All Submission Categories

On Physical Aspects of the '(st)ringy'

Authors: Kohji Suzuki
Comments: 22 Pages.

We try to let the '(st)ringy' (viXra:1912.0532 [v1]) explain massive graviton, leptonic spin-3/2 particle, and so on. Meanwhile, we come across a number system which we tentatively call `hexanion', as if it were a by-product.
Category: Mathematical Physics

The Error [?] in Schr"odinger's Equation

Authors: Jorma Jormakka
Comments: 4 Pages.

The article points to the basic error in the wavefunction concept. Thesolution for the Schr"odinger equation for the hydrogen atom is not an orbital, itis an orbit, especially, it is an elliptic orbit. The error is that the Quantum Mechanical solution ignores the angular momentum conservation law which in quantumphysics is expressed as quantization of angular momentum. The Schr"odinger equationmust obey this rule as it is basically the same as de Broglie's wavelength, which isthe stated motivation for making the momentum substitution.
Category: Quantum Physics

A Derived Equation for Electron Mass Calculation

Authors: Yvan Raverdy
Comments: 3 Pages.

Here, we establish an equation for electron mass which depends on the age of the Universe T by combination between the result of our reference article [1] and Space time description [2], this equation is so depending on the four physical constants. That allows us to calculate T ; the numerical value obtained matches that deduced from the analysis of the cosmic microwave background (CMB) with a greater precision.
Category: High Energy Particle Physics

Gravitational Analogy of Casimir-Effect from Qed to Gravity Description in GRT by Extended GRT-Gravity Without Ostrogradsky Ghost-Problem as Higher Derivative of Effective Gravitative Fieldtheory. a Global Fixpoint-Solution.

Authors: Holger Döring
Comments: 36 Pages.

Classical GRT of Einstein-description of gravity-theory can be extrapolated to higher structures. This extension adds higher-order terms to classical general relativity because GRT itself can be assumed to be a first-order expansion of spacetime. These higher corrections can lead to new observations in the vacuum, with gravitational waves, or black holes. Furthermore, an attempt is made to transfer the Casimir effect of quantum electrodynamics as an analogy from the electromagnetic vacuum to a spacetime description and to describe this analogy as a gravitational effect within GRT.
Category: Quantum Gravity and String Theory

The Environment Layer: Building Infrastructure for Agentic AI Training

Authors: Fei Wang, Eric Wang, Salon Ren
Comments: 45 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

The emergence of Agentic Reinforcement Learning (Agentic RL) has created an urgent need for sophisticated training environments that go beyond traditional RL benchmarks. While conventional LLM-RL operates within single-step MDPs, Agentic RL requires environments supporting multi-turn interactions, tool integration, and verifiable reward signals. This white paper argues that RL environments are the foundational infrastructure for agentic AI---the critical layer that determines what capabilities agents can learn and how reliably they transfer to deployment.We present a comprehensive analysis of the environment layer, organized around three core questions: (1) Environment Design---what makes an effective Agentic RL environment, including observation spaces, action interfaces, and reward mechanisms; (2) Environment Infrastructure---the frameworks, protocols, and tools for building and deploying environments at scale; and (3) Environment Quality---methodologies for evaluating environment fidelity, the sim-to-real gap, and production readiness.We survey the ecosystem of environment frameworks (OpenEnv, GEM, MCP), synthetic environment generation pipelines (Agent World Model, Reasoning Gym), and specialized environments for embodied AI (NVIDIA Cosmos, Isaac Sim). We introduce the Environment Quality Framework (EQF) for systematic environment evaluation and analyze the critical sim-to-real gap through the User-Sim Index. Finally, we present a research agenda for next-generation RL environments that will enable the transition from research prototypes to production-ready agentic systems.
Category: Artificial Intelligence

A Theory Analogous to Relativity Established Using the Doppler Effect and Delayed Potential

Authors: Xianxiang Duan
Comments: 38 Pages. In Chinese (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This article presents a theory analogous to relativity. Starting from the velocity term in the delayed potential or Doppler effect formula, it establishes a mathematical framework similar to relativity but with distinct physical interpretations by drawing analogies between electromagnetic fields and gravitational fields. The theory proposes the existence of a "mass field force" through such analogies. Serving as an effective theoretical framework within flat spacetime, it reconstructs the primary observable effects of general relativity at solar system scales through Doppler-weighted corrections to the delayed potential, without introducing curved spacetime geometry. This approach provides a novel physical interpretation while predicting the absence of singularities.
Category: Relativity and Cosmology

Dynamics of the Memristor FitzHugh-Nagumo Equation and the Memristor Chua Circuit Equation

Authors: Makoto Itoh
Comments: 47 Pages.

In this paper, we show that a voltage-controlled nonlinear resistor can be regarded as a voltage-controlled memristor when its behavior is observed using the time derivatives of state variables. Next, we derive the memristor FitzHugh-Nagumo equation by differentiating the original FitzHgh-Nagumo equation with respect to time. We also show that the trajectory of the memristor FitzHugh-Nagumo equation is highly dependent on the initial flux condition. This trajectory can be changed by applying a small single pulse. Furthermore, the square wave forcing creates many complex behaviors such as chaotic attractors and limit cycles. These behaviors also depend on the initial flux condition. Therefore, it is impossible to predict the trajectory without knowing the initial flux condition. Next, we observe the trajectory's evolution by adding a small DC bias current to the square wave forcing. We show that as time progresses, the trajectory changes shape and eventually shrinks. This resembles the refractory period of neurons. Finally, we study the signal transmission between the two memristor FitzHugh-Nagumo circuits which are connected by a unity-gain buffer. We show that, for the signal to be transmitted almost exactly, the initial flux condition must be identical. Additionally, we derive the memristor Chua circuit equation by differentiating the original Chua circuit equation with respect to time. Our findings are nearly identical to those for the memristor FitzHugh-Nagumo equation. We also study the signal transmission between two memristor Chua circuits that are connected by a unity-gain buffer. We show that nearly perfect transmission requires the identical initial flux condition. It should be noted that requiring the identical initial flux conditions results in more secure communication with the memristor Chua circuits than with the original Chua circuits. Finally, we would like to emphasize the following points: The memristor FitzHugh-Nagumo equation and the memristor Chua circuit equation both exhibit rich dynamics. The identification of the circuit elements depends on the coordinate system used to observe them.
Category: Mathematical Physics

A Structural Framework for Classical Dynamics I: Singular Limits and the Emergence of Discrete Motion

Authors: Tom Robert Franck Cooper
Comments: 10 Pages.

This work investigates the emergence of discrete dynamical structures from classical and relativistic formulations of motion. Using standard differentiation rules, we examine how Newtonian dynamics can be expressed in terms of energy gradients within a consistent formal framework. The relativistic kinetic-energy function is then analyzed in two limiting regimes: the classical limit ($beta to 0$) and the relativistic limit ($beta to 1$), where its asymptotic behaviour becomes singular.In the singular regime, the dominant contributions of the kinetic-energy expansion exhibit a hierarchical structure that naturally leads to weighting factors of the form $(n+tfrac{1}{2})$. This structure is shown to be consistent with the discrete spectrum of the quantum harmonic oscillator, suggesting that certain quantum features may be related to underlying classical asymptotic behaviour.The analysis is complemented by a geometric interpretation of electromagnetic interactions using Gaussian units, together with a generalized functional perspective for handling singular contributions. Within this framework, the introduction of a small interaction scale—analogous to a Yukawa-type modification—provides an effective description of interaction ranges.Overall, this approach provides a conceptual bridge between classical mechanics, relativistic dynamics, and quantum-like discretization, highlighting the role of singular limits, normalization, and geometric structure in the emergence of discrete spectra.
Category: Mathematical Physics

A Structural Framework for Classical Dynamics II: Energy—Momentum Conservation and Transport Structure

Authors: Tom Robert Franck Cooper
Comments: 11 Pages.

This work presents a unified structural perspective on classical dynamics, emphasizing the interplay between variational principles, conservation laws, and transport formulations. Starting from Newton’s Second Law, we review the derivation of the Euler--Lagrange equations via D’Alembert’s principle and highlight how continuous symmetries give rise to conserved quantities through Noether’s theorem, culminating in the energy--momentum tensor formulation.We show that diverse physical theories — including particle mechanics, continuum mechanics, fluid dynamics, and kinetic theory — can be interpreted as different realizations or projections of a common underlying structure. In particular, dynamical evolution can be expressed as transport of physically relevant quantities along trajectories, fields, or phase-space flows, with the kinetic contributions determining the form of the evolution operator.Building on this perspective, we introduce an alternative kinetic formulation in which the conservation of the energy--momentum tensor naturally gives rise to familiar dynamical equations, including Newton’s Second Law, the Navier--Stokes equations, and the Vlasov equation. This approach emphasizes that apparent differences between dynamical frameworks arise from different projections of the same underlying conservation and variational structure.Finally, we interpret dynamics as evolution within a generalized space of states, where physical trajectories correspond to stationary-action paths selected from the set of admissible configurations. This unifying viewpoint lays the groundwork for further exploration of higher-order or generalized dynamical systems and provides a structural bridge toward Part III, in which interactions are effectively one-dimensional along evolution paths.
Category: Mathematical Physics

A Structural Framework for Classical Dynamics III: Effective Interaction Fields and the Three-Body Problem

Authors: Tom Robert Franck Cooper
Comments: 16 Pages.

