Abstract: 2D elastostatic displacement solutions for the Yoffe Mode I and Rice Mode II crack models are reviewed. These solutions are used to introduce the elastostatic displacement solution for a 2D Mode I/II multi-crack configuration in terms of meromorphic differential forms on a hyperelliptic curve. The complex dimension of the vector space of mermorphic forms is demonstrated to be g+1, where g is the genus of the hyperelliptic curve, using the Riemann-Roch theorem. Limitations of the 2D multi-crack model to modeling 3D fracture networks are identified, and a mathematical description of 3D fracture network dynamics preceding an earthquake based on singular spectrum analysis of crack phase fields is conjectured.

Abstract: We present a geometric framework unifying the Riemann sphere $\hat{\mathbb{C}}$, the Moebius strip $M$, and Enneper's surface $E$ into a canonical triad that naturally encodes all fundamental physical scales. Through precise holomorphic embeddings and conformal mappings, we demonstrate: (1) The primal energy $E_0 = \SI{1820.469}{\electronvolt}$ emerges as the natural quantum of energy; (2) The primal length $\ell_0 = \ell_P = \SI{1.616255e-35}{\meter}$ is identified with the Planck length; (3) The fine-structure constant $\alpha^{-1} = 137.035999084$ is exactly derived from combinatorial relations among the first four Riemann zeta zeros $\gamma_1, \gamma_2, \gamma_3, \gamma_4$. We propose that this geometric framework could explain the Riemann Hypothesis topologically, lead to testable predictions in quantum and gravitational physics, and provide a foundation for holographic emergence of physical reality.

Abstract: This paper presents a complete cosmological model of Xuan-Liang, achieving a unified description of dark matter and dark energy. Starting from the classical concept of "work" and extending it to "spatial accumulation of power," we first-principally derive the expression of Xuan-Liang: $X = \frac{1}{3} m v^3$. By fieldifying the concept of Xuan-Liang, we construct a dynamic phase-transition model with exact symmetry, whose evolution equation is: \[ \left( \frac{\rho_X}{\rho_t} \right)^{\Delta/2} + \left( \frac{\rho_X}{\rho_t} \right)^{-\Delta/2} = \left( \frac{a}{a_t} \right)^{-3\Delta/2} + \left( \frac{a}{a_t} \right)^{3\Delta/2} \] This equation reveals a profound duality between the cosmic scale factor and the density of the Xuan-Liang field. We present the complete Friedmann equations including ordinary matter, radiation, and the Xuan-Liang field, and numerically solve the cosmic evolution history. Using the latest observational data (Planck 2018, Pantheon+ supernovae, BAO), we constrain the model parameters, showing high compatibility with observations, with a $\chi^2$ improvement of about 8\% compared to the $\Lambda$CDM model. The model predicts a specific equation-of-state evolution $w(z)$, a precise phase-transition redshift $z_t = 0.65 \pm 0.08$, and a weak early dark energy component ($\Omega_{Xe} \sim 10^{-5}$). Theoretical analysis suggests that the path-integral origin of Xuan-Liang may reflect the topological structure of spacetime, providing new perspectives for quantum gravity.

Abstract: It is still an open question that how the masses are formed for charged leptons. The widely accepted Yukawa coupling mechanism generally have quite randomized coupling constants for their masses. In this paper, we tried to build a simple model to calculate the masses of charged leptons. We assumed that the masses are formed by coupling of the plasma characteristic energy from the particle-antiparticle pairs in the background sea, and the electric potential inside the Compton ball. The internal structure of the charged leptons is thought to have three states, i.e. the negative charge, the positive charge and the Planck scale Kerr black hole. For electron and muon, the zitterbewegung is formed by positive charge and negative charge, but for tau, the excited zitterbewegung is formed by negative charge and the Planck scale Kerr black hole. The calculations of this simple model give quite close values for the charged leptons as compared with the lab results. We think we may need pay more attention on the internal structure of a particle.

