Physical Sciences

Abstract: The mathematization of science is undergoing a structural transformation driven by the rise of computation and data-intensive methods. While classical mathematization relied on explicitly defined laws and formal structures, contemporary scientific practice increasingly encounters mathematical objects that arise as outcomes of dynamical and algorithmic processes. This paper introduces the notion of computationally emergent structures to describe entities generated and stabilized through the interaction of parameterized models, optimization dynamics, and data. We develop a minimal formal framework in which such structures are characterized as asymptotic outcomes of learning dynamics and show that, in over parameterized regimes, they are selected by implicit variational principles not specified a priori. This framework provides a unified account of implicit regularization, kernel regimes, and stability phenomena in modern learning systems. These results show that contemporary learning systems operate according to implicit variational principles in which geometry, dynamics, and data jointly determine effective mathematical structure. They thereby identify a shift from representation to dynamical emergence, extending the scope of mathematization toward a theory of structure formation grounded in computation.

Abstract: Continuous gauge symmetries are usually introduced through Lie groups acting on quantum fields. In this paper we show that the algebraic structure associated with non-abelian gauge symmetry already arises naturally inside the complex group algebra of a finite non-abelian group. The dihedral group D₄, the symmetry group of the square, is used as an explicit example. The complex group algebra C[D₄] decomposes into irreducible matrix blocks under the Artin–Wedderburn theorem. While the character table describes only the subspace of class functions, the full group algebra contains additional intra-class directions invisible to the character table. For D₄ these directions form a three-dimensional subspace which, after elementary normalization, satisfies the Pauli algebra and generates continuous SU(2) transformations inside the two-dimensional irreducible block. The construction is carried out explicitly using only the multiplication table of D₄. The continuity of the complex coefficients allows continuous rotations to arise through exponentials of finite group algebra elements, without requiring the underlying symmetry group itself to be continuous. The mechanism generalizes to any finite group possessing higher-dimensional irreducible representations, where the associated matrix blocks naturally support the corresponding su(N) Lie-algebra structures.

Abstract: Whether the vibrational picture of elementary particles; now nearly half a century old; determines the spectrum we observe has remained without a settled answer. We close the question. At a fixed compactification of the heterotic string, the entire low-energy content follows from the geometry of the chosen Calabi–Yau manifold. Across compactifications, no internal rule singles out the one nature uses: the choice depends on input from outside the framework, of a kind we make explicit. Once that input is supplied, masses and mixings are computable; without it, no derivation is possible. Background. The picture of elementary particles as vibrational modes of a string is forty years old, and a steady catalogue of explicit heterotic constructions has shown that the observed gauge group and three-family chiral content can be reproduced. What no construction has settled is the converse question: does the resonance–particle correspondence, taken as a classification programme, determine the observed spectrum, or merely admit it? The literature has answered case by case; a framework-level resolution has been absent. Methods. We separate the framework into an unconditional mathematical layer (index theorem, Dolbeault cohomology, slope-polystability, Calabi–Yau metric existence, unobstructedness, BRST classification, anomaly cancellation) and a selection layer (choice of compactification datum). The achievable range of topological and cohomological selectors over the resulting landscape is computed by Kodaira–Spencer deformation theory and the special geometry of the complex-structure moduli space. Results. At fixed admissible datum the perturbative massless spectrum is fully determined by bundle cohomology and representation branching (positive direction). No topological or cohomological rule singles out the observed vacuum; the obstruction is a nontrivial class on a positive-dimensional moduli component (negative direction). A closure-completeness theorem unifies the two directions; universality, maximality, rigidity, stable-obstruction, categorical-impossibility, and quantitative-dimension theorems show the result holds for every framework whose predictive data is cohomological and is not improvable from within. A corrected audit lemma, with its converse, identifies the singleton condition any external selector must satisfy. Five residual closure theorems — selector completion, internal no-go, fixed-vacuum Yukawa computability, algebraic exotic lifting with post-stabilisation threshold, and ensemble finiteness — reduce the residual problems to one external axiom.

