Abstract: This paper demonstrates the irrationality of π from dual perspectives of geometric construction and physical reality. The central thesis posits that the irrationality of π is not a mathematical coincidence but a necessary condition for the logically consistent existence of the “ideal circle” as an ideal geometric object. Through retrospective analysis of the Gauss polygon approximation method, combined with the helical propagation model of photons derived from Maxwell’s electromagnetic theory, this paper reveals a profound isomorphism between mathematical irrational numbers and physical reality – the propagation of light essentially constitutes a “non-closing” circular motion, mathematically corresponding to the infinite non-repeating nature of π.

Abstract: Eight Lie algebras of point-symmetric groups and corresponding generators are admitted by the equation of motion, which a general time-dependent quadratic Hamiltonian obtains. We show that invariant quantities obtained by eight algebraic generators are the Wronskian constant, three time-dependent conserved quantities, which are quadratic forms in position and momentum, and the trivial, 0. All obtained invariant quantities are represented by auxiliary conditions, which are two linearly independent solutions of a homogeneous differential equation of the equations of motion. Invariant variables associated with an invariant consisting of the linearity of x and p is defined. It shows that if the motion of the system is oscillatory, the Poisson bracket of the two invariant variables 10 is obtained as i, and in the case of monotonic motion, it is obtained as 1.

Abstract: In this paper, we investigate the Ricci flow from the fundamental perspective of deformation kinematics of Riemannian manifolds, a research direction that has not been systematically explored in existing literature. By combining the geometric framework of Ricci flow with the kinematic theory of continuous medium deformation, we rigorously prove that the Ricci tensor is equivalent to the velocity gradient (strain rate tensor) of the manifold under the Hamilton Ricci flow equation \( \frac{\partial g_{ij}}{\partial t}=-2R_{ij} \). We further extend this core conclusion through detailed tensor analysis, dimensional analysis, and the study of manifold area element evolution, and correct the dimensional imbalance of the original Ricci flow equation by introducing a characteristic kinematic parameter $\kappa$. The results of this paper provide a new kinematic perspective for the study of Ricci flow, enrich the geometric connotation of Ricci flow, and build a bridge between differential geometry and continuum mechanics, which is expected to provide new analytical tools for the study of Ricci flow singularities, long-time behavior and Poincar\'e conjecture related problems.

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: 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: The Collatz ($3x+1$) conjecture remains a formidable enigma in number theory, largely due to the unpredictable, pseudo-random fluctuations of its discrete integer orbits. To bypass the limitations of traditional analytic number theory, this paper introduces a novel interdisciplinary paradigm by translating discrete arithmetic rules into a continuous dynamical sandbox. Specifically, we establish a rigorous topological isomorphism between the $3x+1$ map and the continuous Logistic map $x_{n+1} = 1 - \mu x_n^2$ locked at the superstable period-3 window ($\mu \approx 1.7549$). By constructing a customized Markov partition anchored at the unstable fixed point, the continuous system naturally enforces a "forbidden word 11" grammar, perfectly mirroring the arithmetic constraint that an odd operation must be followed by an even one.Furthermore, by extracting the high-precision eigenspectrum of the Perron-Frobenius transfer operator, we analytically prove a 2:1 ergodic invariant measure for contraction (even) and expansion (odd) states. Crucially, by aligning the theoretical escape rate dictated by the operator's spectral gap with the empirical stopping-time decay of $10^8$ large integers, we demonstrate that the $3x+1$ iteration and one-dimensional dissipative transient chaos belong to the exact same Universality Class. Ultimately, this study transforms an unpredictable Diophantine equation into an inevitable thermodynamic collapse, providing a groundbreaking continuous spectral analysis framework and a potent physical heuristic for the conjecture's global convergence.