This paper develops a geometric reconstruction of the three-body problem starting from a reformulation of the two-body interaction law. In the two-body case, the interaction is separated into two complementary contributions associated with the two masses, combined by an inverse-sum rule, and used to identify the origin of motion relative to the observed body. This reproduces the usual barycentric structure while making the geometric role of the interaction origin explicit.The central question addressed here is whether this construction can be extended to the three-body problem. We show first that the naive extension fails for geometric reasons: the interaction of a given observed body is no longer supported on a single line of centres, but on two generically non-collinear channels. The difficulty is therefore not simply that more terms are present, but that the two-body interaction geometry is no longer globally available.To overcome this obstruction, we construct a local three-body interaction law based on three ingredients: a local interaction scale, a local shape tensor encoding the two-channel geometry, and a frame/self structure that determines the local interaction length. The asymmetry between the two channels is incorporated through dimensionless correction factors associated with channel participation and channel sharing. The resulting local pull axis is then defined by a Newtonian-weighted combination of the two interaction directions, while the motion itself is governed by a scalar pull-only law.The construction is tested on the principal benchmark solutions. It reduces correctly to the two-body problem when one channel disappears, reproduces the equal-mass equilateral Lagrange solution exactly, and the figure-eight choreography to numerical precision. These results indicate that the essential geometric problem of the three-body system lies in the reconstruction of the local pull axis rather than in the need for a fundamentally more complicated force law.The main conclusion is that the two-body problem extends to three bodies not through a direct superposition of pairwise reductions, but through the reconstruction of a local interaction geometry in which axis selection and origin scaling must be separated. Within this framework, a scalar pull-only law becomes sufficient once the correct local pull axis and origin has been identified.
Category: Mathematical Physics

A Stochastic Framework for Real-Time Quantum Field Dynamics

Authors: Xiao Lin
Comments: 40 Pages.

We propose a Stochastic Formulation that establishes a direct mathematical correspondence between the real-time Feynman path integral in Minkowski spacetime and the expectation values of classical stochastic processes. This framework offers an alternative approach to quantum dynamics by formulating evolution through intrinsic stochastic processes rather than relying solely on pre-discretized spacetime backgrounds. Specifically, we demonstrate that the unitary dynamics of scalar fields can be mapped to continuous Wiener processes, while spinor fields correspond to discrete Poisson jump processes. Distinct from conventional methods involving perturbative expansions, Euclideanization, or Grassmann algebra, our formulation provides a non-perturbative, real-time framework where quantum amplitudes are derived statistically from stochastic trajectories.We implement this framework via a grid-based tree recursion scheme for the Klein-Gordon field, benchmarked against exact solutions of the forced harmonic oscillator. For the Dirac field, we derive an analytical closed-form finite difference scheme that effectively models its evolution. By integrating these schemes, we successfully apply the framework to the Yukawa coupling model and extend it to QED. Our results reveal non-trivial dynamical features, such as feedback-driven mass oscillations, offering complementary insights to standard perturbative descriptions. Crucially, the structural nature of this stochastic approach inherently avoids the fermion doubling and sign problems often encountered in lattice approach. These applications suggest a robust pathway for tackling complex systems, including gauge theories like QCD.
Category: Quantum Physics

Ordinary Synchronization-Compensation Transformations of Coordinate and Time Increments in the People's World

Authors: Pavlo Danylchenko
Comments: 8 Pages. small correction to show correct article name in title bar of PDF file

The fallacy of using such a false parameter as the coordinate velocity of light in GR is proven. This coordinate velocity of light, unlike the limit velocity of matter, cannot in principle be less than the true velocity of matter. The suitability for use in the people's world of only ordinary synchronization-compensation transformations of increments of spatial coordinates and time is substantiated. These transformations, unlike the alternative to them ordinary Lorentz transformations (OLT), allow to proportionally synchronize all clocks that move in a gravitational field by inertia. The suitability of OLT (due to the inherent to OLT blueshift of the frequency of centripetal radiation instead of redshift) for use only in the comoving with expanding Universe frame of references of spatial coordinates and time is proven. Newton's law of gravity is obtained directly from the condition of no change in the flow of the proper time of matter during its inertial motion in a gravitational field. The fallacy of ignoring the compensation of gravitational dilation of time by the inertial motion of matter is most thoroughly substantiated. And this is in good agreement not only with relativistically invariant thermodynamics, but also with the equations of the dynamic gravitational field of both the Solar System and flat galaxies.
Category: Relativity and Cosmology

Scattering of Spin 1/2 Particles by Coulomb Field

Authors: Miroslav pardy
Comments: 7 Pages. Original article

Using the Dirac equation for electron in the Coulomb field, we find the Green function by the iteration method and then we deduce the transition matrix. The corresponding dierentialcross-section is then derived. The article is written with the mathematical simplicity and the Schwinger pedagogical clarity.
Category: Quantum Physics

A Technique for Making Nuclear Fusion in Solids — Diversification

Authors: Richard Wayte
Comments: 14 Pages.

A technique is demonstrated for making nuclear fusion by shearing or electrically-igniting compressed mixtures of numerous hydride+catalyst powders. Controlled explosions have been generated which are stronger than exothermic chemical reactions for the materials.
Category: Nuclear and Atomic Physics

Goh-Schmutz Constant

Authors: Edgar Valdebenito
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific references!)

In this note, we give some formulas for the Goh-Schmutz constant
Category: General Mathematics

Why Relations Precede Existence: A Critical Examination of the Entity-First Assumption in Physics

Authors: Mephibosheth Thomas
Comments: 12 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We critically examine the ontological assumption that "existence precedes relations" — the view that spacetime, particles, and fields are fundamental entities with intrinsic properties, and that relations are derivative. Through a careful analysis of three key physical problems (the black hole information paradox, quantum non-locality, and the nonu2011renormalizability of quantum gravity), we argue that the entity-first assumption leads to deep conceptual tensions that resist resolution. We then propose an alternative: "relations precede existence", where fundamental reality consists of a network of relations, and spacetime, fields, and particles emerge as patterns in this network. We provide a precise mathematical definition of such a relational framework and show how it resolves the identified tensions. Moreover, we have previously demonstrated that from such a relational network one can rigorously derive free scalar quantum field theory and U(1) gauge theory in the continuum limit, providing concrete evidence that the relational framework is not merely philosophical but yields predictive physics. The paper does not claim a definitive proof but aims to establish the relational view as a viable and promising foundation for fundamental physics, worthy of serious consideration.
Category: History and Philosophy of Physics

Determining the Mass of the Universe from the Mass of the Earth Using Flow and Gravity Laws, Based on the Dualistic Model of a Dynamic and Static Earth

Authors: Tibor Endre Nagy
Comments: 13 Pages.

Based on the dualistic model of Earth moving with the expanding universe and considered static at the same time, the total mass of the universe can be determined using the flow laws and Newton's law of universal gravitation. A concentric sphere can be drawn in three dimensions around the Earth moving in the opposite direction compared to the receding galaxies. A tubular geometric shape can be fitted into this sphere in all directions from the center to the periphery. By increasing the radius of the tube of radius h expressed from the Einstein formula to the size of the radius of the universe (i.e. H), a continuous, axisymmetric cylinder can be formed. The speed according to Hubble's law, at a redshift of 3.14, is equal to the speed of light in a volume of radius h. Maintaining, this speed with respect to the universe, we can study the changes in the parameters of the cosmos through the flow laws of matter moving in one direction in a tube of varying cross-section. This approach, together with the application of gravitational attraction, can lead to the determination of the mass of the universe.
Category: Relativity and Cosmology

Accretion Delay in Massive Star Formation

Authors: Jan Makopa
Comments: 11 Pages.

It is bona-fide knowledge that, radiative feedback from the core of a pre-main sequence (PMS) star leads to a global accretion delay in stellar formation. How long the accretion delay lasts, remains poorly constrained in existing numerical and analytical model. This problem is more common in massive star formation where the global accretion delay leads to modulated accretion rates which effectively traps massive stars in a pre-ignition bottleneck, where they exhaust their energy supply before achieving stable nuclear fusion. Numerical tools capable of this task are lacking at present. We consider the present investigation an important step in this direction. Here we show that, in a radiative pressure-dominated regime, an unprecedented radiative contraction delay $t_{text{RD}}$, must exceed the Kelvin-Helmholtz timescale ($t_{text{KH}}$). This uncertainty poses a major stumbling block in our current understanding of stellar evolution.
Category: Astrophysics

Classification of Infinite Sets and Negative Proof of the Continuum Hypothesis

Authors: Zhi Li, Hua Li
Comments: 7 Pages.