Abstract: This paper explores the role of dimensional analysis as the fundamental grammar that decides which physical expressions can be meaningfully compared before any dynamics is established. We develop this grammar inside the Finite Ring Cosmology framework. In this setting, spatial and temporal dimensions arise as frame readings of a finite symmetry space of admissible reference frames, while conventional units such as metres and seconds enter as observer-assigned modular domains. The shell structure itself is invariant under changes of observer frame and unit convention, although its numerical values change with the chosen scale. The resulting unit-domain algebra reproduces the familiar rules of dimensional analysis: quantities can be added only within the same domain, products and ratios move between domains, and physical invariants are precisely the expressions with neutral total domain. The construction gives a finite-domain reading of the constants that connect mechanics, quantum phase, and gravitation. The speed of light appears as a finite, observer-invariant upper boundary relating spatial and temporal scale assignments: its numerical value changes with units, but the boundary does not. The Planck relation forces energy to share the temporal recurrence domain: h converts frequency measured relative to the chosen time scale into frequency measured relative to the complete phase cycle, not into a separate modular unit domain. The mass domain is derived from energy and speed, and Newton’s constant is identified as the conversion domain for gravitational geometry.

Abstract: A four-dimensional anisotropic metric branch is defined for traceless gauge-side stress tensors of non-null Rainich type. The construction is motivated by the Alena Tensor correspondence between flat force-density and curvilinear geometric descriptions, but the branch data are fixed locally. The branch has a 2+2 form with one anisotropy field. This field is obtained both from the gauge stress and from a normalized vorticity-flux closure, while the remaining tensor-force term is represented by the Levi-Civita geometry only when a trace-adjusted branch tensor satisfies the Codazzi condition. Under these restrictions, the closed branch gives a common geometric language for elementary-mode data. Mass is read as scalar curvature response, electric charge as transverse-frame holonomy, and color as a three-dimensional multiplicity space of equivalent self-dual branch modes. After a two-dimensional weak space is adjoined, preservation of the top form gives S(U(3) \times U(2)), and the even exterior algebra gives the anomaly-free representation content of one Standard Model generation, including \nu_L^c. The same branch normalization fixes a primitive coupling $g_B$; with G_F and one intermediate scale E_B=2.64\times10^{14}\,\mathrm{GeV} as inputs, the resulting branch-level one-loop evaluation gives a consistency check for m_W, m_Z, m_h, and \alpha_s(m_Z). The numerical Yukawa matrices, confinement energy, and global family spectrum are left as dynamical problems.

Abstract: The solution theory of Sylvester-type equations finds wide applications in control theory, robotics, and image processing. This paper systematically surveys, classifies and summarizes the existing research results of three classes of Sylvester-type equations: matrix equations, tensor equations, and operator equations. It extracts nine mainstream research methods and clarifies the internal correlations among these methods as well as their applicable equation types. This work establishes a complete framework for solving Sylvester-type equations and, together with four prior review articles, forms a comprehensive framework for linear equations. It not only provides a systematic theoretical foundation and a clear research thread for subsequent researchers, but also offers valuable methodological insights for further investigations in related fields.

Abstract: We construct electromagnetic Kantowski–Sachs (KS) solutions in covariant teleparallel F(T) gravity using the coframe/spin–connection (CSC) formalism. In the restricted branch considered here, the Maxwell conservation laws (CLs) impose strong restrictions on the anisotropic scale factors and lead to the scaling ρ_{em} A_3^{−4}. We derive the corresponding symmetric and antisymmetric field equations (SFEs and AFEs) and formulate a reconstruction scheme in which F(T) is determined from the KS dynamics rather than imposed a priori. Power-law (PL) and exponential (EXP) coframe ansätze generate distinct invariant reconstruction branches, including scaling cosmologies, teleparallel de Sitter (TdS) regimes, and KS black-hole-interior-like reconstruction branches. The resulting models are organized using the Coley–Landry invariant classification and analyzed through leading-order stability conditions F_T>0 and F_{TT}>0.

Abstract: We are concerned with the Riemann problem for the Aw-Rascle (AR) traffic flow model with variable velocity offset. The model describes the traffic flow under different road conditions. The stationary wave is introduced in the traffic flow, which is determined by an ordinary differential equation. The resonance phenomenon and coalescence of waves are analyzed. Uniqueness of the Riemann solution is also discussed. We further study the interaction of elementary waves case by case, under the framework of the characteristic analysis method. Numerical simulations are also given to verify our analysis.