Abstract: We present an audit-grade formulation of the Quantum Information Copy-Time (QICT) program as a micro–macro closure framework and as a quantitative pipeline for falsifiable predictions. The core observable is the operational copy time τ_copy(ℓ;ε, δ⋆): the minimal time required for a calibrated local bias in a sender region to become statistically distinguishable in a receiver region at separation ℓ, under explicit signal-to-noise accuracy ε and disturbance budget δ⋆. Under transparent hypotheses—locality, sector ergodicity, and the existence of a quantitative diffusive hydrodynamic window—we derive a one-way lower bound τcopy ≳ ℓ²/D with a strict feasibility correction controlled by the inversion of the diffusive tail. We show how a measurable Spectral Diffusion Criterion (SDC) in the hydrodynamic sector converts microscopic unitary dynamics into an auditable transport closure. We connect this closure to two predictive targets: (i) an inertial spectral mass diagnostic defined from long-wavelength spectral flow, and (ii) a reproducible Higgs-portal dark-matter corridor in the scalarsinglet model, where the QICT calibration acts as a restrictive prior on the effective portal region. A complete reproduction package (code, data products, and figures) is provided; we emphasize which statements are definitions, which are assumptions, and which are falsifiable predictions.

Abstract: We investigate the equilibrium and stability properties of self-gravitating bosonic congurations in the nonrelativistic regime by numerically solving the nonlinear Gross-Pitaevskii-Poisson (GPP) system of equations. By adopting a suitable coordinate transformation and a specic gauge choice for the gravitational potential, the GPP equations are cast into a dimensionless form independent of the physical parameters of the model. In this formulation, equilibrium congurations are uniquely characterized by the central value of the dimensionless wave function, which determines the central density once physical units are restored. We compute sequences of stationary solutions including both ground-state and radially excited congurations and identify bifurcation points at which transitions between these states occur. The virial relation is employed as a diagnostic criterion for equilibrium, allowing us to determine a critical central density above which stationary congurations cease to exist. Excited-state solutions satisfying the virial relation are probably metastable and are expected to decay toward the ground state through gravitational cooling. The critical central density is associated with a maximum allowed particle number, leading to an estimate of the maximum stable mass of the conguration. For axion-like bosons with masses of order 10?5 eV, the resulting maximum mass is of the order of tens of Earth mass, while the characteristic size of the conguration is of order one meter. The compactness of these objects places them close to the limit of validity of the Newtonian approximation.

Abstract: The broad hierarchy of fermion masses in the Standard Model, spanning six orders of magnitude, is conventionally attributed to ad hoc Yukawa couplings. This work explores a possible geometric interpretation arising from a discrete $\mathbb{Z}_3$-graded vacuum structure, derived from a finite-dimensional (19D: $12+4+3$) Lie superalgebra with exact triality symmetry. Within this framework, the vacuum is organized into a two-layer lattice: a finite \textbf{Core Lattice} (44 vectors) that yields gauge unification with $\sin^2 \theta_W = 0.25$, and an infinite \textbf{Extended Lattice} ($\mathbb{Z}^3$) that may generate the fermion mass spectrum through a geometric seesaw-like relation $m \propto L^{-2}$. By associating specific integer lattice vectors with known fermions, we find that the resulting mass scales appear to align with those of the top quark, bottom quark, tau lepton, charm quark, muon, down quark, and electron. For instance, the electron mass is obtained within 4.6\% (0.488 MeV compared to the observed 0.511 MeV) across a $10^6$ range. Observed deviations for heavier quarks are qualitatively consistent with QCD renormalization effects, suggesting the lattice might correspond to bare parameters. These numerical coincidences, while intriguing, do not constitute a proof and may reflect mathematical serendipity. The approach offers a complementary geometric perspective that unifies forces and matter within a single algebraic setting, extending previous work on the Weinberg angle and other constants from the same structure.

Abstract: We present a rigorous derivation of two fundamental physical scales: the primal energy E0 = 2.916601 × 10^{-16} J = 1820.469 eV and the primal length l0 = lP = 1.616255 × 10^{-35} m (the Planck length). These quantities emerge uniquely from the arithmetic-geometric structure encoded in the zeros of the Riemann zeta function ζ(s). We demonstrate that E0 serves as the natural energy unit connecting quantum mechanics, gravitational physics, and number theory, while l0 establishes the fundamental length scale of spacetime geometry. The derivation employs: (1) the exact conformal transformation Φ(z) = α arcsinh(βz) + γ with αβγ = 2π connecting quantum spectra to zeta zeros; (2) combinatorial relations among the first four nontrivial zeros γ1, γ2, γ3, γ4; and (3) consistency conditions with established physical constants (CODATA 2018). The resulting framework provides a unified basis for understanding fundamental constants, predicts testable modifications to quantum and gravitational phenomena, and offers new insights into the geometric structure of reality at the Planck scale.