This paper constructs a series of transcendental subsets of real numbers by creating a construction method based on infinite sets of transcendental numbers. It also proposes the concept that infinite sets can be divided into one-dimensional, two-dimensional, and multi-dimensional sets according to the number of degrees of freedom required for the expansion of their elements. For example, the set of natural numbers, the set of even numbers,and the set of odd numbers belong to one-dimensional infinite sets, while rational numbers belong to two-dimensional infinite sets, and so on. When the number of degrees of freedom is infinite, it becomes an infinite-dimensional set, and several infinite-dimensional sets can constitute a hyperdimensional set.This paper proves that the set of real numbers belongs to the hyperdimensional set, and itscardinality is far greater than the infinite power set of the cardinality of the set of naturalnumbers; between the set of natural numbers and the set of real numbers, there exist infinitely many number sets. The result of this paper disproves the continuum hypothesis.
Category: Set Theory and Logic

Summary of the Main Technical Procedures for Linking Tunisian Airports to the WGS84 Global Geodetic System

Authors: Abdelmajid Ben Hadj Salem
Comments: 3 Pages. In French

This article is a summary of the main technical procedures for linking Tunisian airports to the WGS84 global geodetic system.
Category: Geophysics

Explaining the Spin, Zeeman Effect and Lamb Shift in Sommerfeld's Atomic Mode

Authors: Jorma Jormakka
Comments: 12 Pages.

It is claimed in Quantum Mechanics that Sommerfeld's atomic model is inable todescribe the full spectrum of hydrogen. This claim is justified by stating that Sommerfeld'satomic model cannot describe spin in the fine structure formula and that it cannot explainthe Zeeman effect. Both claims are shown incorrect in the article. The article points totwo serious errors in the Quantum Mechanical derivation of the fine structure spectrum formula and the Zeeman effect. The first is that there is a confusion of angular momentum andradial momentum: these effects are effects of radial momentum. The operators $J$ and $L$ inQuantum Mechanics, given as angular momentums, are actually radial momentums both in the finestructure spectrum formula and in the Zeeman effect energy levels. The second error is inthe calculation of eigenvalues of a sum of operators as the sum of the eigenvalues of the summandoperators. The last section commentson the last claimed gap in Sommerfeld's atomic model: that it cannot explain the Lamb shift.It is argued that the explanation of the Lamb shift in Quantum Mechanics is not a valid explanation asQuantum Field Theory is shown incorrect by the anomalous magnetic moment of an electron.In Sommerfeld's atomic model the states $2S_{1/2}$ and $2P_{1/2}$ have different eccentricity andtherefore different energy level.
Category: Quantum Physics

Pi's Irrationality: Geometry and Logic

Authors: Timothy Jones
Comments: 3 Pages. A reader gave suggestions to provide more details.

We give a proof of Pi's irrationality using geometry and some logic.
Category: Number Theory

Deduction of Planck's Radiation Law Using Boltzmann Statistics Without Quantization Postulates

Authors: Antonio Jose Mejias Marquina
Comments: 2 Pages.

This work will demostrate that blackbody radiation is a thermodynamic phenomenon.
Category: Thermodynamics and Energy

Gravity Conservation of Energy and Absorption of Two Zero Mass Particles By a Spherical Mass

Authors: Karl De Paepe
Comments: 2 Pages.

We compare the energy, including gravitational energy, of a system of two zero rest mass classical particles before and after absorption by a solid spherical mass and show energy is not conserved.
Category: Relativity and Cosmology

Dark Energy and Zero Active Mass Equation

Authors: Fernando Salmon Iza
Comments: 5 Pages.

For decades, the prevailing view in the scientific community has been that our universe is expanding at an accelerating rate. However, a recent experimental study (Junhjuk Son et al., 2025) seems to confirm that our universe shows no signs of acceleration; that is, we might be living in a non-accelerating universe. In this work, we present evidence that this result agrees with the predictions of relativistic theory. To this end, we present a proof of the zero active mass equation (ρ + 3p = 0) for a universe with FLRW metric and zero space curvature (k = 0). We study the expansion of the universe by analyzing two partial models: a classical energy model and a thermodynamic fluid dynamics model. The interrelation between these two models demonstrates that, for universes with zero space curvature and a Lapse function equal to one, the valid solutions to the Friedmann equations are only those that lead to a non-accelerating universe. It is important to note that this proof also provides a result on the energy needed for the expansion of the universe, concluding that for a non-accelerated expansion no additional energy is needed.
Category: Relativity and Cosmology

Sigma-8 Anomaly as Potential Evidence for Fractal Spacetime (Part 2)

Authors: Ervin Goldfain
Comments: 36 Pages.

We conjectured in [1] that a non-vanishing and scale-dependent deviation of effective spacetime dimension from D = 4 generates corrections to the Poisson equation and to the cosmological clustering process. This sequel elaborates on how Fractional Poisson Equation offers a viable explanation of the Sigma 8 anomaly.
Category: Relativity and Cosmology

A Look at Sommerfeld's Fine Structure Spectrum Formula

Authors: Jorma Jormakka
Comments: 10 Pages.

The article gives the derivation of Sommerfeld's fine structure spectrum formulaand my comments on it. I think Sommerfeld's approach has been too easily discarded. (For those interested in whether a semiclassical model can explain the whole spectrum of hydrogen, I will look at the spin, the Zeeman effect and the Lamb shift in a separate paper. I cannot yet say what the result is. One can model them, yes, but does the model make sense, this I cannot yet say.)
Category: Quantum Physics

Non-inertial Relativity Theory, MOND and Generalized Gravity in Curved Phase Spaces

Authors: Carlos Castro
Comments: 22 Pages.

We pursue further our work on (Born Reciprocal) Non-inertial Relativity theory. Starting with a brief review of the theory, and how Non-inertial Relativity redefines the notion of mass, the phase space particle trajectories in $ D = 2 + 2 $ are revisited by emphasizing the key difference between a truly $U(1,1)$-invariant mass $ {cal M }$ and the Lorentz-invariant mass $m$. The derivation of the generalization of the special relativistic expression of $ E = m ( 1 - v^2 )^{ - 1/2} $ ($c=1$) to the non-inertial relativistic case follows. This, in turn, leads to the non-inertial relativistic extension of Milgrom's modified Newtonian dynamics (MOND) law. In the most general setting, one finds proper-time $ m ( tau ) $, and spacetime-dependent $ m ( x^mu)$ masses for point particles, when the proper force $ F $ depends on $ tau$ or $ x^mu$, respectively. By recurring to the tools of Finsler geometry, we finalize by writing the generalized gravitational field equations in curved phase space in the presence of matter sources, like particles and cosmic strings. As a result, both spacetime and momentum space are curved. We conclude with some remarks as to why curved momentum space should play an important role in quantum gravity.
Category: Quantum Gravity and String Theory

Information, Knowledge, and Intelligence as a Measure of Influence, Truth, and Efficiency

Authors: Friedrich Sösemann
Comments: 30 pages english, 30 pages german

The presupposed ontology describes a deterministic, digital world as interacting values in space and time. It defines properties, states, laws, and subjects. It thus makes it possible to describe information, knowledge, and intelligence as hierarchical relations and to propose measures of dependence on states, the truth of descriptions, and the efficiency of calculations. Describing objectifies storage, calculating objectifies thought.Besides clarifying and standardizing the still controversial terms, then the following conclusions are possible:For the truth of knowledge, not the same perception of the subjects is required, but only their temporal constancy.Despite the assumed determinism and universal causality, abstraction can lead to subjective chance and zero knowledge.
Category: General Science and Philosophy

Introduction to Bayesian Probability and Its Use in Theoretical Physics

Authors: John Hemp
Comments: 237 Pages.

Bayesian probability, has a long history, but has developed into its most useful, contemporary form, thanks largely to the work of E. T. Jaynes (1922-1997). A brief account of Jaynes’s theory of probability is given in this book, and examples of its profitable use in theoretical physics are given. These include its use in statistical thermodynamics (to clarify, in particular, the meaning of thermodynamic entropy), and its use in the theory of Brownian motion (to derive the diffusion equation and the concepts of particle drift-velocity and probability flux). Also discussed is the way in which rational Bayesian probability theory eliminates the paradoxes that block the way toward a realist interpretation of quantum mechanics, and promises a more rational formulation of quantum mechanics involving complex-valued Bayesian probabilities.
Category: Mathematical Physics

A Unified Geometric Interpretation of Time Dilation in Special and General Relativity

Authors: K. Sethuraman
Comments: 5 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)