Abstract: The paper introduces a fundamental shift in the representation of physical reality, moving from a particle-based paradigm to a Recursive Complex Representation of 5 scaling levels (RCR) of a “hypophenomenal” geometric model. A unique scaling base ξ is defined that is deduced from the Planck constant and the gravitational constant G. The model posits that space is not a static container but a Plenum (David Bohm’s name for vacuum) of Planck contiguous cells (P-cells) whose vibrations constitute the fundamental energy of the universe. Masses are “trapped light”, viewed as localized vibrational resonances of the signal c that maintain a portion of the signal within contiguous groups of P-cells, and can propagate along the plenum by keeping their internal vibrational configuration. The fine-structure constant α acts within a universal renormalizing factor, strictly related to the scaling factor ξ = (GL_P)^1/5. Experimental masses, across 60 orders of magnitude (from the neutrino to the Sun) are retrieved from their ξ logarithmic localization with respect to Planck’s mass. The fundamental equations of Planck, Einstein, and de Broglie are not independent postulates, but natural geometric emergences of the signal’s vibrational dynamics. Gravitational force and constant G are formulated in terms of matter aggregation and dynamic curvature of the signal in the Plenum.

Abstract: The hydrogen atom has historically played a foundational role in the development of quantum mechanics, where its discrete energy spectrum is conventionally derived from solutions of the Schrödinger wave equation. In this work we present an alternative formulation in which the hydrogen spectrum emerges without invoking the Schrödinger equation. We consider a semi-quantum framework in which the electron is treated as a classical particle governed by Poisson-bracket dynamics while interacting with a quantized electromagnetic field described using second quantization. The electron moves in the Coulomb potential generated by the proton and simultaneously couples to quantized electromagnetic modes through minimal coupling. The Coulomb system possesses a hidden dynamical symmetry characterized by the Runge–Lenz vector, which enlarges the rotational symmetry to an group for bound states. Within this framework, interactions between the classical particle and the quantized field induce an effective commutation structure in the particle’s phase space. Once this structure emerges, the algebra of the conserved quantities associated with the Runge–Lenz vector becomes identical to the operator algebra used in Pauli’s symmetry-based derivation of the hydrogen spectrum. Consequently, the discrete hydrogen energy levels arise naturally from the combined effects of Coulomb symmetry and particle–field interaction, offering a physically transparent interpretation of atomic quantization and suggesting that wave–particle duality may arise dynamically from interactions with quantized electromagnetic fields.

Abstract: This article deals with sixth-degree polynomial equations and, more specifically, describes the necessary conditions under which the solutions of these polynomials can be expressed in closed-form radicals. For this purpose, eight different cases of sixth-degree polynomial equations will be discussed in detail. It is concluded that if a specific relation exists between the constant term and the remaining coefficients of a sixth-degree polynomial equation, then its solutions can be obtained by formulas expressed as functions of the initial coefficients and ultimately represented in terms of radicals. Each case of the polynomial discussed in this article is presented in the “problem–proof” format.

Abstract:

We provide a coordinate-free characterisation of phase boundaries in field theory by proving that a complex scalar field on a globally hyperbolic spacetime with boundary admits a stratified covariant phase space. The stratification is governed by a diffeomorphism-invariant functional $P$ partitioning spacetime into strata, together with a finite-energy selection rule: in the dense stratum $\{P \geq P_\star\}$, a diverging phase-stiffness functional $\kappa(P)$ forces any finite-action tangent vector to satisfy $\delta\theta = 0$, reducing the admissible variation class to amplitude fluctuations alone. We show that this selection rule simultaneously enlarges the presymplectic kernel of the augmented symplectic form $\Omega^{\mathrm{aug}}_\Sigma$ and suppresses the central extension of the boundary charge algebra: $K_{dens} = 0$. The Phase Boundary Characterisation Theorem establishes that these two effects are algebraically equivalent, identifying $\mathcal{H}$ as the unique degeneracy locus of $\Omega^{\mathrm{aug}}_\Sigma$ --- a purely coordinate-free characterisation independent of the specific trigger functional. The Iyer--Wald--Zoupas ambiguity in the boundary symplectic density is resolved by explicit mixed boundary conditions, and the algebraic structure on each stratum is compatible with standard quantization procedures applied independently per stratum.