Abstract: A graph-theoretical approach to the analysis of motion and rest in many-body systems is developed. Point bodies are represented as vertices of a complete bi-colored graph, termed the motion–rest graph (MRG). Two vertices are connected by a rust-colored edge when the corresponding bodies are at rest relative to each other, that is, when their mutual distance remains constant in time; bodies moving relative to each other are connected by a cyan edge. It is shown that the logical structure of the relation “to be at rest relative to each other” determines the combinatorial structure of the graph. For one-dimensional motion in classical mechanics and special relativity, this relation is reflexive, symmetric, and transitive, and therefore defines an equivalence relation. As a result, rust edges form disjoint complete cliques corresponding to rest-clusters, and the MRG becomes a semi-transitive complete bi-colored graph completely determined by the partition of bodies into equivalence classes. It is proved that any such graph on five vertices necessarily contains a monochromatic triangle. For two- and three-dimensional motion, the transitivity of relative rest generally fails because constant mutual distance does not imply equality of velocities in the presence of rotational degrees of freedom. In this case the MRG is non-transitive, and the Ramsey threshold becomes the classical value R(3,3)=6. The approach is extended to mixed sets containing moving bodies and reference points, including the center of mass of the system. Generalizations to general relativity and quantum mechanics are also discussed. In general relativity, transitivity of relative rest is generically lost because global rigid congruences do not generally exist. In quantum mechanics, exact transitivity survives only at the level of idealized delocalized eigenstates, whereas for physically realizable localized states the notion of mutual rest becomes only approxi-mate. The results demonstrate that the interplay between kinematics, logical properties of relational motion, and Ramsey-type combinatorial constraints gives rise to unavoidable ordered substructures in many-body systems.

Abstract: We study the Euclidean prime-shell stage of the Finite Ring Continuum programme through a framed finite field and establish three main components. First, after fixing an extended frame, the shell supports a finite orbital complex combinatorially equivalent to the two-sphere; the labeled complex is frame-presented, while its cellular isomorphism class is shell-invariant. Second, its core Euclidean symmetry axes appear with increasing framing depth: additively framed negation, affinely framed reciprocal inversion, and Euclidean conjugation in the extended frame. Third, standard density and Lipschitz estimates show that sufficiently large prime-shell grids reproduce the observer-side outputs of any fixed bounded Euclidean experiment to any prescribed tolerance. The Fourier formalism is recorded strictly as a discrete Fourier transform over the shell ring, while contact with conventional continuum Fourier language is treated only as an observer-side large-prime approximation.

Abstract: We present a strictly finitist formulation of Schr\"odinger-type and Dirac-type dynamics in the Finite Ring Continuum, together with exact information counts for reversible and compressive shell maps. The construction uses a symmetry-complete prime field, its quadratic extension, and the Frobenius involution to define finite Hermitian state spaces and finitist Hamiltonians. On Euclidean shells, continuum time evolution is replaced by a Cayley update that preserves the Hermitian form exactly and therefore produces periodic trajectories. On the Lorentzian extension, we construct explicit gamma matrices, a finitist Dirac operator, its associated Klein-Gordon factorization, and a covariant lifted boost action. To connect the formalism with entropy and information theory without leaving strict finiteness, we measure finite maps by their image counts and exact loss factors. This separates reversible transformations, which preserve distinguishability exactly, from shell power maps, which merge states by a computable arithmetic factor. All results are finite, algebraic, and exact; no limits, differential calculus, or continuum structures are used.

Abstract: The Pauli exclusion principle is traditionally introduced in quantum mechanics as a postulate encoded in the antisymmetry of the fermionic wavefunction. While extraordinarily successful, this formulation leaves open a deeper question: why must nature forbid the perfect overlap of identical fermions? In this work, we propose a reinterpretation of Pauli exclusion within the framework of Viscous Time Theory (VTT), where physical law emerges from the geometry of informational state space under constraints of memory, recoverability, and causal trace preservation. We propose that the coincidence of two identical fermionic states can be interpreted, in informational-geometric terms, as a loss of injectivity of the causal mapping, i.e., to an informational singularity where distinct histories become non-separable. To prevent this collapse of recoverability, the joint state manifold naturally develops a “diagonal barrier”: a forbidden submanifold where the informational cost diverges and admissible trajectories are repelled. Within this perspective, antisymmetry of the wavefunction appears not as the cause of exclusion, but as its mathematical symptom. Within this perspective, Pauli exclusion can be interpreted as a geometric and informational constraint rather than a primitive quantum axiom. The framework further suggests a unified interpretation of the difference between fermions and bosons: the former may be viewed as carriers of identity-bearing, non-overwritable informational structure, while the latter correspond to additive excitations that do not threaten causal injectivity. In this way, the exclusion principle appears as a consequence of informational geometry in a universe characterized by viscous time and memory.

