Abstract: We present a unified and rigorous resolution of the Kakeya conjecture across all dimensions using a novel geometric algebra framework. By extending classical 2D and 3D formulations to general ℝⁿ, we construct directional sweep configurations governed by self-similar fractal structures embedded within Clifford (geometric) algebra. Through this framework, we derive explicit lower bounds for the minimal measure of Kakeya sets in ℝⁿ and prove that these bounds are precisely captured by the Riemann zeta function ζ(n − 1). We show that the directional integral over unit sphere rotations, framed through the spectral partition function, yields closed-form volume expressions analogous to those found in quantum statistical mechanics. The results validate not only the non-zero volume of Kakeya sets in all dimensions, but also rigorously establish the exact minimum volume through spectral and algebraic techniques. Our method offers an elegant and generalizable alternative to existing harmonic analytic and algebraic geometric approaches and opens a new bridge between analysis, number theory, and geometric measure theory.

Abstract: We present a first-principles derivation of the hydrogen \( 2S_{1/2}-2P_{1/2} \) Lamb shift correction from the spectral geometry of Riemann zeta zeros. The framework reveals an exact scaling factor 366 connecting pure mathematical expressions to physical observables. Starting from the first four non-trivial zeros \( \gamma_1, \gamma_2, \gamma_3, \gamma_4 \) of \( \zeta(1/2 + i\gamma_n) = 0 \), we derive: (1) the Lamb shift correction \( \Delta\nu_{\text{Lamb}} = 7.314 \) kHz, (2) the exact scaling factor \( 366 = 8\pi^2(\gamma_4/\gamma_1)^2 \), and (3) demonstrate that this factor emerges necessarily from the Enneper-Möbius geometric framework underlying fundamental constants. The derivation is mathematically self-contained, numerically exact to computational precision, and provides a unified geometric origin for \( \alpha^{-1} = 137.035999084 \), \( \ell_P = 1.616255\times10^{-35} \) m, and quantum electrodynamic corrections.

Abstract: We show that Quantum Mechanical Hilbert space can be paradoxical under some group action and explore its physical consequences. (1) Is there a more natural way of resolving the paradox of Wigner's friend without invoking the Heisenberg's cut?; (2) We notice the qualitative similarities between the paradox and paradoxical sets and use it as a motivation to rigorously prove that the Hilbert space $\mathcal{H}$ of the harmonic oscillator is paradoxical under the group action $\mathcal{U} \leq U(\mathcal{H})\times U(1)$; (3) This paradoxical nature of the Hilbert space $\mathcal{H}$ provides the natural resolution for the paradox by using the Axiom of Choice instead of the Heisenberg's cut; (4) Finally, we show that due to the very same paradoxical nature of $\mathcal{H}$, certain class of quantum gravities naturally emerge from Quantum Mechanics that mediates a self-decoherence of the system.

Abstract: Recent experimental demonstrations of the Montgomery effect have revealed the possibility of controlled, lensless self-focusing and three-dimensional reconstruction of optical fields in free space. In this work, we present a geometric interpretation of this phenomenon based on the Kuznetsov tensor formalism, offering an alternative theoretical framework that extends beyond conventional wave-interference descriptions.Within this approach, the propagation of a coherent optical field is treated as an evolution of field configurations in an effective configuration space with a dynamically modified metric. The spatial phase modulation imposed at the initial plane induces phase-dependent singularities encoded by the Kuznetsov tensor, which alters the geometry of the configuration space. As a result, light propagation follows geodesics of the effective metric rather than straight trajectories in Euclidean space.We show that the characteristic features of the Montgomery effect—namely, periodic self-reconstruction, discrete refocusing planes, and the robustness of complex structured beams—naturally arise as consequences of the geometric evolution governed by a modified flow equation involving the Kuznetsov tensor. The observed refocusing planes correspond to stable critical points of a configurational entropy functional, explaining the sharp re-emergence of optical structures without the use of physical focusing elements.Furthermore, the successful reconstruction of vortex beams and multi-spot arrays indicates the preservation of topological invariants of the optical field, which is naturally described within the tensor-geometric framework. This interpretation provides a unified explanation for the stability and repeatability of the Montgomery effect and establishes a direct conceptual link between structured light, geometric self-organization, and effective curvature of configuration space.The proposed framework offers new theoretical insights into lensless optical manipulation and suggests pathways for extending Montgomery-type effects to metasurfaces and volumetric optical architectures.