Abstract: The Collatz (3x + 1) conjecture remains one of the most challenging open problems in number theory, largely due to the unpredictable, pseudo-random fluctuations of its discrete integer o rbits. This paper introduces an interdisciplinary approach by translating discrete arithmetic rules into a continuous dynamical sandbox. Specifically, we construct a symbolic analogy between the 3x + 1 map and the Logistic map f (x) = 1 − µx² locked at the superstable period-3 window (µ ≈ 1.7549). By building a customized threshold partition anchored at the unstable fixed point, the continuous system naturally enforces a “forbidden word 11” grammar, mirroring the arithmetic constraint that an odd operation (T(n) = 3n + 1) must produce an even number. Through the eigenspectrum of the Perron-Frobenius transfer operator, we demonstrate a 2:1 ergodic measure ratio for contraction (even) and expansion (odd) states—a direct geometric consequence of the period-3 attractor structure. We validate the ro-bustness of the spectral quantities through convergence studies across multiple discretization schemes. Null-model controls show that the sandbox captures aspects of the global stopping-time distribution that a generic forbidden-11 Markov chain does not, while run-length analysis reveals that local arith-metic statistics (ν2-valuations) are better reproduced by the simpler null model. This mixed result delineates the sandbox as a partial surrogate: useful for global transient statistics, but not a replacement for the actual arithmetic dynamics. This study offers a heuristic framework positioning coarse-grained transient dynamics as a null-model approach for Collatz statistics, with explicitly characterized failure modes.

Abstract: Similar to electroweak interaction, strong force and electromagnetism can have similar Higgs mechanism mediated interaction. Thus, gluons can acquire mass. And, neural colored gluons have larger mass than colored gluons. Total, we can have eight gluons without red-anti-red gluon. The puzzle of proton or neutron mass can be solved. We can also derive a new SU(5) model to include all the above eight gluons, three W/Z bosons, photon, Higgs boson, three generations of leptons and quarks to make a new 5x5 SU(5) model. Wightman axioms can be fulfilled in this new SU(5) without causing proton decay crisis. We can also add the 4x4 four dimensional spacetime tensor integrating mass-energy density, light pressure, electric fields, and magnetic fields as well as four gradients to make a new contravariant SO(10) model. Weyl tensor and Ricci tensor related to the new SO(10) model are also given. Thus, grand unified theory or theory of everything can be obtained, that is compatible with four dimensional spacetime without extra-dimension needed in string theories.

Abstract: Hodge theory is a powerful framework for modeling and analyzing vector fields, as found in fluid and electro- dynamics. However, its full power and modern formulation, especially on manifolds, remain largely inaccessible to non-specialists due to the substantial prerequisite in modern mathematics. This article aims to fill this gap by providing a comprehensive, self-contained introduction to Hodge theory, specifically tailored for an audience versed in traditional vector/tensor calculus and seeking to understand the modern formulation of this theory. We choose orientable closed surfaces as our pedagogical setting, due to their conceptual simplicity, mathematical tractability, and physical relevance (as boundaries of three-dimensional regions). Crucially, this setting is perfect for elucidating the complete structure of Hodge theory, particularly its topological and geometric aspects---elements frequently absent or obscured in treatments rooted in classical physics. We accomplish this through a parallel development of the modern exterior calculus and a dedicated 2D vector calculus on surfaces, followed by a series of specifically designed analytical and numerical examples. Furthermore, recognizing the fragmented historical development of the subject---a primary source of the conceptual gap for modern readers---we include a concise historical synopsis to bridge this divide.

Abstract: Anchor-based bipartite graph methods provide linear scalability for multi-view clustering, but most of them construct graphs in the original feature space, where high dimensionality distorts the proximity between samples and anchors and degrades graph quality. In addition, the K-means step commonly used to discretize spectral embeddings produces different cluster assignments across random seeds. To address these limitations, this paper proposes Projection-Enhanced Bipartite Graph Learning (PEBGL), a unified framework that jointly performs subspace projection, bipartite graph construction, consensus graph fusion with adaptive view weighting, and discrete label assignment. Every subproblem admits a closed-form or deterministic solution, so the algorithm runs in linear time and produces reproducible cluster labels for any fixed initialization. Experiments on six benchmark datasets demonstrate that PEBGL achieves consistently competitive accuracy across all evaluation settings and improves over the strongest baseline by up to 4.8 percentage points. These results confirm the effectiveness and generality of the proposed framework.