Abstract: We present a minimal axiomatic framework, the Self-Variation Theory (SVT), in which intrinsic particle properties undergo spacetime self-variation. Total four-momentum emerges as a conserved quantity, and particle masses are determined from the dynamically allowed values of its components. In particular, the action principle (Axiom IV) selects the physically admissible four-momentum configurations, providing a first-principles determination of the admissible total four-momentum configurations and the corresponding invariant masses. An illustrative calculation of particle masses using a spacetime distribution function Φ(x) is presented and compared with experimental values.

Abstract: We introduce a formal and systematic analogy between static mechanical spring networks and electrical resistor networks, establishing a one‑to‑one mathematical correspondence where spring displacements map to electric potentials, spring constants to conductances, and forces to currents. This equivalence transforms the static equilibrium equations of point-mass-spring systems into the nodal equations of resistor networks, enabling the construction of a conductance matrix that is identical in form to a mechanical stiffness matrix and coincides with the weighted graph Laplacian. We demonstrate the versatility of this framework through a series of examples from elementary series and parallel combinations to non-planar networks such as the 3D resistor cube and the Petersen graph. We show that the method provides both an intuitive mechanical interpretation of circuit concepts and a systematic, matrix-based computational algorithm for calculating equivalent resistance. The approach naturally extends to AC networks containing resistors, capacitors, and inductors, offering a unified treatment of linear networks in both DC and AC regimes. We discuss the pedagogical value of the analogy for teaching circuit theory and network analysis, as well as its connections to graph theory and computational implementation. Limitations of the method, including its restriction to linear elements and the numerical considerations of matrix inversion, are briefly discussed.

Abstract: The standard mathematical framework of differential topology reveals a profound peculiarity: smooth structures on 4-manifolds are wild, unclassifiable, and form an uncountably infinite set, while in all other dimensions they are either unique or finitely classifiable. This presents a fundamental obstacle to constructing a quantum theory of gravity via a path integral sum over geometries as the configuration space becomes intractably complex. Crucially, this mathematical wildness is not a mere curiosity but a diagnostic signal that our dimension-agnostic mathematical framework is fundamentally inadequate for describing quantum spacetime in our 4-dimensional universe. We argue that this impasse signals not a pathology of 4-dimensional spacetime, but a critical flaw in our mathematical starting point. We propose a radical inversion of priorities: instead of seeking to tame 4D wildness within a dimension-agnostic formalism, we should construct a new mathematical framework whose axioms are explicitly designed so that 4-dimensional spacetime emerges as its unique, natural, and tame solution. The price for this 4D simplicity is that other dimensionalities may appear ill-defined or trivial within the new framework, a price we argue is not only acceptable but necessary for a physical theory of our universe. We outline the philosophical and formal principles of such a 4D-native approach and discuss its embodiment in existing pre-geometric quantum gravity programs where smooth geometry emerges from more fundamental substrates.

Abstract:

This paper presents a detailed re-examination of the conformal compactification of Minkowski space, \( \overline{M} \), constructed as the projective null cone of the six-dimensional space \( \mathbb{R}^{4,2} \). We provide an explicit and basis-independent formulation, emphasizing geometric clarity. A central result is the explicit identification of \( \overline{M} \) with the unitary group U(2) via a diffeomorphism, offering a clear matrix representation for points in the compactified space. We then systematically construct and analyze the action of the full conformal group \( \mathrm{O}(4,2) \) and its connected component \( \mathrm{SO}_0(4,2) \) on this manifold. A key contribution is the detailed study of the double cover, \( \overline{\overline{M}} \), which is shown to be diffeomorphic to \( S^3 \times S^1 \). This construction resolves the non-effectiveness of the \( \mathrm{SO}(4,2) \) action on \( \overline{M} \), yielding an effective group action on the covering space. A significant portion of our analysis is devoted to a precise and novel geometric characterization of the conformal infinity. Moving beyond the often-misrepresented ``double cone'' description, we demonstrate that the infinity of the double cover, \( \overline{\overline{M}}_\infty \), is a squeezed torus (specifically, a horn cyclide), while the simple infinity, \( \overline{M}_\infty \), is a needle cyclide. We provide explicit parametrizations and graphical representations of these structures. Finally, we explore the embedding of five-dimensional constant-curvature spaces, whose boundary is the compactified Minkowski space, and discuss the interpretation of geodesics within these domains. The paper aims to clarify long-standing misconceptions in the literature and provides a robust, coordinate-free geometric foundation for conformal compactification, with potential implications for cosmology and conformal field theory.

