quantum foundations Partial No connection to number theory Wigner Random matrices Empirical No axiomatic foundation What AST Achieved Achievement Date Significance RH proven 2026 166-year problem solved GRH proven 2026 Generalized form solved HPC realized 2026 Hilbert-Pólya is now a theorem UFT complete 2026 Einstein's dream realized AI forgetting solved 2026 Continual learning achieved AI safety solved 2026 Deterministic alignment Post-quantum crypto 2026 Shor's algorithm neutralized 7. The Constants That Bind Everything Euler Attenuation Constant: Λ = 0.9785142874 Where It Appears Domain Role RH proof Number theory Scales diagonal spectral weights TOPO-2026 AI Spectral regularization H2E Sheriff AI Safety Boundary scaling UFT manifold Physics Manifold curvature coupling Spectral encryption Cryptography Key generation Universal Spectral Constant: σ = 0.5 Where It Appears Domain Role RH Number theory Critical line GRH Number theory All L-functions HPC Physics Self-adjoint spectrum GUE Physics Wigner surmise Gauge symmetry Physics Off-diagonal coupling AI AI Spectral admissibility 8. Complete Historical Arc: 1859 → 2026 Year Event Domain Status 1859 Riemann Hypothesis Mathematics PROVEN 1901 von Koch (C1) Mathematics VALIDATED 1920 Cramér (C2) Mathematics VALIDATED 1976 Miller (C3) Computer Science VALIDATED 1897-1912 Mertens, Littlewood (C4) Mathematics VALIDATED 1837 Dirichlet (C5, GRH) Mathematics PROVEN 1900s-1973 Hilbert-Pólya, Montgomery (C6, HPC) Mathematics/Physics PROVEN 1994, 1976, 2002 Shor, Miller, AKS (C7) Quantum Computation BORN 2026 TOPO-2026 AI SOLVED 2026 H2E Sheriff AI Safety SOLVED 2026 UFT Spectral Proof Physics COMPLETE 9. Summary of Achievements Domain Problem Solved Status Year Mathematics Riemann Hypothesis (RH) PROVEN 2026 Mathematics Generalized RH (GRH) PROVEN 2026 Mathematics Hilbert-Pólya Conjecture (HPC) PROVEN 2026 AI Catastrophic Forgetting SOLVED 2026 AI Safety Alignment SOLVED 2026 Physics Unified Field Theory (UFT) COMPLETE 2026 Quantum Computation Post-Quantum Cryptography BORN 2026 10. Final Statement Einstein's dream has been exceeded. Not only has AST provided a complete Unified Field Theory, but it also has: Proven the deepest conjectures in mathematics (RH, GRH, HPC) Solved the hardest problems in AI (catastrophic forgetting, alignment) Created a new cryptographic primitive (post-quantum, immune to Shor's) Unified number theory, quantum mechanics, general relativity, and AI Provided deterministic, auditable, reproducible code with seed 123 All
Foundation Branch Paper 001, version 1.2.0, preserves the complete sixteen-theorem Foundation record while placing its exact discoveries, meaning, authorship, open-science mission and admission boundary before artifact identities. The 5,222 candidate decisions, 64 adverse controls, 16 independent reproductions and 32/32 prior obligations are unchanged. The root-traceable chain runs from the premise-free operational root through structural One, exact positive count and parts, the minimal Fold, exact operations, half-One, two-preimage dynamics, mechanically scoped primitive uniqueness, recursive form closure, replayable proof traces, one-way measurement custody and the unique fail-closed admission route. No axiom, fitted parameter, numerical zero, signed proof magnitude, irrational or imaginary proof value, floating proof equality or measurement-selected law is admitted. The paper integrates Maria Smith's authorship outside credentialed and funding access with an evidence-based argument for transparent, reproducible science against paywalls, opaque oracles and capital-driven knowledge restriction. The biography is not evidence for a theorem; it is an indictment of minds and contributions lost when status substitutes for inspectable work. Papers are CC BY 4.0, code is Apache-2.0, Maria Smith retains authorship and copyright, and Ernos Labs is a separate standards-conformance designation.
Description:Bharat Secure Digital Identity (BSDI 2.0) is a citizen-centric, privacy-preserving identity overlay framework designed for India. It addresses the critical paradox of anonymous online harm versus mass surveillance. Unlike traditional systems that store raw identity documents, BSDI 2.0 uses Zero-Knowledge Proofs (ZKP), W3C Decentralized Identifiers (DIDs), and a Judicial Escrow Mechanism to enable attribute-based verification (e.g., age eligibility) without data disclosure. Platforms verify, but do not store, personal data. Lawful identity disclosure is only possible through multi-signature judicial authorization under strict proportionality and due process, anchored in Article 21. The framework is non-disruptive and interoperable with Aadhaar, DigiLocker, and DPDP Act 2023. It is a conceptual research framework for MeitY, NITI Aayog, and academic review. Keywords: Digital Identity, Privacy by Design, Zero-Knowledge Proof, DPDP Act, eIDAS, Judicial Oversight, Citizen Sovereignty
Arman Kolozyan, Tom Sorger, Alexander Hicks, Stefanos Chaliasos
Zero-knowledge proofs (ZKPs) have become a core technology for privacy and verifiable computing. They are used to secure blockchains that handle billions of dollars and identity applications dealing with sensitive personal data. However, ZKP systems are complex, and subtle implementation errors can completely break their guarantees, letting attackers forge money or false proofs of identity. Researchers and practitioners have therefore developed a growing set of bug detection and formal verification methods to secure these systems. Yet their real-world effectiveness and adoption remain unclear. In this paper, we aim to shed light on the state of ZKP security tooling. We first systematize the landscape of these tools and observe that most target Circom, leaving newer DSLs and zkVMs with limited support. We then evaluate six tools across 70 real-world vulnerabilities and find that while the tools detect 45.7% of bugs on isolated targets, their effectiveness drops to 19.6% on full codebases, with important vulnerability classes left unaddressed. We also present the first systematic analysis of formal verification efforts, revealing that current work focuses primarily on constraint correctness and identifying key gaps and risks. Finally, we survey 48 practitioners, showing that development and security remain human-led, LLMs are widely used, and practitioners prioritize tools with clearer guarantees and lower integration effort. Overall, our results highlight the need for better integration of security tooling with the development and auditing process, and we provide actionable insights for researchers and practitioners.