Abstract: We present a comprehensive derivation of the geometric factor $K$ that establishes a mathematical bridge between the first four nontrivial Riemann zeta zeros and fundamental physical constants. Through high-precision computation (200+ digits) we demonstrate that $K$ decomposes into two exactly inverse components: the \textit{geometric seed} $K_g \approx 0.008353870129$ and the \textit{completion factor} $C \approx 119.700000000$, with $K_g \cdot C = 1$. This identity reveals that the Planck length $\ell_P = \sqrt{G\hbar/(c^3 K)}$ is intrinsically determined by arithmetic relationships among $\gamma_1, \gamma_2, \gamma_3, \gamma_4$. The framework provides first-principles derivations of $\ell_P = \SI{1.616255e-35}{\meter}$, $E_0 = \SI{1820.469}{\electronvolt}$, and $\alpha^{-1} = 137.035999084$, all emerging from the same geometric structure. The work resolves previous apparent inconsistencies and establishes a mathematical foundation for the geometric origin of physical scales.

Abstract: We reformulate the classical three-body problem within the algebra of octonions and the geometry of the exceptional Lie group G₂. By embedding the Newtonian configuration space into a seven-dimensional non-associative manifold, the apparent chaos of three-body motion becomes a geometric property of the associator torque rather than a random instability. A small informational-viscosity parameter , derived from the Viscous Time Theory (VTT), is introduced to regularize energy divergence and confine chaotic diffusion. The resulting G₂–Lie variational integrator preserves phase volume while dynamically damping entropy flux through coherence feedback . Numerical simulations confirm that near-collision singularities are resolved without artificial damping, and long-term energy drift remains below . This framework provides a coherent bridge between non-associative geometry, variational mechanics, and informational physics, suggesting that the stability of gravitational systems arises from the preservation of informational structure rather than purely dynamical constraints.

Abstract: This paper presents a complete Unified Field Theory of Xuan-Liang, constructing a comprehensive the-oretical framework from fundamental physical concepts to cosmology and emergent gravity. Starting from the basic definition of Xuan-Liang X = 1/3 mv3, through rigorous mathematical-physical derivation, we establish the unified equation of Xuan-Liang theory: R M h 1/2 X ∧ ⋆X + αR Ω ∧ ⋆X + ⟨ΨX, DΨX ⟩Ω i = χ(M)ρmin X R M Ω + β R ∂M Φobs This theoretical framework contains two core aspects: Xuan-Liang fluid theory achieves unified description of dark matter and dark energy, and the emergent gravity mechanism reveals the natural origin of Einstein’s field equations from Xuan-Liang fluid dynamics. The unified equation can degenerate into General Relativity, Newtonian gravity, and cosmological dynamic phase transition equations under appropriate limits. Main innovative contributions include: 1. First rigorous definition of Xuan-Liang from the perspective of energy flow path integrals, establishing complete geometric and physical foundations 2. Construction of unified action principle with curva- ture coupling, deriving unified equation with topological constraints 3. Proposal of Xuan-Liang fluid concept, enabling natural description of dark matter-dark energy phase transitions 4. Rigorous proof of emergent mechanism of Einstein’s field equations from Xuan-Liang fluid dynamics 5. Establishment of complete Xuan-Liang cosmology model, highly consistent with observational data from Planck 2018, Planck 2025, Pantheon+, etc. 6. Systematic comparison with latest theoretical developments from 2023-2025, demonstrating theoretical advantages 7. Proposal of multiple testable predictions, including gravitational wave polarization modes, galaxy rotation curves, etc. Numerical simulations show that Xuan-Liang theory is highly compatible with key observational data such as CMB powerspectra, BAO observations, and supernova distance moduli (χ2 red = 1.02), outperforming the ΛCDM model (χ2 red = 1.08). The theoretically predicted phase transition redshift zt = 0.65 ± 0.08 provides clear targets for future observational tests.

Abstract: We investigate several equivalence notions arising in the study of Calabi-Yau manifolds and their interactions with ideas from string theory. The focus is on bimeromorphic equivalence in complex geometry, Morita equivalence in noncommutative geometry, and twisted K-theory as a receptacle for D-brane charges in backgrounds with flux. Using tools from derived categories, Fukaya categories, and operator K-theory, we analyze how these equivalences appear across geometric, categorical, and physical frameworks. Particular attention is given to Fujiki class C manifolds, Hilbert C*-modules, and the role of homological mirror symmetry in relating these structures. Several examples and applications are discussed, illustrating how string-motivated constructions provide a unifying perspective on equivalence phenomena in Calabi-Yau geometry.