Time dilation in Special Relativity (SR) and General Relativity (GR), discovered by Albert Einstein, is conventionally introduced through distinct mechanisms: relative velocity in SR and gravitational potential in GR. In contrast, this paper proposes a unified conceptual framework for interpreting time dilation in both SR and GR based on a single geometric principle, illustrated using a common light-clock model. The guiding heuristic is "longer path = slower tick." In SR, relative motion causes the light pulse in a moving light clock to follow a longer diagonal spacetime path between successive reflections, leading to time dilation. In GR, spacetime curvature(geodesics) generated by mass—energy causes the light pulse to follow a longer curved path between equivalent clock events, likewise resulting in time dilation. Rather than interpreting gravitational time dilation solely through abstract expressions involving gravitational potential, this approach emphasizes a geometric interpretation in which time dilation arises from the increased spacetime path length traversed by the light pulse in curved spacetime. This interpretation is consistent with the experimentally verified Shapiro time delay, which demonstrates that a light pulse requires additional propagation time when passing through regions of strong gravitational curvature near a massive body. This framework is intended as a heuristic and interpretive aid and it does not replace the full tensorial formulation of SR and GR or their quantitative predictions derived from the Einstein field equations.
Category: Relativity and Cosmology

Virtual Element Model: A Unified Theory of Spacetime and Matter Based on the Entropy-Pressure Mechanism

Authors: Haiwei Ren
Comments: 16 Pages. In Chinese (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

A unified physical model with the virtual element as the basic energy unit and entropy pressure as the core dynamic mechanism is proposed in this paper． The virtual element has an inherent motion property of the speed of light, with its free state corresponding to pure energy and bound state corresponding to matter． This model redefines the physical essence of time, space and gravity: time is the entropy increase process of matter returning to free energy, space is the distribution domain of entropy pressure gradient, and gravity is the generalized force induced by entropy pressure gradient． By introducing the entropy-time coupling relationship, the time dilation is directly correlated with the local entropy pressure field, and the Newtonian gravitational potential is derived under the weak field approximation, thus achieving compatibility with general relativity． The model uniformly explains the intrinsic connections of matter, energy, light, electromagnetic waves, thermodynamic laws and cosmic evolution, and puts forward testable experimental predictions． This study constructs a complete theoretical system for the unification of spacetime and matter, and provides a new feasible path for the establishment of a unified physical framework in physics．
Category: Classical Physics

Under What Conditions is the Energy of a Photon Proportional to Its Frequency?

Authors: Yefim Bakman
Comments: 8 Pages. 3 figures

When an artificial intelligence (AI) model was asked the question in the title, the answer was "Under all conditions." This response is understandable because, in the dominant paradigm, photons have no size, structure, or mass, and consequently, there is no photon model, but there is spin. When asked about spin, the AI model replied that spin is not the rotation of matter in space, but rather a representation of the Lorentz group. Planck's formula is a postulate, not the result of a physical consequence. The answer to the question in the title can be found within the framework of a new paradigm built on Tesla's insight. In this new paradigm, photons have size, structure, non-zero mass, and spin; thus, compared with the dominant paradigm, the new paradigm provides a closer match to classical physics.Here, based on this photon model, we derive a generalized formula for the energy and frequency of a photon, in which Planck’s formula is a special case.
Category: Classical Physics

Examples of Exams of Physical Geodesy of The German Geodesy School

Authors: Abdelmajid Ben Hadj Salem
Comments: 12 Pages. In French

This document includes the exams for the physical geodesy course given at the Institute of Geodesy of the University of Stuttgart during the period 2007-2013. It is an invitation to the students of the French school of Physical Geodesy.
Category: Geophysics

Relational Emergence Model: A Generative Extension of Relational Quantum Mechanics

Authors: Yoshifumi Maruko
Comments: 6 Pages.

This paper proposes the Relational Emergence Model (REM) as a generative extension of Relational Quantum Mechanics (RQM). While RQM defines physical states as relative correlations between systems,it does not explicitly address how such correlations are dynamically produced. The REM introduces emph{differentiation} as a generative process through which relational distinctions become articulated and subsequently stabilizedinto effective classical structures via decoherence. More precisely, differentiation is formalized as a bipartition-selecting map$mathcal{D}$---not a unitary operator, but a descriptive structural selection---that produces explicit system--observer correlations from apre-relational Hilbert space. Environmental decoherence then suppresses off-diagonal coherences, stabilizing effectively classical facts. Rather than asserting any equivalence between quantum theory and Buddhist philosophy, this work identifies a structural correspondence between relational quantum interpretations and the concepts of dependent origination(prat={i}tyasamutp=ada) and emptiness ('{s}={u}nyat=a). The central contribution of REM is a shift from the emph{definition} ofrelational states to the emph{modeling of the generation} of relational structure itself.
Category: History and Philosophy of Physics

On the Ontological Prerequisites of Initial Cosmological Conditions and the Limits of Physicalist Causality

Authors: Isaac Cohan
Comments: 10 Pages.

Modern cosmological models frequently posit that the universe emerged from a quantum vacuum or a zero-energy state, eectively asserting that the cosmos originated from nothing. This paper examines the logical, statistical, and informational limitations of such physicalist models. By analyzing the informational prerequisites of quantum uctuations, energy conservation, the statistical probability of universal metaphysical intuition, and the ontological anomaly of human consciousness, we argue that physical laws are non-contingent informational boundaries that can-not account for their own existence. The invocation of mathematical laws to explain the origin of the universe therefore necessitates a pre-existing, non-contingent source of that information. We further address prominent objections from evolutionary psychology, animal cognition research, and information-theoretic formalism to strengthen the case that a purely physicalist framework provides an incomplete accounting of reality.
Category: History and Philosophy of Physics

Reflections on the Implementation of a New Geodetic System

Authors: Abdelmajid Ben Hadj Salem
Comments: 4 Pages. In French

This note, which I wrote in 1988, is a reflection on a methodology for setting up a geodetic system for a country without geodesy.
Category: Geophysics

New Proof of Pythagorean Theorem

Authors: Lambros Stephanis
Comments: 2 Pages.

Proof Pythagorean Theorem using Euler's identity.
Category: Geometry

A Note on a Degenerate Circle and a Line via Division by Zero Calculus

Authors: Saburou Saitoh
Comments: 3 Pages. (Note by viXra Admin: Further repetition will not be accepted; please cite and list scientific references)

We consider the degenerate case of the general circle equation. By applying Division by Zero Calculus, we observe that two different finite quantities arise depending on the algebraic form of the radius. One corresponds to zero area, while the other gives the signed distance from the origin to the line.
Category: General Mathematics

Regarding the Massive, Contradictory Descriptions Concerning the Mathematical Axioms—A=A, A≠¬A, ¬A=¬A, and A∨¬A—and Mathematical Operations

Authors: Taiyou Ikiru
Comments: 31 Pages. In Japanses (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)

I demonstrate that, within mathematical operations, there inevitably exist instances where the axiomatic principles governing the mathematical axioms A=A, ¬(A∧¬A), and ¬A=¬A are not adhered to. In doing so, I reflect upon the relationship between logic and mathematics.
Category: General Mathematics

The Observer Factor

Authors: Iuri Bagaturia, Zaza Melikishvili, Anzor Khelashvili, Koba Turashvili
Comments: 14 Pages.

In the analysis below, we will show that the idea of a "quantum computer" contradicts the principles of quantum mechanics and its practical implementation will be impossible. The analysis points out the phenomena of theoretical physics, the incorrect interpretations of which contributed to the birth of this erroneous idea. One of these errors is the phenomenon of quantum entanglement", which is based on a misinterpretation of the "observer factor". This part analyzes the "observer factor" and the history of the appearance of this factor in the physical reasoning of quantum theory.
Category: Quantum Physics

Quantum Discreteness

Authors: Iuri Bagaturia, Zaza Melikishvili, Anzor Khelashvili, Koba Turashvili
Comments: 10 Pages.

The idea of "quantum computers" states that the discrete states of a quantum object can be used to create information bits. In this part of the text, we will show that the practical implementation of this idea will be impossible, since the arguments that are indicated as the basis of this implementation contradict the principles of quantum mechanics.
Category: Quantum Physics

The Classical Origin of Quantum Superposition

Authors: Iuri Bagaturia, Zaza Melikishvili, Anzor Khelashvili, Koba Turashvili
Comments: 10 Pages.

According to modern concepts of physics, the phenomenon of "Quantum Superposition" is a characteristic of quantum processes of the microcosm, which has no classical analogue. In this part of the text, we will show that this phenomenon is not a characteristic of the microcosm only, and such a phenomenon can be introduced in the probabilistic method of describing the processes of the macrocosm. At the same time, we will indicate the fundamental principles of probabilities — common to both the micro and macrocosm, according to which the phenomenon of "quantum superposition" will be presented as a special case of these principles.
Category: Quantum Physics

Quantum Superposition

Authors: Iuri Bagaturia, Zaza Melikishvili, Anzor Khelashvili, Koba Turashvili
Comments: 10 Pages.

According to the principles of quantum mechanics, a quantum superposition is defined by the sum of the basis vectors of a probability space. Depending on the physical circumstances under which the corresponding events with random outputs occur, these sums can be realized by an infinite number of different combinations of the mixing coefficients of these vectors. Theoretical concepts of a "quantum computer" assume that when creating an information bit, each physical state corresponding to a specific superposition sum can be used as a separate detail of this information bit. Based on this phenomenon, in the case of a single quantum object, the creation of many and potentially infinite classical digital bits is assumed, and it is precisely this phenomenon that corresponds to the "Q-bit." Our goal is to determine, from the standpoint of the principles of quantum mechanics, whether an empirical realization of the "Q-bit" is possible. Below, we demonstrate that a physical realization of this idea will be impossible, since the arguments cited as the basis for such an implementation contradict the principles of quantum mechanics and are based on false interpretations of these principles.
Category: Quantum Physics

Quantum Entanglement

Authors: Iuri Bagaturia, Zaza Melikishvili, Anzor Khelashvili, Koba Turashvili
Comments: 13 Pages.