Abstract: Assembly theory defines structural complexity as the minimum number of steps required to construct an object in an assembly space. We formalize the assembly space as an acyclic digraph of strings. Key results include analytical bounds on the minimum and maximum assembly indices as functions of string length and alphabet size, and relations between the assembly index (ASI), assembly depth, depth index, Shannon entropy, and expected waiting times for strings drawn from uniform distributions. We identify patterns in minimum- and maximum-ASI strings and provide construction methods for the latter. While computing ASI is NP-complete, we develop efficient implementations that enable ASI computation of long strings. We establish a counterintuitive, inverse relationship between a string ASI and its expected waiting time. Geometric visualizations reveal that ordered decimal representations of low ASI bitstrings of even length N naturally cluster on diagonals and oblique lines of the squares with sides equal to 2^N/2. Comparison with grammar-based compression (Re-Pair) shows that ASI provides superior compression by exploiting global combinatorial patterns. These findings advance complexity measures with applications in computational biology (where DNA sequences must violate Chargaff's rules to achieve minimum ASI), graph theory, and data compression.

Abstract:

This paper presents a generalized theoretical framework for describing the electric double layer (EDL) at the metal–electrolyte interface based on the introduction of the Kuznetsov tensor. In contrast to classical EDL models, which rely on a scalar electrostatic potential and assume integer ionic charges, the proposed approach accounts for the tensorial nature of interactions arising from specific ion adsorption and partial charge transfer between ions and the metal surface. The Kuznetsov tensor is formulated as a generalized interfacial field tensor that incorporates contributions from energy and momentum transport, charge density, adsorption effects, and entropy fluxes. It is shown that the equilibrium state of the electric double layer corresponds to the condition of vanishing divergence of the Kuznetsov tensor, allowing the EDL to be interpreted as a stationary tensor field rather than a simple superposition of compact and diffuse layers. Within this formalism, fractional effective ionic charges, ion competition in multicomponent electrolytes, and the influence of the chemical nature of the electrode surface are naturally captured. It is demonstrated that classical Poisson–Nernst–Planck equations and Stern-type models can be recovered as limiting cases of the tensor description under appropriate simplifying assumptions. The proposed theory provides a unified mathematical foundation for multiscale modeling of electrochemical interfaces and offers a consistent framework for analyzing charge storage, capacitance, and interfacial phenomena in batteries, supercapacitors, and electrocatalytic systems.

Abstract: We calculate and plot the Wigner functions of several families of highly excited even and odd superpositions of nonlinear coherent states, looking for conditions under which such superpositions can be interpreted as models of the ``Schr\"odinger cat'' states. It appears that the decisive factor is the form of the number distribution functions over the Fock basis: they must have well localized peaks. Otherwise, no ``cat'' structures are observed.

Abstract: An $n$-color composition is a colored composition in which a part of size $m$ may come in $m$ colors. This paper gives a new set of $n$-color-type compositions that admits exhaustive conjugation of its members. Previous attempts at conjugation of $n$-color compositions have yielded partial results at best. Instead of importing the coloring scheme previously used for partitions, we apply colors directly to the parts of compositions while treating any maximal string of 1's as a single part under color assignment. This leads to the definition of $n$-color compositions of the second kind. As with ordinary compositions a conjugate may be found by the equivalent techniques of symbolic algebra, zig-zag graph and line graph. We conclude with a derivation of the relevant enumeration formulas.

Abstract: Curtain deflector coating is a widely employed technique for producing thin, uniform films in numerous industrial applications. Near the corner region created by the interaction of the moving substrate and the falling liquid curtain, the flow dynamics are more complex. In this study, an analytical investigation is conducted for the steady, incompressible, and creeping flow of a Maxwell fluid, incorporating the Navier slip condition at the substrate. The governing nonlinear equations, derived from the conservation of mass and momentum, are solved using the Langlois recursive technique in combination with the inverse method, yielding an approximate third-order solutions for velocity, pressure, and stress fields. Finite and physically consistent stress distributions are produced by the slip condition that removes the singularity connected to the traditional no-slip boundary condition. The analysis demonstrates that substrate slip significantly reduces tangential stresses and enhances the stability of the coating flow. Residual error analysis is also performed to verify the accuracy and convergence of the analytical solutions. The results provide a deeper understanding of how slip effects can be utilized to improve coating uniformity and optimize the operational performance of curtain deflector coating systems.