Abstract: Hilbert’s Sixth Problem challenges us to rigorously axiomatize physics, particularly the bridge between microscopic dynamics and macroscopic laws. Yet, a conceptual gap remains: probability is usually treated as a fundamental assumption rather than a derived consequence of physical evolution. To address this, we introduce a Viscous Time Theory (VTT) framework governing evolution through admissibility, coherence, and recoverability. Applying an informational action principle, probability naturally emerges as an induced statistical measure over bundles of admissible trajectories. We validate this approach by analyzing a viscous-time kinetic transport operator, mapping out its contraction semigroup structure, spectral gap, and hypocoercive convergence. We further extend the model to nonlinear interaction kernels and evaluate its hydrodynamic scaling limit. Our analysis proves this diffusion-driven operator achieves strict spectral stability, exponential entropy decay, and global nonlinear stability. Furthermore, the macroscopic scaling limit rigorously yields nonlinear diffusion dynamics for coherence density. Ultimately, this provides an analytically tractable layer connecting microscopic evolution to macroscopic behavior. It demonstrates that probability, irreversibility, and transport laws can cohesively emerge from informational geometry, advancing the structural program envisioned by Hilbert.

Abstract: The hierarchy problem---the seventeen-order-of-magnitude separation between the electroweak scale and the Planck scale---remains one of the most compelling open questions in theoretical physics. Within the Standard Model, quadratically divergent quantum corrections to the Higgs mass require extraordinary fine-tuning at every loop order, with no underlying physical explanation. We propose a geometric suppression mechanism in which the electroweak scale arises naturally from the local curvature geometry of singular cycles within a six-dimensional Calabi--Yau compactification of Type~IIB superstring theory. When toroidal cycles degenerate, string-theoretic $D$-brane defects form at the resulting singular loci, and monopole-brane recoil governed by the Nambu--Goto action produces massless spin-2 gravitons that propagate into the higher-dimensional bulk. The local singularity energy density, controlled entirely by the curvature scale of the collapsed cycle, determines the electroweak mass scale without free parameters. A complementary brane-instanton mechanism generates the hierarchy exponentially from a geometric action of order thirty-seven, naturally reproducing the observed ratio of electroweak to Planck scales. We derive an explicit four-dimensional effective action from Kaluza--Klein reduction, demonstrate three independent consistency limits, compare the mechanism with Randall--Sundrum warped geometry and supersymmetric approaches, and outline a programme for embedding the proposal in explicit compactification models.

Abstract: This paper develops a unified and comprehensive framework for Hopf-like fibrations on Calabi–Yau spaces, with emphasis on when topological fibration data is compatible with Ricci-flat Kähler geometry and with compactification constraints from string/M-theory. We prove obstruction statements for smooth compact settings by combining characteristic-class constraints, Leray–Serre transgression, and rational formality, and we contrast these with constructive local models in hyperkähler and singular regimes where circle and higher-sphere fiber structures remain geometrically meaningful. New contributions (v4). This version resolves all major open problems identified in the prior literature and in earlier versions of this manuscript. We prove: (1) a complete finite classification of Hopf-like fibrations on compact CY₃ orbifolds (16 admissible isotropy types, ≤ 47 diffeomorphism classes); (2) the sharp constant C(n) = n/(4π²) in the Ricci-flat Hopf inequality; (3) an exact spectral gap formula for CY submersions; (4) a complete classification of MCF singularities preserving Hopflike structure (Types I/II/III, with Type III being conifold transitions); (5) finiteness and explicit count (2741 for the quintic) of Hopf-like flux vacua; (6) a Hopf-like analogue of the Cardy formula with logarithmic corrections from CFT twist operators; (7) a foundational p-adic theory of Hopflike fibrations with crystalline Euler class and p-adic instanton sums; (8) a constructive proof of the Cobordism Conjecture for CY₃ compactifications via Hopf-like geometric transitions; (9) an L-function factorization theorem establishing a Hopf-like BSD analogy (proved for K3 surfaces). These results together constitute a resolution of the main structural questions in Hopf-like fibration theory on CY manifolds, from both geometric/topological and string-theoretic perspectives. The manuscript includes explicit diagnostic workflows—minimal-model growth estimates, low-degree homotopy exact-sequence tests, and spectral-page bookkeeping—designed for reproducible analysis. The main conclusion is precise: strict Hopf behavior is severely limited on smooth compact Calabi–Yau manifolds, while robust Hopf-like structures naturally appear in local, singular, and effective-field-theory phases, and these are now completely classified.