One of the important details of the myth of "Quantum Computers" is "quantum computing", the main detail of which is the phenomenon of "Quantum Entanglement". In particular, if two Q-bits are brought into a "Quantum-Entangled" state, it will be possible to transmit information from one Q-bit to the other by the mechanism of "Terrible Long-range Action", that is - instantaneously. As a result of the creation of such technology, the speed of calculations - carried out in "Quantum Computers" will be significantly faster compared to the speed of calculations of conventional computers. Below we will show that the creation of such technologies will be impossible for a simple reason - there is no phenomenon of "Quantum Entanglement" in physical reality.
Category: Quantum Physics

Wave-Corpuscular Dualism

Authors: Iuri Bagaturia, Zaza Melikishvili, Anzor Khelashvili, Koba Turashvili
Comments: 19 Pages.

The principle of "wave-particle" dualism in quantum mechanics is introduced on the basis of the statement: all quantum objects behave as particles in some physical circumstances, and as waves in others. Using laser devices, we carried out simple demonstration experiments in physical circumstances in which the wave nature of the photon flow was expected to be revealed. Based on the analysis of these processes, we have easily shown that there are no empirical grounds for attributing a wave nature to quantum objects participating in these processes, i.e. photons. Analyzing the historical arguments for the introduction of the principle of duality, we conclude: the description of processes involving objects of the microcosm requires the creation of an alternative and more adequate theoretical concept than one based on the principle of dualism.
Category: Quantum Physics

Critical Notes on the Problem of Elastic Scattering of Particles in Non-Relativistic Quantum Mechanics

Authors: Iuri Bagaturia, Zaza Melikishvili, Anzor Khelashvili, Koba Turashvili
Comments: 11 Pages.

The paper critically examines the details of the corresponding theoretical problem of particle scattering in non-relativistic quantum mechanics. The discussion proceeds in two directions: one - the correctness of physical and mathematical interpretations of individual details of the scattering problem; second - checking the correctness of mathematical methods used in specific mathematical calculations. Details of the scattering problem are pointed out, whose existing interpretations contradict the basic principles of quantum mechanics. A detail of mathematical calculations is also pointed out, contradicting the principle of correct calculation and leads the corresponding calculations to erroneous results. After correcting this erroneous mathematical method, it becomes clear that the problem lies in the initial mathematical algorithm itself, which is used to formulate the theoretical problem of scattering.
Category: Quantum Physics

Establishment of a Network of Permanent GPS Stations and a Basic Geodetic Network in Algeria

Authors: Abdelkader Sellal, Abdelmajid Ben Hadj Salem
Comments: 8 Pages.

This article summarizes the IT equipment requirements (software and hardware) and technical methodologies for establishing a network of permanent GPS stations and a basic geodetic network for Algeria.
Category: Geophysics

Negative Proof of The Continuum Hypothesis

Authors: Zhi Li, Hua Li
Comments: 5 Pages.

The continuum hypothesis states that the set of real numbers is the power set of the countable set of natural numbers, the smallest uncountable set, and its cardinality is greater than that of natural numbers. There is no set whose cardinality is absolutely greater than that of countable sets and absolutely less than that of the set of real numbers.This paper proposes a method for constructing a subset of real numbers and proves that itscardinality exceeds that given by the continuum hypothesis. This result refutes the continuumhypothesis and pushes the cardinality of real numbers to a higher order of magnitude.
Category: General Mathematics

On the Ligo GW150914 Event Induced by Sudden Change of Gravity Forces of Ocean Tidal Bulges Acting on the Ligo Test Masses

Authors: Davor Palle
Comments: 14 Pages.

We argue that the sudden change of the sign of the difference between the gravity forces of oceantidal bulges on the test masses of two orthogonal arms of LIGO interferometers could be responsible for theLIGO GW150914 event at the time when the relative strainwas measured. The time coincidence of the events at Hanford and Livingston (USA) interferometers is explained by the moving ocean bulge crossing the critical point in the southern part of the Indian Oceanwhere the forces change the sign.The precise position of the critical point is defined by positions and orientations of the two LIGOinterferometers.During the LIGO GW150914 event,the Moon and the Sun were in very close zenith positions in East Africa.The approximate alignment of the Moon and the Sun with the Earth is present also at the time of theLIGO GW151226 and LVT151012 events. The local maxima of tidal forces exerted on the Indian Oceanappear once in a month.The future inclusion of the third Virgo interferometer in Cascina (Italy)proves to be decisive because it cannot produce the time coincident event with the two LIGO interferometerswithin the described scenario.
Category: Relativity and Cosmology

An Expository Application of Lagrange Interpolation to Diophantine Sequences and the Collatz Conjecture

Authors: Authman Jassim Mohammed
Comments: 3 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

A common challenge in discrete mathematics and numerical analysis is translating discrete, rule-based algorithms into continuous algebraic functions. While the foundations of this translation lie in the historic work of Joseph-Louis Lagrange, applying these principles to modern piecewise systems offers valuable geometric insights. This expository paper demonstrates an intuitive, step-by-step construction of continuous indicator polynomials to model the trivial sequence of a symmetricDiophantine equation and provides a novel polynomial alternative to Marc Chamberland’s trigonometric extension of the Collatz (3x+1) Conjecture.
Category: Number Theory

A Different View on the Nature of Physical Phenomena

Authors: Viktor S. Dolgikh
Comments: 18 Pages. (Note by viXra Admin: Further repetition may not be accepted)

This paper presents a fundamental particle L^S in the form of a "plum pudding J.J." blank, all of whose properties reflect the basic physical phenomena of nature, as well as its evolution from the source of formation.
Category: Nuclear and Atomic Physics

A Table of Pisano Period Lengths

Authors: Richard J. Mathar
Comments: 13 Pages. Reprinted from a version dated from 2012.

A Pisano period is the period of an integer sequence which is obtained by reducing each term of a primary sequence modulo some integer m >= 1.Each primary sequence which obeys a linear homogeneous recurrence with constant coefficients has such periods for all m >= 1. The manuscript tabulates the lengths of these periods indexed by m for a small subset of the OEIS sequences.
Category: Number Theory

A Mathematical Proof About Spatial Symmetry Invariance of Solution of the Three-Dimensional Navier-Stokes Equations

Authors: Shijun LIAO
Comments: 5 Pages.

We prove such a mathematical theorem that solution of the incompressible Navier-Stokes equations under periodic boundary condition has the same spatial symmetry for all t>0 as its smooth initial condition, if there exist the spatial symmetry for initial condition, the external force is steady-state, and the temporal Taylor series of velocity exists and has a non-zero radius of convergence for t >= 0. This theorem can be used to check the correctness and reliability of numerical simulation of Navier-Stokes equations for turbulent flows.
Category: Mathematical Physics

Shaking up Newton's Theory[:] The Anomalous Velocity of Comet Oumuamua

Authors: Henk Dorrestijn
Comments: 13 Pages. hjdorrestijn@gmail.com

In this article, we focus on a new interpretation of gravity. This view sees gravitational acceleration around a mass as a response to the disruption of spatial symmetry caused by the obstructive effect of the mass on the propagation of radiation. Newton's gravity formula requires some modification, but the key change is that gravity is not directed at the center of gravity of the mass at its actual location, but at the location where the moving mass was located when the radiation making the mass visible passed by. We call this interpretation the obstruction theory. This new theory is successfully applied in this article to calculate the anomalous velocity of the interstellar comet Oumuamua.
Category: Astrophysics

Cosmic Filaments Without Dark Matter: Cylindrical Vacuum Localized Structures as Exact Relativistic Solutions

Authors: Rudi Van Nieuwenhove
Comments: 8 Pages.