Abstract: The paper reexamines Stevenson’s technique for solving Schrödinger’s “Kepler problem” in a spherical space in terms of formally complex hypergeometric polynomials. A certain advantage has been achieved by reformulating the genetic 'dual principal Fro-benius solution’ (d-PFS) problem as the Dirichlet problem for the given second-order or-dinary differential equation (ODE) rewritten in its 'prime' form. It was demonstrated that the cited polynomials match Askey’s hypergeometric expressions for the re-al-by-definition Romanovski/pseudo-Jacobi polynomials (‘Romanovski-Routh’ polyno-mials in our terms). The formulated Dirichlet problem was then reduced to the two more specific cases representing the Sturm-Liouville problems (SLPs) with infinite and respec-tively finite discrete energy spectra. The exact solvability of the former SLP (with the Li-ouville potential represented by the ‘trigonometric Rosen-Morse’ potential) was proven by taking into account that the Romanovski-Routh polynomial of degree n must have exactly n real zeros (with no upper bound for the eigenvalues). As the direct consequence of this proof, we then found that the mentioned d-PFS problem in general and therefore the second SLP with the finite discrete energy spectrum are exactly solvable via qua-si-rational solutions (q-RSs) composed of the Romanovski/Routh polynomials with de-gree-dependent indexes.

Abstract: We develop a topological classification of admissible reconstruction operations in generative systems where extended structure is built through repeated local extension subject to compatibility constraints. Reconstruction is formalized as a feasibility--governed process rather than a dynamical or metric one, with admissibility determined by the accumulation of obstruction under composition. Using loop diagnostics, we identify global incompatibilities that are invisible to local extension rules but become unavoidable under closed composition. Under mild and realization--independent assumptions, including indefinite continuation and finite interface capacity, we show that persistent nontrivial obstruction is possible only when it is supported on codimension--2 subsets of the reconstructed domain. This result induces a small number of topological universality classes distinguished by the existence and stability of loop--detectable obstruction. The framework is model--agnostic and applies equally to discrete, combinatorial, and continuum reconstructions, providing a topological explanation for the ubiquity of codimension—2 defects in generative systems.

Abstract: The nature of time and its role in physical evolution remain central open questions in theoretical physics, particularly in the presence of irreversibility. In this study, a geometric framework for time evolution is introduced based on a multivector time generator acting on the phase plane. Rather than extending time as a parameter, this approach focuses on the structure of the time derivative and its associated symmetries. Using geometric algebra, the generator decomposes naturally into scalar, bivector, and vector components. The bivector part generates Hamiltonian, symplectic evolution and corresponds to reversible dynamics, while the scalar part produces uniform contraction or expansion and provides a geometric interpretation of irreversibility and entropy production. In addition, vector components generate reversible but anti-symplectic transformations, such as reflections, revealing symmetries that are not captured by the standard Hamiltonian or complex-time approaches. The general solutions of linear systems follow an exponential form, and the reversible generator admits a natural classification into elliptic, hyperbolic, and nilpotent cases, yielding a clear geometric interpretation of oscillatory, overdamped, and critically damped behavior. The framework further clarifies the status of complex time, showing that it arises only as a restricted case when vector components are absent, and the time derivative admits a Wirtinger representation. Outside this regime, the time evolution is a multivector and cannot be described by a single complex parameter. Overall, the proposed framework provides a unified geometric language for analyzing reversible and irreversible dynamics, and highlights the central role of the time generator in shaping temporal evolution.

Abstract: We present a systematic geometric analysis of sectional curvature structures on fibred Calabi-Yau manifolds using the theory of Riemannian submersions and O'Neill's curvature decomposition formulas. Emphasis is placed on elliptic, toroidal, and K3 fibrations arising in complex dimensions one through four. We derive explicit curvature decompositions for horizontal, vertical, and mixed planes and establish quantitative bounds that relate sectional curvature to the tensorial data governing the fibration geometry. These results clarify how rich local curvature phenomena and anisotropies can arise despite the global Ricci-flatness of Calabi-Yau metrics. The framework developed here provides a unified geometric perspective on curvature behavior in fibred Calabi-Yau manifolds and supports both analytical investigations and computational approaches to curvature estimation in explicit geometries.