Abstract: We introduce a Resonance--Particle Classification Framework (RPCF) that attempts to bring some order to the correspondence between the vibrational modes of strings and the particle content of the Standard Model. Building upon earlier work connecting hypercomplex manifold geometry with particle genesis via chiral permutation cycles and upon a detailed study of Calabi--Yau compactifications with Euler number $\chi = \pm 6$ that reproduces three net chiral generations, the present manuscript develops a unified classification in which closed-string, open-string, and Ramond--sector modes are mapped, respectively, to the gravitational sector, gauge bosons, and fermionic matter of the Standard Model. We introduce a hierarchical labelling scheme based on mode number, boundary condition, and symmetry representation, and we show how Calabi--Yau topology constrains the degeneracy of these resonances to yield exactly three particle families. The Atiyah--Singer index theorem and its cohomological refinements are used to quantify generation multiplicity, while the Standard Model gauge group $\SU(3)\times\SU(2)\times\mathrm{U}(1)$ is recovered from appropriate bundle holonomy choices on the compactification manifold. We further discuss composite resonance interference as a pathway toward an effective description of hadronic states. It must be stressed at the outset that this work is a conceptual proposal, not a completed derivation. Exact vibration--particle correspondences are not established here; they remain a genuine open problem. The analysis is intended to illuminate structural patterns and suggest productive research directions, not to assert a confirmed physical identification.

Abstract: Keith numbers form a rare class of integers defined by a digit-generated linear recurrence in which the original number reappears within its own sequence. Although known for several decades, their structural properties and the mechanisms underlying their extreme sparsity remain poorly understood.In this work, we introduce a dynamical reformulation of Keith sequences by embedding the digit recurrence into a discrete state-space system governed by a companion matrix. Within this framework, the recurrence trajectory can be interpreted as an orbit of a finite-dimensional linear dynamical system. This representation enables the introduction of trajectory observables—including informational inertia, an admissibility field, and a stability functional—which characterize the evolution of the sequence.Using this formulation, we analyze the spectral structure of the recurrence operator and show that the reappearance of the original integer corresponds to a transient intersection between the expanding trajectory and a fixed identity hyperplane in state space. Representative numerical scans over increasing integer ranges confirm that such identity-return events are extremely rare and occur only under tightly constrained dynamical conditions.These results suggest that Keith numbers can be interpreted as non-generic return events in a linear dynamical system determined by digit-based initial conditions. The proposed framework provides a dynamical explanation for their empirical sparsity and offers a basis for studying digit recurrences using tools from dynamical systems, spectral analysis, and computational number theory.

Abstract: We analyze the spherically symmetric complex diffusion and special type of the complex reaction-diffusion equations with the self-similar Ansatz. These equations are form invariant to the free Schrödinger equations and to Schrödinger equations with power-law space dependent potentials. The self-similar Ansatz couples the spatial and temporal variables together instead of the usual separation, therefore new type of solutions can be derived. For both cases analytic solutions are presented which are the Kummer's and the Whittaker functions with complex quadratic arguments. The results are analyzed in depth. In the second case the role of the complex angular momenta is investigated as well.

Abstract: We investigate the mechanisms by which natural systems encode data across multi-dimensional spaces. Integrating principles from information theory, probability, and geometry, we propose that certain Lie Groups govern these encoding processes. We first demonstrate that evenly distributed information becomes computationally unsolvable in higher dimensions. If we do not notice the "curse of dimensionality," it is because nature likely uses geometric positional notation at a rudimentary level. By extending the definition of representational cost to m dimensions using Benford’s Law, we identify a cost minimum at powers of Euler’s number (е^m). We introduce the "Lie Squad" (B3, F4, G2, A2, A1, and E6), a set of six compact simple Lie groups whose irreducible representations coincide with this ideal cost when m matches the group’s algebraic rank. These irreps facilitate a fundamental, rank-invariant number system based on balanced ternary, uniquely encoding integers as the difference of two natural numbers in bijective notation. Finally, we examine the Weyl orders of the Lie Squad members to show that Weyl divisors yield a logarithmic scale consistent with Benford’s Law and the universal number system proposed.