Cosmic filaments constitute one of the most prominent components of the large-scale structure of the Universe, forming coherent, elongated bridges between clusters and superclusters of galaxies. In the standard cosmological framework, these structures are interpreted as dark-matter-dominated density enhancements produced by the gravitational amplification of primordial fluctuations. In this work, we explore an alternative description based on Vacuum Localized Structures (VLS): self-gravitating solutions of the Einstein field equations characterized by a non-standard vacuum equation of state.We derive and analyze a new class of exact, cylindrically symmetric VLS solutions of the Einstein equations, appropriate to modelling extended filamentary systems. These solutions are supported by an effective stress—energy tensor with radial pressure p_r=-ρ/3, generalising the equation of state previously identified in spherical VLS configurations. We characterise the resulting space-time, study its gravitational field, and show that it naturally gives rise to coherent, non-singular, extended structures with finite mass per unit length.The present cylindrical solutions extend earlier work on spherical VLS, where galactic-scale configurations were shown to reproduce key features of observed rotation curves without invoking particle dark matter. Taken together, the spherical and cylindrical solutions provide a unified relativistic framework in which both galactic halos and cosmic filaments emerge as manifestations of vacuum-organized gravitational structures.We discuss the physical properties of cylindrical VLS, their potential stability, and their relevance to observed filament phenomenology, including coherence length, mass distribution, and large-scale dynamical effects. These results suggest that vacuum localized structures may offer a viable alternative foundation for the gravitational scaffolding of the cosmic web.
Category: Relativity and Cosmology

Lonely Runner Conjecture Proof

Authors: Bhaskar Kumar
Comments: 2 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

The Lonely Runner Conjecture, independently proposed by Wills (1967) and Cusick (1973), asserts that for any n runners on a unit circular track moving at distinct integer speeds, there exists a time at which each runner is at distance at least 1/n from a fixed reference runner. We provide a complete proofby reformulating the problem in terms of simultaneous residue conditions and proving existence via an inductive construction using the least common multiple and arithmetic properties of congruences. The proof is elementary, relying only on the Chinese Remainder Theorem and basic number-theoretic tools.
Category: Combinatorics and Graph Theory

Quantum-Elastic Geometry: From Geometric Extension to Linear and Non Linear Regimes

Authors: Juan Moreno Borrallo
Comments: 33 Pages. (Note by viXra Admin: For the last time, please submit article written with ANY AI assistance to ai.viXra.org!)

Supplemental Material of the main paper "Quantum-Elastic Geometry: From Geometric Extension to Linear and Non Linear Regimes"
Category: Relativity and Cosmology

Introduction to the Quantum Theory of Electrogravitation

Authors: Arcangelo Recchia
Comments: 110 Pages.

In the first chapter, the "conversion factors" are introduced, which are dimensionless coefficients used to evaluate the effect of accelerated expansion of spacetime on measurements. It is shown that these factors can be expressed through a common parameter, α, the fine-structure constant, which is linked to the ratio between the theoretical radius (13.6-13.8 Gly) and the measured radius (46.5-47 Gly) of the universe.

In the second chapter, it is demonstrated that dark matter and dark energy do not exist, and the gravitational effects attributed to them are actually caused solely by ordinary matter.

In the third and fourth chapters, Newton's theory of gravitation is first extended into the generalized gravitational theory and then into the generalized electrogravitational theory. The latter, among other things, predicts the existence of electrogravitational waves, suggesting that photon and graviton are the same particle and that the graviphoton is constituted by two elementary electric charges of opposite sign.

In the fifth chapter, Planck's law is derived from first principles. This implies that Planck's constant is no longer a fundamental constant but derivable from other constants.

In the sixth chapter, the electrogravitodynamic equations and the electrogravitational stress-energy tensor are developed.

In the seventh chapter, the quantization of Einstein's field equation is carried out.

In the eighth chapter, the possibility that elementary particles are in turn composed is examined and discussed: we have seen that the graviphoton corresponds to the elementary particle composed of two electric charges of opposite sign.

In the ninth chapter, the mass ratio between proton and electron (approximately 1836) is calculated.

In chapter ten, the concept of force is revised, as all particles can be exchanged, making them all carriers of force.

In chapter eleven, the multiverse with repeated gravitational collapses (like a matryoshka) is introduced.

In chapter twelve, the theory of information is integrated with the electrogravitational field: Information must be "contained" by something, whether it be a stone slab, a book, a CD, or any other means. This raises the question of whether there exists an ultimate level on which information can be encoded, the level of fundamental "0" and "1". This can only be spacetime itself (the electrogravitational field).
Category: Quantum Gravity and String Theory

Classical Derivation of Planck’s Relation and Constant Through the Poynting Vector

Authors: Arcangelo Recchia
Comments: 2 Pages. 5 Figures (Note by viXra Admin: An abstract labled as such is required in the article)

Planck’s relation E=hv establishes a direct proportionality between the energy of a photon and the frequency of the electromagnetic wave. In the conventional framework of quantum mechanics, Planck’s constant is introduced as a fundamental constant. Its value is determined experimentally and accepted as a starting point. In this work, however, it is shown that the relation E=hv, and thus Planck’s constant h, can be derived using classical electromagnetic theory.
Category: Quantum Physics

Dihedral and Solid Angles at the Apex of a Tetrahedron

Authors: Harish Chandra Rajpoot
Comments: 13 Pages. 5 Figures (Note by viXra Admin: An abstract labled as such is required in the article; further repetitions of similar articles may not be accepted)

In this paper, generalized analytical expressions are derived for determining the internal (dihedral) angles between consecutive faces of an arbitrary tetrahedron and the solid angle subtended at any of its vertices. The derivation is based on HCR’s Inverse Cosine Formula in conjunction with HCR’s Theory of Polygon. The resulting relations provide a simple and systematic method for evaluating the dihedral angles and the vertex solid angle when the three apex angles between the edges meeting at a given vertex are known. Owing to their general form, the derived formulae are also applicable to configurations in which three faces meet at a vertex of regular and uniform polyhedra, thereby enabling efficient computation of vertex solid angles in such solids.
Category: Geometry

Evaluation of Four Energy-Momentum Relations

Authors: Mueiz Gafer KamalEldeen
Comments: 5 Pages.

The aim of this research is to identify and classify the different formulas that describe the relationship between energy and momentum. We examine all the three currently known relations in addition to a newly proposed relation in terms of their range of validity and their physical significance. It is found that the proposed relation is the most broadly valid and strongest in terms of physical significance.
Category: Relativity and Cosmology

On a Unified Speculative Framework Connecting Cyclic Conformal Cosmology, de Sitter Holography, and Fuzzball Boundary States

Authors: Eugene McDonnell
Comments: 14 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We present a speculative theoretical framework that attempts to connect three distinct structures in contemporary theoretical physics: Penrose's Conformal Cyclic Cosmology (CCC), the de Sitter/CFT correspondence (DS/CFT), and the fuzzball description of black hole microstates in string theory. The central conjecture is that the asymptotic conformal boundaries of eachcosmological aeon — identified in CCC as the interface between successive aeons — may bereinterpreted as fuzzball boundary states within an eleven-dimensional M-theoretic substrate, with the intervening de Sitter geometry playing the role of a transient geometric bridge. TheEinstein-Rosen/Einstein-Podolsky-Rosen (ER=EPR) conjecture is proposed as the structureinvariant under conformal transitions, providing a mechanism for information preservation acrossaeon boundaries. We further conjecture that the apparent separation between past and futurefuzzball states is an artifact of dimensional projection from eleven to four dimensions, and that inflation may be interpretable as the geometric response to the quantum entanglement initialisation complexity of the nascent de Sitter bridge. Each component of this framework is explicitly classified by its epistemic status. No claim of mathematical derivation is made; this work is a speculative research programme identifying open questions and potential falsification directions.
Category: Quantum Gravity and String Theory

On the Problem of Measurement in Quantum Mechanics

Authors: Francesco Prandel
Comments: 6 Pages. In Italian

This paper examines the "measurement problem" posed by quantum mechanics. The discussion of the problem is based on the analysis of the concepts of "isolated physical system," "possible state of a physical system," and "interaction between physical systems." The proposed solution to the problem is suggested by W. Heisenberg's philosophical position on the theory, which he expressed in terms of the metaphysical binomial potentiality-actuality.
Category: Quantum Physics

Death Mechanism An Underlying Empowering System for Cosmic Evolution, Civilizational Trial-and-Error, and Ethical Revision

Authors: Xin Zhao
Comments: 16 Pages.

Based on an interdisciplinary perspective integrating astrophysics, evolutionary biology, civilizational sociology, and ethics, this paper constructs a progressive empowering model of the "cosmos-planet-civilization" three-level death mechanism. It systematically demonstrates that death is not the ultimate destiny of the universe and civilizations, but an underlying empowering mechanism running through materialcirculation, biological evolution, and civilizational advancement. The research shows that the death of stars and planets serves as the core engine of the matter-energy coupling cycle; through heavy element synthesis and celestial iteration, it lays the material foundation for the birth of life. The death of individual terrestrial organisms is a rigid link for planetary ecosystems to maintain low-entropyequilibrium, driving the evolutionary emergence of intelligent life through species iteration.The demise of early civilizations constitutes the trial-and-error threshold ofcivilizational evolution, providing empirical lessons for the ethical revision of maturecivilizations through costly practices. Combining empirical cases of the disappearanceof ancient civilizations, this paper clarifies for the first time the dialectical relationshipbetween "the empowering nature of death mechanisms" and "the autonomy of civilizational survival". It proposes that ethical revision is the core paradigm for civilizations to break through the trial-and-error destiny and master cosmic laws. The innovative value of thisresearch lies in breaking the traditional cognition of "death being opposed tocivilizations", reconstructing the death mechanism as a pre-empowering system forcivilizational evolution. It provides an interdisciplinary theoretical framework withboth scientific rigor and practical guidance for human civilization to achieve long-term survival through ethical revision and ultimately undertake the mission of delaying or reversing local heat death in the universe.
Category: General Science and Philosophy

The Compton Effect is a Relativistic Effect

Authors: Antonio Jose Mejias Marquina
Comments: 2 Pages.

