This Volume VII explores quantum-enhanced and fractal-inspired approaches to mathematical discovery as an advanced frontier that extends beyond the classical neuro-symbolic hybrids examined in Volume III. Building on the series' commitment to preserving human insight while pursuing rigorous foundations, it examines two complementary lines of inquiry. First, it surveys quantum algorithms for search and optimisation over mathematical spaces—including Grover-type search, quantum walks, and variational quantum eigensolvers applied to proof or conjecture landscapes—and assesses their potential to accelerate exploration of vast, high-dimensional mathematical structures while identifying fundamental limitations arising from noise, decoherence, and the encoding problem. Second, it investigates fractal and self-similar architectures for conjecture generation, drawing on scale-invariant and recursive structures that appear across number theory, geometry, and dynamics, and considers how such architectures might generate conjectures with intrinsic multi-scale coherence. The volume then analyses the deeper epistemic implications of non-classical computation for notions of proof, surveyability, and understanding that have been central to the Leiden Declaration and previous volumes in this series. Also available on Academia.edu: https://www.academia.edu/attachments/133179980/download_file?s=portfolio
This Volume VII explores quantum-enhanced and fractal-inspired approaches to mathematical discovery as an advanced frontier that extends beyond the classical neuro-symbolic hybrids examined in Volume III. Building on the series' commitment to preserving human insight while pursuing rigorous foundations, it examines two complementary lines of inquiry. First, it surveys quantum algorithms for search and optimisation over mathematical spaces—including Grover-type search, quantum walks, and variational quantum eigensolvers applied to proof or conjecture landscapes—and assesses their potential to accelerate exploration of vast, high-dimensional mathematical structures while identifying fundamental limitations arising from noise, decoherence, and the encoding problem. Second, it investigates fractal and self-similar architectures for conjecture generation, drawing on scale-invariant and recursive structures that appear across number theory, geometry, and dynamics, and considers how such architectures might generate conjectures with intrinsic multi-scale coherence. The volume then analyses the deeper epistemic implications of non-classical computation for notions of proof, surveyability, and understanding that have been central to the Leiden Declaration and previous volumes in this series. Also available on Academia.edu: https://www.academia.edu/attachments/133179980/download_file?s=portfolio
This paper provides a rigorous academic review of the seminal work by Mamun and Grok xAI (2025) on advancing quantum matrix computations through the integration of geometric mean subroutines and Chebyshev-accelerated chain multiplications for nonlinear and iterative challenges in machine learning and quantum information. Following a thorough summary of its salient contributions, we compare and contrast the approach with contemporary research in quantum genetic algorithms (QGAs), variational quantum eigensolvers (VQEs), and tensor network methods. Strengths in polylogarithmic complexity and hybrid fault-tolerance are highlighted, alongside limitations in NISQ-era decoherence. Extending the framework, we propose novel integrations with fractal-recursive ontologies for non-integer dimensional simulations, the Principle of Co-Adaptation for bivariational dynamics in adaptive quantum circuits, and Souriau's axiomatic geometric model for positive-type state representations in high-dimensional entanglement. These extensions unify quadratic tensor structures with emergent fractal geometries, enabling scalable applications in quantum gravity simulations and autopoietic cognitive systems. Implications for post-quantum cryptography and geopolitical quantum strategies are discussed, advocating for interdisciplinary policies to harness transformative potential while mitigating risks. Also available on Academia.edu: https://www.academia.edu/attachments/131744773/download_file?s=portfolio
This paper employs fractal mathematics principles to forecast Q-Day—the threshold at which fault-tolerant quantum computers can practically compromise public-key cryptography—over an 18-month horizon from December 2025. Synthesizing insights from recent advancements in quantum ecosystems, algorithmic hybrids for nonlinear matrix operations, and interdisciplinary frameworks integrating quantum entanglement with fractal geometries in high-dimensional computing, the analysis models technological progress as self-similar iterative processes. Utilizing iterated function systems and fractional dimensions (approximately 1.618, akin to the golden ratio), the fractal model projects accelerated convergence toward utility-scale systems, predicting Q-Day by late 2027 with a 70% probability within the specified timeframe. Key drivers include the end of the Noisy Intermediate-Scale Quantum (NISQ) era, polylogarithmic speedups in quantum algorithms, and recursive self-improvement in autopoietic multi-agent systems. Implications encompass urgent transitions to post-quantum cryptography, geopolitical divergences in investment strategies, and existential risks from quantum-AI synergies, advocating for coordinated policies in commercialization, international collaboration, and ethical regulation to mitigate vulnerabilities while harnessing transformative potential in cybersecurity, healthcare, and environmental monitoring. Also available on Academia.edu: https://www.academia.edu/attachments/125804561/download_file?s=portfolio