Abstract: The dynamic interaction between self-propelled constant-speed motion and surface topology is a fundamental problem in active matter physics and the control of surface-climbing robotics. While tangential trajectories on curved manifolds are well-documented, the curvature-coupled dynamics normal to the surface remain underexplored. In this paper, we present a rigorous analytical framework for the normal dynamics of a point mass constrained to move with a strictly constant tangential projection speed (\( V \)) over a smooth two-dimensional manifold. By applying the Weingarten map (Shape Operator) within a Newtonian framework, we derive the governing equation for the normal distance \( D: D'' \approx V^2 k_n - F_N/M \), where \( k_n \) is the normal curvature along the instantaneous trajectory and \( F_N \) is the applied normal force (e.g., gravity or adhesion). This reveals a purely geometry-induced inertial lift term, \( +V^2 k_n \), generated by the non-holonomic constraint of maintaining constant speed on a curved path. We establish the analytical threshold for surface detachment (\( V^2 k_n > F_N/M \)) and demonstrate that this effect is highly anisotropic on non-spherical surfaces. The core kinematic identity linking the normal acceleration to the inner product of velocity and the normal vector's derivative is formally verified using the Lean 4 theorem prover. Our findings provide a generalized mathematical tool for predicting the lift-off of active particles and calculating the minimum adhesion requirements for autonomous robots navigating complex topological surfaces.

Abstract: Modern physics rests on two pillars: general relativity and quantum field theory. However, they are not yet unified, and observations of dark matter and dark energy suggest shortcomings in existing theories. This paper presents a comprehensive reconstruction and extension of the Xuan-Liang unified field theory. Starting from first principles, we define Xuan-Liang as the line integral of power along a path, filling the geometric hierarchy of physical quantities (mass, momentum, kinetic energy, Xuan-Liang). A key advancement is the generalization of the Xuan-Liang concept to multi-velocity components, i.e., the Xuan-Liang of a complex system (such as a galaxy) is the product of its various characteristic velocities (e.g., rotation, revolution, bulk motion). This naturally leads to a modified Newtonian potential of the Yukawa form: Φ(r) = −GM r [1 + δ(1 − e−r/λ)], where the coupling strength δ and characteristic scale λ arise from multi-velocity coupling. Based on the action principle, we rigorously derive the unified field equations, demonstrating their self-consistency and their reduction to general relativity, Newtonian gravity, and cosmology. The theory’s explanatory power is demonstrated through applications: (i) it perfectly fits galaxy rotation curves from dwarf galaxies to the Milky Way and Andromeda, spanning a huge mass range, without requiring dark matter particles; (ii) it provides a dynamical dark energy model whose energy density smoothly transitions from matter-like behavior (w ≈ 0) at high densities to cosmological-constantlike behavior (w ≈ −1) at low densities, consistent with cosmic acceleration; (iii) it predicts testable modifications to black hole thermodynamics and strong-field gravity, including changes in black hole shadows and gravitational wave signals. The multi-velocity construction not only resolves the theoretical inadequacy of singlevelocity Xuan-Liang in explaining galactic dynamics but also builds a mathematically self-consistent, experimentally testable unified framework. Finally, we discuss prospects for quantization and a roadmap for future observational tests.

Abstract: An operational meta-model is presented for transitions among quantum, classical, relativis- tic, and thermodynamic descriptions without forcing a single master state space. Unification is performed at the level of observable predictions: each formalism produces an output in a common space Y defined by a feature map Φ (moments, spectra, correlations, or other functionals). Convex weights are assigned via a standard soft selection rule (softmax / Gibbs form) from losses, with entropic regularization and a complexity penalty (AIC/BIC/MDL) to reduce bias toward overly expressive models. Physics-facing priors are encoded through calibrated dimensionless knobs for decoherence/classicality, relativistic severity, and ther- modynamicity, yielding a regime-aware gating layer. A simple out-of-catalog diagnostic (surprise/residual monitoring) is included to flag persistent mismatch that may indicate missing observables or missing models. A minimal case study template (harmonic oscillator with a bath) and an acid test in the NISQ regime (critical decoherence) are outlined as reproducible validation pathways.