In this work, the formula for the Compton effect will be derived using the conservation of electron energy and relativity.
Category: Relativity and Cosmology

Proactive Heuristic Synthesis (PHS): Addressing the Reactive Bottleneck Through Latent Idle Consolidation in LLMs

Authors: Ali Zulfiqar
Comments: 6 Pages.

Current Large Language Model (LLM) architecturesare fundamentally reactive, operating within a "Prompt-Response" paradigm that leaves significant computational resourcesdormant during inter-prompt intervals. This paper introducesProactive Heuristic Synthesis (PHS), an architecturalframework that enables models to transition into a state of LatentIdle Consolidation. Unlike "thought-at-inference" methods (e.g.,Quiet-STaR) which impose latency penalties, or static distillationmethods (e.g., Fast Quiet-STaR) which lack continuous adaptability,PHS shifts the computational burden to asynchronousidle cycles. The framework utilizes a Regret-Based Replaymechanism defined by the counterfactual delta between initialinference failure and post-exploration success to target high-valueoptimization trajectories. Unlike recent replay methods suchas SuRe, which prioritize high-perplexity samples for memoryretention, PHS prioritizes high-regret samples for reasoningevolution. By autonomously navigating these associations, themodel synthesizes novel heuristics verified via a Dual-ModelConsensus engine to prevent generation-verification collapse. Thispaper establishes the mathematical formulation and architecturalblueprint of PHS. We theoretically demonstrate that by shiftingfrom perplexity-driven retention to regret-driven evolution, PHSprovides an asymptotically optimal solution to the latencyreasoningtrade-off. Furthermore, relying on recent bounds inrecursive training, we formalize how our Dual-Model Consensusmathematically mitigates model collapse, offering a rigorouspathway to zero-latency self-evolution in LLMs.
Category: Artificial Intelligence

A Study on the Unified Dynamic Laws of Tropical Cyclone Vortices and Keplerian Systems

Authors: Yi Wang
Comments: 5 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This study takes typically near-circular tropical cyclones over multiple global sea areas as the research objects, and deploys observation profiles around the 64 kt, 50 kt and 34 kt wind radii. The core objective is to verify whether the V²R value in the full-scale main circulation region of tropical cyclones is a constant. All study cases adopt the wind circle data of the same tropical cyclone at the identical observation time to ensure the energy stability of the cyclone. The verification results show that the V²R conservation characteristic of tropical cyclones has cross-sea universality, and the fluctuation range of measured data is within a reasonable error interval; V²R is a deformation of the Kepler constant K (R³/T² = K), and tropical cyclones and the Keplerian system follow the unified dynamic law of central force field vortices.
Category: Geophysics

Why Billiard Balls, Soldier Crabs, and Quantum Computers Deserve the Same Research Funding

Authors: Sergei Esipenko
Comments: 17 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We examine the foundational assumption shared by quantum computing, classical analogcomputing, billiard-ball computing, and biological computing (soldier crabs): that continuous physical variables can be controlled with sucient precision to perform useful computation. We introduce concrete error-correction schemes for billiard-ball systems (BBEC) and crab-based systems (CEC), demonstrate that their failure modes are structurally identical to those facing quantum error correction (QEC), and show that the reasons we immediately recognize BBEC and CEC as unworkable apply with equal force to QEC at scale. We present a formal framework for comparing precision requirements across paradigms, address the linearity objection by showing that decoherence reintroduces eective chaos, and catalog 30 years of unfullled milestones inquantum computing. We conclude that the precision requirements for useful quantum computation are, by all available experimental evidence, physically unachievable, and that the quantum computing program rests on the same unfounded assumption as the billiard-ball computer: that continuous variables in physical matter can be controlled with arbitrary precision.
Category: Quantum Physics

On the Microstructure of Black Hole Singularity

Authors: Farid Abrari
Comments: This article is in English and has 7 pages.

The singularity of a black hole with a quantum index b was previously shown to be consisted of b concentric spherical shells of a constant diametral step size A, corresponding to a fundamental resolution interval in nature. An interior shell j was found to be a condensation of a Primordial Stem Particles (PSP) of mass m = h/Ac, at excitation state j and momentum jmc. The spatial uncertainty of the constituents at this excitation state was found to be jA, therefore, confining the particles to the very surface of the shell of diameter jA. The number of constituents N_1 = A^2/(4L_p^2) of the innermost shell j = 1 was found to be the quanta of particle count in the singularity, such that particle count on a shell of index j was N_j = (3j^2 − 3j + 1) N_1. Invoking Tammes’s conjecture, we have shown in this article that the microstructure of the black hole singularity is consisted of a distinct set of curved hexagonal lattices that are wrapped into the spherical shells. It is shown that the distance of the constituents on the innermost shell is d_1 = √ (8π/√3) Lp. The microstructure gets progressively finer towards the outer shells. In the outermost shell of the singularity of the supermassive black holes, the lower limit d_infty is approaching to √(8π/3) Lp. It is also noted that a centered hexagonal tessellation naturally emerges from N_j law which is a remarkable hint of consistency in the proposed theory of black holes.
Category: Condensed Matter

Confirming the Riemann Hypothesis

Authors: Francesco Aquilante
Comments: 3 Pages.

The concept of the textit{quantum Riemann sum} ($Q$-sum) is introduced as a theoretical framework to bridge the gap between the physics of discrete energy quantization and the analytic continuation of divergent series. By identifying the $Q$-sum generator as a Todd operator, we demonstrate that the Riemann zeta function emerges as the spectral signature of a complex dynamical system. We show % derive a closed-form energy identity revealing that the non-trivial zeros $ho=sigma+igamma$ correspond to states of vanishing boundary flux, where the system’s hermitian potential and anti-hermitian flow reach perfect equilibrium,for which charge-parity symmetry % governed by charge-parity symmetry, that is conserved if and only if requires the states to reside on the critical line $sigma = 1/2$. Consequently, the {em Riemann Hypothesis} of 1859 is now revealed not merely as a proven arithmetic theorem, but as a necessary condition for spectral stability and symmetry conservation in quantized systems.
Category: Number Theory

On the Nonexistence of the Identity C 3 − B 3 = a 3 for Pythagorean Triples

Authors: Kurmet Sultan
Comments: 3 Pages.

We show that for any Pythagorean triple (a, b, c) satisfying a^2+b^2 = c^2 with a, b, c ∈ N, the equality c^3 −b^3 = a^3 cannot hold. The proof is elementary and relies only on algebraic manipulation and the structure of the Pythagorean equation, without invoking Fermat’s Last Theorem.
Category: Algebra

Extraction of Complex Analytic Cycles

Authors: B. Wang
Comments: 10 Pages.

Let $X$ be a compact K"ahler manifold. We first define the positivity of homology classes in $H_{2k}(X;mathbb Q)$.From the positivity, we extract complex analytic cycles. Precisely, we show if $tauin H_{2k}(X;mathbb Q)$ is positive, i.e. $tau $ is represented by a closed, strongly positive current, then there are a complex analytic cycle $V$ with positive rational coefficients and a positive current $S$ of bidimension $(k, k)$ such thatbegin{equation}tau=[T_V+S]end{equation}where $T_{bullet}$ denotes the current of integration over the chain $bullet$, and $[bullet]$ denotes the homologyclass represented by $bullet$.
Category: Geometry

Why Mass Doesn't Increase with Velocity

Authors: David M. Bower
Comments: 3 Pages.

Ever since the advent of special relativity over one hundred years ago, there has been considerable debate over the concept of "relativistic mass", i.e., whether mass increases with velocity. Although the equations of Einstein’s version of special relativity (ESR) can be interpreted to show that mass increases with velocity (and that such an increase of mass with velocity seems logically required to explain that light speed is a limiting velocity), assuming that mass increases with velocity is associated with an insurmountable logical contradiction (in view of a previously published paradox involving ESR). This insurmountable contradiction suggests that it is meaningless to postulate that mass increases with velocity
Category: Relativity and Cosmology

Psychology Is An Unfalsifiable Pseudo-Science—A First-Principle Deconstruction of The Epistemological Failure Of Psychology

Authors: Leland Xing
Comments: 26 Pages.

This paper presents a first-principle deconstruction of psychology, arguing that it operates fundamentally as an unfalsifiable pseudo-science. We highlight that clinical psychology relies entirely on subjective self-reporting and circular diagnostic frameworks, its severe reproducibility crisis, the mathematically invalid statistical practices within the field, and how descriptive behavioral categories like ``mental illnesses'' are improperly used as root causes in a circular logical loop. We further dismantle common defenses of the field—including philosophical attempts to redefine the scientific method without the requirement of falsifiability and the retroactive rationalizations used to defend psychological practices. We conclude that the foundational mechanics of psychology are indistinguishable from astrology. Hence, we propose that society cease classifying psychology as a science, as expanding the definition of science to accommodate psychological methodologies would logically require granting the exact same scientific legitimacy to astrology.
Category: Social Science

Occluded Person Re-identification via Spatio-Semantic Topology Guidance and Geometry-Aware Semantic Alignment

Authors: Xiaohao Xie, Wenhua Jiao, Caoyu Chen
Comments: 10 Pages.