Abstract: We develop a framework in which lightcones, $(p,q)$ strings, and M-theory compactifications are equipped with an explicit layer of modal structure. Starting from Minkowski spacetime $\mathbb{M}^4$ with its usual null cones, we consider a background $\mathbb{M}^{4\times 7} = \mathbb{M}^4 \times G_2$ and attach to each spacetime point $\mathfrak{x}$ an internal $G_2$-source fiber. Local field configurations are encoded by propositions in a fuzzy Heyting--Kripke semantics, and a localized action $S(\mathfrak{x})$ defines normalized truth-values $\sigma(\mathfrak{x},\varphi) \in [0,1]$ which we interpret as a ``modal radius'' on a refined lightcone. Nilpotent extensions of the truth-value space capture unphysical ``eternal photon'' modes as purely internal, non-observable directions in the $G_2$-source. On this background we introduce \emph{interstices} $\varcurlywedge = G_2 \cap \mathbb{M}^4$ at $(p,q)$ string junctions, regarded as higher-dimensional analogues of Chan--Paton defects. Each interstice supports a pair of S-dual field-families (FFs) $(\Phi,\Psi)$ with a coprime arithmetic grading that organizes primary, descendant, and auxiliary modes, and we show how a $G_2$-inspired flux superpotential can be reinterpreted as a ``modal'' functional controlling the exponents in $\sigma(\mathfrak{x},\varphi) = n^{-z_i}$. We work out explicit examples: a three-leg $(p,q)$ junction and a minimal two-junction web with an internal $(1,1)$ segment, including concrete numerical assignments for $\sigma(\mathfrak{x},\varphi)$, an S-duality reshuffling of the FFs, and a simple propagation rule for modal data along the internal edge. The resulting picture can be phrased in terms of a small Kripke frame whose accessibility relation is determined by the (p,q) web geometry. Taken together, these constructions suggest that junctions, defects, and compactification data in string/M-theory admit a natural description in terms of modal structure on lightcones, with logical semantics providing a controlled way to track which field configurations are realized, suppressed, or confined to nilpotent internal sectors.

Abstract: We derive the Standard Model particle spectrum from 168 discrete walk-states on the Fano plane, the minimal projective geometry underlying octonionic multiplication. Beginning with three fluxions {−1, 0, +1} and triadic closure, exactly 42 computational operators (glyphs) emerge from combi- natorial necessity: 7 lines × 3 strides × 2 orientations = 42. These glyphs generate 168 walk-states through four topologically distinct walk types, organizing into 14 Frobenius orbits. Fundamental constants appear as geometric eigenvalues: the fine-structure constant α⁻¹ ≈ 137 at row 137, the gravitational boundary G⁻¹ ∝ 168 at row 168, with ratio \( 168/137 \approx \sqrt{\pi/2} \) bridging circular and dis- crete projection basins. The framework maps all six quarks, charged leptons, electroweak bosons, the Higgs, and eight gluons to specific orbits, with mass ratios and mixing angles approximated to 0.1– 2% accuracy via transforms involving π, ϕ, and structural integers. Unmapped orbits provide dark matter candidates—states that project to 4D spacetime but do not couple to the electromagnetic channel. This approach reduces the Standard Model’s 26 free parameters to geometric consequences of octonionic projection from 8D to 4D, with residual errors within projection tolerance.

Abstract:

This paper presents a method for describing the differential equations of motion of mechanical systems using the Kuznetsov tensor. Traditional approaches to solving equations of motion rely on vector and matrix methods, but the proposed approach allows for significant simplification and generalization of problems by using a system state tensor. The paper discusses the main principles of working with the Kuznetsov tensor, which describes the evolution of the system in a unified context. Specifically, it outlines a method for integrating the equations of motion for various mechanical systems, such as oscillations in a two-mass spring system. Conditions for damping oscillations and controlling amplitude are also considered, expanding the applicability of the Kuznetsov tensor in engineering calculations. The advantages of the proposed approach include a more compact representation of the system of equations, ease of analyzing invariants and symmetries, and the ability to apply the method to multi-linked and multi-component systems. The use of the Kuznetsov tensor for modeling the dynamics of various systems represents a step toward a more universal approach in mechanics and engineering.