Abstract: This paper integrates a theorem-bearing dual-time core with an explicit reconstruction of Lorentzian geometry from stabilized event statistics. Starting from a dual-time architecture --- generative inner time, completed outer time, and a completion/projection interface between them --- the manuscript develops admissible-history monads, completion and observation functors, Hilbert-fiber realizations, quadratic carrier algebras, compact phase symmetry, a stabilization interface from completed histories to candidate event structures, and a factorized geometry map from stabilized measures to weighted discrete or continuum Lorentzian geometry. The central representation-theoretic result is retained and sharpened in context: a strongly continuous action of \textit{S}\textsuperscript{1} on a real Hilbert space induces, on each nontrivial irreducible sector, a canonical orthogonal complex structure, unique up to conjugation. A further rigidity statement remains decisive: every symmetric bilinear form invariant under the compact phase action on such a sector is a scalar multiple of the Euclidean one. Hence no nondegenerate split form can be preserved on the same compact phase sector. This gives a concrete mathematical reason that phase and causal readout must be carried by different algebraic branches. The new synthesis supplied here is that completed observable histories can feed a stabilization stage whose output is a probability measure on candidate event structures, and that this measure in turn determines effective causal and metric observables. Around this bridge the paper proves local continuity of the reconstruction data, threshold recovery of a sharp effective order, consistency of volume-based and chain-based proper-time estimators, a continuum recovery theorem showing that reconstructed order together with reconstructed volume determines a Lorentzian metric up to diffeomorphism and coarse graining in a manifoldlike regime, and an exact light-cone-strip benchmark converging to flat $(1+1)$-dimensional Minkowski geometry. The resulting contribution is structural and constraint-based rather than elemental. Familiar ingredients are placed in one controlled emergence chain: compact recurrence yields complex Hilbert phase sectors, completion and projection yield observable irreversibility, stabilization yields persistent event statistics, and reconstructed order plus volume yield Lorentzian geometry. The paper does not derive gravitational dynamics. It does show that complex phase structure and spacetime geometry can arise in one framework at different levels, with a mathematically forced separation between their invariant quadratic carriers.

Abstract: We investigate the consequences of modeling space–time as a finite-information substrate composed of Planck-scale quantum memory cells. In this framework, physical laws traditionally formulated in a continuous setting emerge as effective descriptions of an underlying discrete, finite-capacity system. We introduce the concept of finite-information deviations, defined as systematic differences between ideal continuum dynamics and their realizations on a discrete informational structure. These deviations arise from bounded Hilbert space dimension, local imprint accumulation, and coarse-graining of quantum information. We demonstrate that such deviations manifest as additional contributions to effective field equations, naturally reproducing phenomena previously attributed to dark matter, dark energy, and quantum gravitational corrections. The framework preserves global unitarity and locality while introducing structured, scale-dependent modifications to continuum physics. We outline observational consequences and discuss the role of these deviations as fundamental features rather than imperfections of physical law.

Abstract: We introduced a possible electric charge forming mechanism, which includes the quasi normal mode calculations for a Kerr black hole with area of and spin 2 and the states generation by the preon model. We think the electric charge may exist as energy in the 3+1 space time with no need for additional dimension. And we think there might possibly be a tiny Kerr black hole net in our universe, which is sparse for electric charges and will select out the energies corresponding to electric charges as the only possible propagating wave energies. This net may at least be another possible way to have electric charge quantization except confinement, especially when we have to treat it as quantized energy propagating in the 3+1 space time. We also showed that electric charge may be the source of a Berry curvature to curve the 3+1 space time to form the conventional electromagnetic field. Observation considerations have also been given. Future gravitational wave detections may offer opportunities to check the ideas proposed here.

Abstract: The article explores an epistemological framework for understanding existence, symmetry, complexity, and randomness as emergent phenomena arising when a large-but-finite complex totality is represented through lower-complexity observational subsystems. We propose that existence is not a binary property, but an epistemological category determined by the measure of a system's structural symmetry over time. Chaos, randomness, and infinity are reinterpreted as epistemic markers --- thresholds of comprehension rather than fundamental properties of reality. Through this lens, we examine fractals, cellular automata, and quantum uncertainty, arguing that apparent uncertainty emerges from the compression of finite universal structure into observable forms. The article argues that all localized systems, from particles to cognitive processes, are projections of the universe's total informational structure. This paradigm reframes emergence, not as the accumulation of local interactions, but as the revelation of global coherence through representational compression. By situating existence and complexity within this framework, the manifesto outlines a programme-level foundation for understanding the interconnectedness of phenomena and the unity of the universe as a singular, self-reflective process.