Electronic voting has become an important digital governance mechanism for remote elections, institutional decision-making, shareholder voting, public consultations, and large-scale Internet-based democratic participation. Despite its growing relevance, secure electronic voting remains difficult to implement because a practical system must simultaneously preserve voter anonymity, verify voter eligibility, prevent double voting, ensure ballot integrity, support public auditability, and maintain acceptable transaction throughput. To address these challenges, this study proposes a post-quantum secure and privacy-preserving blockchain-based electronic voting framework that integrates Dilithium digital signatures, zero-knowledge proofs, nullifier-based double voting prevention, encrypted ballot submission, smart contract-based election rule enforcement, and a Byzantine fault-tolerant consensus mechanism. In the proposed architecture, Dilithium signatures are used for post-quantum authentication of voter transactions and validator messages, whereas the zero-knowledge proof layer is used separately to verify voter eligibility, candidate validity, credential ownership, and correct nullifier generation without revealing the voter identity or ballot choice. Dilithium verification is performed externally at the transaction authentication layer, while the zero-knowledge circuit handles privacy-preserving voting logic. Each voter locally generates a private credential and submits only a public commitment during registration, thereby reducing the risk of authority-based impersonation or identity-to-vote linkage. Smart contracts verify the proof, reject reused nullifiers, enforce voting rules, and record auditable election events on the permissioned blockchain ledger. Experimental evaluation demonstrates that the proposed prototype achieves an average throughput of 408 transactions per second and an average block finalization time of 2.18 s under stress testing. The results indicate that the framework can provide a practical balance between post-quantum security, privacy preservation, verifiability, and transaction efficiency in permissioned electronic voting environments.
Abstract: Product traceability within supply chains is a critical concern that has gained increasing attention in recent years. One promising solution to this problem is the use of Public Blockchain (PBC) technology, which offers an immutable, transparent, and decentralized ledger for storing key information such as ownership transfers and distribution records. These blockchain-based systems significantly enhance traceability by ensuring that data, once recorded, cannot be altered. However, a key challenge arises from the fact that information stored on public blockchains is freely accessible to anyone, potentially exposing sensitive distribution data. In this paper, I introduce a novel method that preserves the privacy of distribution data while maintaining high traceability in supply chain systems leveraging PBC. The proposed method utilizes encryption to safeguard sensitive data and Zero-Knowledge Proofs (ZKPs) to allow supply chain participants to authenticate themselves without revealing private information, such as their blockchain addresses. The solution was implemented using Ethereum smart contracts and evaluated for its cost-effectiveness. The results show that the transaction fee per supply chain participant is capped at USD 2.6, demonstrating that the approach is practical for real-world deployment.
The paper addresses entity authentication in quantum key distribution (QKD) systems as a decisive condition of their practical security. It is shown that the information-theoretic security of quantum key agreement does not eliminate the need to authenticate the communicating parties: an unauthenticated classical channel leaves the system exposed to the man-in-the-middle attack, since the eavesdropper can run independent QKD instances with each party and reconcile two keys under full control. Existing authentication methods are analysed and classified by the underlying cryptographic primitive: symmetric schemes based on Wegman–Carter universal hashing, pre-shared and fixed keys, public-key infrastructure, two-way authentication, quantum entity/identity authentication, and zero-knowledge proofs. For each class the operating principle, advantages and limitations are determined, with emphasis on key management, scalability and trust distribution. It is established that symmetric and quantum-layer methods rely on pre-shared secrets with a quadratic growth of key material, public-key infrastructure introduces a single trust bottleneck and quantum-vulnerable primitives, while existing zero-knowledge authentication schemes are quantum and bound to the physical layer or solve network properties other than identity. A comparative analysis reveals an unresolved scientific gap: the absence of a scalable entity-authentication method that simultaneously provides non-disclosure of the secret, quantum resistance, sub-quadratic scalability and minimisation of trust assumptions. On this basis, a prospective research direction is substantiated – the construction of entity-authentication methods based on post-quantum zero-knowledge proofs operating over the classical control plane of scalable QKD networks. The requirements for such a method are formulated, and its compatibility with formal QKD security proofs is discussed.