Abstract: We study holonomy-induced deviations arising from Levi-Civita parallel transport on Calabi-Yau manifolds of complex dimensions one through four. Using the Ricci-flat Kähler structure and the associated SU(n) holonomy reduction, we develop a unified framework for deviation operators that applies uniformly across dimensions. General expressions are formulated in terms of path-ordered transport, curvature endomorphisms, and non-Abelian Stokes techniques, clarifying how nontrivial holonomy effects persist despite vanishing Ricci curvature. A dimension-by-dimension analysis is presented, covering elliptic curves, K3 and abelian surfaces, Calabi-Yau threefolds, and Calabi-Yau fourfolds. We identify which holonomy contributions are suppressed by type constraints in Kähler geometry, which arise only at higher order, and how these features depend on the complex dimension. The paper is intended both as a reference for explicit holonomy and deviation computations and as a bridge to applications involving geometric phases and compactification effects.

Abstract: We investigate a random field of mutually dependent random variables ("spins"), indexed by a finite one-dimensional lattice, called in physical sciences the one-dimensional Ising model, in which the random variables can take only ±1 values (see the text for a precise definition). One of the couplings, termed a "bond," that describes the mutual influence of two adjacent random variables is altered—it does not equal the others, thereby introducing a single "defect" bond. This defect bond represents a localised perturbation within an otherwise uniform system. Utilising the recurrence relations of Chebyshev polynomials and the bijective  map between the number of spins and the polynomial index, we present a new method for calculations and  systematically explore, using it, the system’s properties across different chain lengths and boundary conditions. As an application, we derive analytical expressions for the dependence of the average values of the random variables on their position within the chain, which we refer to as the "local magnetisation profile". From the findings related to the system with a defect bond, we present a novel result for this profile under free (Dirichlet) boundary conditions and re-derive the corresponding result for antiperiodic boundary conditions.

Abstract: The Weak Equivalence Principle (WEP) postulates that the trajectory of a free-falling test body is independent of its internal structure. However, this universality formally applies only to structureless point masses. In this work, we re-examine the dynamics of extended spinning bodies within the Schwarzschild spacetime framework. Starting from the premise that gravity manifests as spacetime curvature---geometrically analogous to a slope---we derive that tangential motion modifies the effective geodesic path of an object.We demonstrate analytically that for a rotating body with angular momentum parallel to the local gravitational field, the geometric factors in the gravitational and inertial sectors of the radial geodesic equation exhibit an exact cancellation at the critical orbital velocity $v_c = \sqrt{GM/R}$. This implies that ``weight''---defined as the force required to deviate an object from its natural geodesic---is not an intrinsic invariant but a dynamic quantity dependent on the geometric alignment between the velocity vector and the spacetime curvature.We extend this finding to the microscopic regime, proposing that polarized atomic nuclei should exhibit weight anisotropy dependent on spin orientation. The thermodynamic consequence---a ``Geometric Buoyancy'' effect causing spontaneous stratification of spin-polarized gases---is derived and formally verified using the Lean~4 theorem prover.

Abstract: The fermion mass hierarchy in the Standard Model spans six chief orders of magnitude and is conventionally explained by arbitrary Yukawa couplings. Here we explore a purely mathematical construction—a discrete vacuum geometry derived from a finite-dimensional 19D (12+4+3) $\mathbb{Z}_3$-graded Lie superalgebra with exact triality symmetry—and examine whether simple integer lattice vectors embedded in its extended $\mathbb{Z}^3$ lattice happen to produce mass ratios and other parameter values resembling those observed in nature. Using a geometric scaling $m \propto L^{-2}$ where $L$ is the Euclidean norm of selected lattice vectors, and anchoring to the top-quark mass (173 GeV), the framework yields the following curious numerical proximities: electron 0.49 MeV (4.6\% agreement), muon 118 MeV (12\%), qualitative up/down quark mass inversion $m_u < m_d$, exact Weinberg angle $\sin^2 \theta_W = 0.25$, a Higgs-related scale ratio of 0.727 (0.3\%), strong/weak coupling ratio $\approx 0.95$ (near equipartition), CKM CP phase $\approx 65.3^\circ$ (5\%), and neutrino mixing angles of exactly $45^\circ$ (maximal atmospheric) with $\cos^2 \theta_{12} = 1/3$ (exact tri-bimaximal solar angle). These alignments, along with geometric patterns resembling tri-bimaximal neutrino mixing, are presented as intriguing mathematical coincidences within an abstract algebraic framework and do not constitute evidence of physical relevance or predictive power. The construction offers a speculative geometric perspective that unifies gauge and flavor aspects in a single algebraic setting, extending previous work on the same structure, while emphasising that the observed numerical matches may reflect serendipity rather than deeper significance.