This volume systematically reinterprets the security foundations of lattice-based post-quantum cryptography (PQC) for the theoretical post-post-quantum era through the mathematical lens of fractal geometry and fractal algebra. The analysis asserts that cryptographic hardness boundaries are not smooth Euclidean manifolds but complex fractal objects exhibiting scale invariance. Specifically, the research demonstrates four key fractal phenomena: 1) The distribution of viable short vectors in Learning With Errors (LWE) lattices possesses a non-integer box-counting dimension (typically KSCCN Mamun, SM (2026) ……………………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Fractal Geometry Interpretations of Post-Post-Quantum Cryptography (Vol-VIII) K S C 1.8–2.3), yielding a tighter, scale-invariant bound on average-case hardness against quantum oracles. 2) Heavy-tailed Lévy-stable noise generates scale-free random walks (Lévy flights) with infinite Hausdorff measure, geometrically overwhelming the finite Euclidean capacity of Gottesman-Kitaev-Preskill (GKP) quantum error-correcting codes at every resolution. 3) The physical limitations of quantum attacks are bounded by thermodynamic strange attractors governing Quantum Random Access Memory (QRAM) erasure costs. 4) Noisy Intermediate-Scale Quantum (NISQ) variational algorithms fail due to chaotic optimization paths across non-integer dimensional barren plateaus. Also available on Academia.edu: https://www.academia.edu/attachments/132780584/download_file?s=portfolio
Quantum-AI hybrid algorithms represent the most practical near-term pathway for leveraging quantum computing's probabilistic power alongside classical AI's scalability and optimization strengths. In the noisy intermediate-scale quantum (NISQ) era, these hybrids—primarily variational quantum algorithms (VQAs) such as the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA)—delegate hard subproblems (e.g., high-dimensional sampling, optimization under quantum constraints, or feature extraction in entangled spaces) to quantum processors while classical optimizers or neural networks handle parameter tuning, error mitigation, and large-scale data processing (Ferguson et al., 2026; Acampora, 2026). This exploration builds on the 2025 NSF AI+MPS community paper by integrating accelerated timelines: mass deployment of utility-scale quantum computing by August 2027 and AGI substantially before 2030. Hybrid approaches become critical enablers for MPS grand challenges in Mamun, SM (2026)…………………………………………1 Quantum-AI Hybrid Algorithms: Bridging Classical Intelligence, Quantum Acceleration, and Scientific Discovery in the MPS Domains (Vol-I) chemistry (molecular ground states), materials (inverse design), physics (many-body simulation), and astronomy (anomaly detection under uncertainty). Also available on Academia.edu: https://www.academia.edu/attachments/132469868/download_file?s=portfolio
The convergence of quantum mechanics, fractal algebra, and cybernetic systems represents a frontier in understanding emergent intelligence and self-organizing phenomena. This paper synthesizes recent advancements in high-dimensional quantum entanglement, exemplified by the 37-dimensional Greenberger-Horne-Zeilinger (GHZ) paradox experiment, with the Latent Collaboration framework (LatentMAS) in multi-agent systems (MAS). We elucidate the mathematical foundations of the GHZ paradox and high-dimensional quantum computing (HDQC), while exploring their connections to cybernetic organisms defined through fractal structures and autopoiesis, as conceptualized by Maturana and Varela. Enhanced discussions on quantum biology reveal entanglement's role in biological efficiency, drawing parallels to autopoietic self-maintenance in synthetic hives. We further integrate enactivism's sensorimotor roots, emphasizing cognition as embodied action grounded in perceptual-motor contingencies, and investigate radical enactivism variants, such as Hutto and Myin's Radical Enactive Cognition (REC), which rejects representational content for basic minds. Details from Von Neumann's theory of self-reproducing automata highlight logical models of replication. Aggregating theoretical contributions from Good, Vinge, Kurzweil, Bostrom, and Yudkowsky, alongside empirical data, we assess implications for technological singularity timelines. Also available on Academia.edu: https://www.academia.edu/attachments/125654522/download_file?s=portfolio
This paper presents a rigorous mathematical and ontological synthesis of solvable quantum systems and emergent fractal geometries. By leveraging the quadratic tensor framework, we demonstrate that disparate families of quantum models—including Clifford circuits, stabilizer codes, Gaussian states, and free-fermion systems—can be unified as quadratic functions over abelian groups or super-commutative Hopf algebras. This algebraic unification permits classical simulation with O(n2) complexity via a generalized Schur complement contraction mechanism, dissolving the boundaries between discrete-variable qudits and continuous-variable bosonic modes. However, physical reality frequently transcends these smooth, integer-dimensional algebraic bases. We integrate a fractal-recursive ontology, identifying that quantum trajectories in path integrals and energy spectra in quantum materials (e.g., Hofstadter's butterfly) exhibit non-integer Hausdorff dimensions and self-similar architectures that challenge traditional linear causality. We further explore the transition from quadratic to i-th order tensors, identifying the Clifford hierarchy's correspondence to higher-order polynomial maps and the resulting computational boundaries of the Gottesman-Knill theorem. Also available on Academia.edu: https://www.academia.edu/attachments/131609415/download_file?s=portfolio