Abstract: I formulate a precise obstruction to identifying the four-dimensional Standard Model with the holographic dual of a four-dimensional de Sitter (dS$_4$) universe. Under mild and physically motivated assumptions showing no such duality exists. This result delineates a sharp sense in which a dS/SM boundary correspondence is excluded in our universe, while leaving consistent non-boundary frameworks open.

Abstract: We develop a systematic framework for studying Buggy Spaces, anomalous loci that arise in the moduli spaces of non-compact Calabi–Yau manifolds and obstruct standard geometric, categorical, and physical descriptions. These loci appear naturally in toric constructions, orbifold limits, and mirror symmetry, where familiar tools such as derived categories, stability conditions, and enumerative invariants exhibit discontinuities, ambiguities, or outright failure. Rather than treating these phenomena as isolated pathologies, we show that they form a coherent and structurally rich class of moduli-space defects. We introduce precise criteria for identifying Buggy Spaces and propose a classification scheme based on geometric degeneration, categorical instability, and physical inconsistency. Using explicit examples in various dimensions, we demonstrate how Buggy Spaces manifest in both A-model and B-model settings, and how they influence wall-crossing behavior, mirror maps, and the topology of moduli spaces. We further examine their consequences for string theory compactifications and gauge-theory realizations, where Buggy Spaces signal obstructions to naive effective descriptions. Our results indicate that Buggy Spaces encode subtle links between geometry and physics that are invisible in smooth or compact settings. By isolating and organizing these anomalies, we provide a unified perspective on several previously disconnected phenomena in mirror symmetry and string theory. We conclude by outlining open problems and directions for future work, including implications for non-compact moduli stabilization, derived categorical dynamics, and the structure of the string theory landscape.

Abstract: A moving permanent magnet induces currents in nearby conductors and superconducting loops. This electrodynamic back-action produces forces that are history-dependent and, in general, cannot be represented by a conservative potential depending only on instantaneous separation. We derive a compact state--space model for the axial motion of a magnetic dipole coupled to an $N$-turn loop with inductance $L$ and resistance $R$. Using Faraday's law for the flux linkage $\Lambda(x)=N\Phi(x)$, the coupled dynamics are \[ L\dot i + R i = -\Lambda'(x)\dot x,\qquad F_{\mathrm{em}}(t)= i(t)\,\Lambda'(x(t)), \] which defines a passive, causal magnetomechanical memory element. Linearizing about an operating point $x_0$ yields an exact complex dynamic stiffness \[ K_{\mathrm{em}}(\omega) \equiv -\frac{F_{\mathrm{em}}(\omega)}{X(\omega)} = \frac{\Lambda'(x_0)^2\, i\omega}{R+i\omega L} = k_\infty\,\frac{i\omega\tau}{1+i\omega\tau}, \quad k_\infty=\frac{\Lambda'(x_0)^2}{L},\ \tau=\frac{L}{R}. \] We show that this back-action is exactly equivalent to a mechanical Maxwell element, derive closed-form expressions for added stiffness and added damping, and provide direct identification formulas for $(\tau,k_\infty)$ from measured complex stiffness. The dipole--loop geometry further admits an analytic design rule: the coupling gradient $|\Lambda'(x_0)|$ is maximized at $x_0=a/2$ where $a$ is the loop radius. Finally, we connect commonly proposed \emph{jerk-like} and \emph{absement-like} terms to controlled low- and high-frequency asymptotic expansions of the same passive kernel (with explicit validity limits). All predictions are validated by reproducible Python simulations, and code to generate figures and data is provided.

Abstract: In this paper, we examine the inherent mathematical and physical inconsistencies of strain gradient theories. It is shown that strain gradients are not proper measures of deformation, because their corresponding energetically conjugate stresses are non-physical and cannot represent the state of internal stresses in the continuum. Furthermore, the governing equations in these theories do not describe the equilibrium or motion of infinitesimal elements of matter properly. In first strain gradient theory (F-SGT), there are nine explicit governing equations of motion for infinitesimal elements of matter at each point; three force equations, and six unsubstantiated artificial moment equations that violate Newton’s third law of action and reaction. This shows that F-SGT is not an extension of rigid body mechanics, which then is not recovered in the absence of deformation. Moreover, F-SGT would require the existence of six additional fictitious symmetries of space-time according to Noether’s theorem, and a complete revision of the well-established concept of static indeterminacy in introductory mechanics. The inconsistencies of F-SGT also manifest themselves in the appearance of strains as boundary conditions.