Identifying pedestrians under heavy occlusion (Occluded Re-ID) remains highly challenging, primarily because obstacles inevitably corrupt human structural integrity and induce severe spatial-semantic mismatching. Current approaches either struggle to recover fragmented topological features or blindly trust fragile pose estimators, making them highly vulnerable to complex background interference. To overcome these bottlenecks, we present textbf{SSGA}, a unified multi-modal enhancement framework that seamlessly couples topology restoration, cross-modal feature calibration, and semantic-driven decoding. Specifically, a Spatial Guided Graph Convolutional Network (SG-GCN) is first formulated to repair corrupted local structures by embedding physical spatial constraints into visual patch representations. Moreover, to tackle cross-modal mismatching, we propose the Spatio-Semantic Dual-Metric Greedy Alignment (SSDA) strategy. By anchoring visual embeddings to reliable skeletal cues under strict geometric boundaries, SSDA effectively eliminates semantic ambiguity such as symmetrical limb confusion. Furthermore, a Geometry-Aware Semantic Matching (GASM) module is designed to employ learnable semantic queries for dynamically extracting part-level features, which forces the network to highlight visible body regions and filter out occlusion noise. Comprehensive evaluations across five standard benchmarks validate the superiority of our SSGA framework, which establishes new state-of-the-art results and yields substantial improvements particularly on the severely occluded Occluded-Duke and Occluded-ReID datasets.
Category: Artificial Intelligence

A Multi-Background Normalization and Dynamic Meta Feature Mining Approach for Person Re-Identification

Authors: Xiaohao Xie
Comments: 10 Pages.

Person re-identification (ReID) aims to retrieve pedestrians across cameras, facing challenges from differences in perspective, background, and lighting, which introduce noise and hinder key feature extraction. Existing methods, often relying on normalization or generative data augmentation, suffer from limitations such as neglecting camera label information or the unreliability of two-stage learning. To address this, we propose a one-stage architecture, M-MBNNet, consisting of MBN (Multi Background Norm) and MetaRep (Meta-Representation for Adaptive Metric) modules. MBN uses a camera-wise Assignment Gate and Multi-aggregation Norm to align and normalize backgrounds, reducing interference and enhancing person-relevant feature robustness. MetaRep bridges representation and metric learning, leveraging mutual information (quality measures) to dynamically adjust asymmetric metrics for consistent multi-task convergence. It also incorporates curriculum learning to dynamically emphasize either inter-class separability or intra-class compactness. M-MBNNet offers a systematic approach to extracting key pedestrian features and resolving cross-camera differences through active alignment and adaptive optimization. We achieve strong results on two baselines—one mainly for representation and one for metric learning—demonstrating the method's scalability.
Category: Artificial Intelligence

Old Quantum Theory or Quantum Mechanics: a Look at the Spectrum of the Hydrogen Atom

Authors: Jorma Jormakka
Comments: 10 Pages.

The article compares the derivation of the principal energy states in the hydrogen atomas it is done in Quantum Mechanics and in old quantum theory. The method in old quantum physics makes a lot more sense. Section 3 explains why Schr"odinger's momentum substitution only worksin one dimension and there it is basically the same as Bohr's quantization rule for the angularmomentum, which is the same as de Broglie's hypothesis in the application where this hypothesisworks. The conclusion of the article is that Quantum Mechanics and Quantum Field Theory incorrectlyhave generalized Schr"odinger's one-dimensional momentum substitution to a many dimensional momentum substitution.
Category: Quantum Physics

Phoenix Numbers

Authors: Walter A. Kehowski
Comments: 135 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

If the digits of number n (in base b) can be split p|q so that n=rq rp$, where rp and rq are the numbers formed by reversing the digits of p and q, respectively, then n is called a phoenix number in base b. If q has k digits in base b, then n is called a k-Phoenix number in base b. For example, 1.1.n.n|b is 1-Phoenix in base b=n^2-1. The existence of infinitely many k-phoenix numbers in any base b is claimed, but not proven.
Category: Number Theory

Global Regularity of 3D Incompressible Navier-Stokes Equations via Deterministic Harmonic Resolution

Authors: Barry L. Guevremont
Comments: Corrected version with fixed Lean 4 code, added images (coffee_cup.png and math_monster.jpg), and minor formatting improvements. 10 pages. Physics - Mathematical Physics category.

For any smooth, divergence-free initial velocity field uu2080 ∈ Hˢ(ℝ³) with s ≥ 1/2, there exists a unique, global-in-time smooth solution u(x, t). We prove this by establishing that the Leray-Hopf energy inequality is a strict equality and that the velocity gradient satisfies a uniform L∞ bound for all t ≥ 0. This closes the supercritical scaling gap and satisfies the Beale—Kato—Majda criterion, thereby proving global regularity of the 3D incompressible Navier-Stokes equations.
Category: Mathematical Physics

The Duality of Geometric Primality: Subgroups and Quotients Shape Polygons and Fractals

Authors: Youssef Ayyad
Comments: 35 Pages. (Header blocked by viXra Admin)

This article presents a systematic exposition of the duality between two complementary geometric characterizations of prime numbers, united through Pontryagin duality for finite cyclic groups. The first characterization is internal: an integer n ≥ 2 is prime if and only if everyequitable partition of the vertices of the regular n-gon Pn is fixed classwise only bythe identity rotation. We establish a canonical bijection between equitable partitionsof Pn and subgroups of the cyclic group Cn ∼= Z/nZ (Theorem 3.6). The second characterization is external and fractal: an integer n ≥ 2 is prime if and only if the regular polygon Gn is a primitive leaf in the universal subdivisiontree—that is, Gn has no ancestors other than the root G1. We prove that Gn appears in the fractal hierarchy F(m) generated by Gm if and only if m | n (Lemma 4.4), establishing a bijection between fractal ancestors of Gn and quotients of Cn (Theorem 4.8). These two characterizations are not merely equivalent reformulations of the arithmetic definition of primality; they are canonically dual. The duality is mediated bythe Pontryagin isomorphism Φ : Sub(Cn) → Quot(Cn) sending a subgroup H ⊆ Cn to its quotient Cn/H. For each proper divisor d | n with 1 < d < n, this duality pairs:u2022 Internally: the subgroup Hn/d ⊆ Cn of order n/d, realized geometrically as an equitable partition Pd of Pn into d classes; u2022 Externally: the quotient Cn/Hn/d ∼= Cd, realized geometrically as the ancestor polygon Gd in the fractal tree. These correspondences fit into a commutative diagram (Theorem 5.1) that unifies the two geometric visions: the same divisor d manifests simultaneously as a symmetriccoloring of the polygon (internal witness) and as a smaller polygon from which Gn descends (external witness). We develop this duality, with complete proofs, extensive examples, and visualizations. The framework then generalizes naturally: for arbitrary finite groups, rings,and modules, the duality between substructures and quotient structures can be rendered visible through dual geometric realizations. This suggests a broad research program in dual geometric algebra, where abstract algebraic simplicity—the absence of non-trivial substructures or quotients—acquires concrete geometric meaning.
Category: Algebra

Asymptotic Behavior of the Erdos Consecutive Sum Sequence A003462

Authors: Jehan Singh
Comments: 4 Pages. CC BY 4.0 license (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

We study the self-generating consecutive-sum greedy sequence (a_n)_{n ≥ 1}, defined by a₁ = 1, a₂ = 2, and for k ≥ 3, a_k is the least integer greater than a_k−1 expressible as a sum of at least two consecutive earlier terms. We prove that the auxiliary sequence b_n := a_n − n is nondecreasing and unbounded, showing that infinitely many positive integers are omitted. We also provide an explicit Binet-type formula for a_n, enabling exact computation of terms, and discuss implications for the sequence’s asymptotic growth.
Category: Combinatorics and Graph Theory

Relativistic Energy Conservation in Gravitational Motion Across Scales

Authors: Oliver R. Jovanovic
Comments: 13 Pages.

I present an empirically grounded, effective description of gravitational dynamics. In my view, a full quantitative description would require the formalism of general relativity, a closely related extension, or an appropriate replacement, which is beyond the scope of this work.When a rubber ball is pressed against the floor, its internal energy—and consequently its mass—increases by Δm = ΔE/c²; a similar effect occurs when a body is lifted. Building on this principle, the gravitational formulation proposed here avoids singularities at the centers of black holes. In regions where rs exceeds r, gravity becomes repulsive, which can be interpreted as an effective contribution similar to dark energy. The framework also provides a mechanism consistent with quark confinement, while all resulting equations remain closely aligned with those of general relativity.
Category: Relativity and Cosmology

A Geometric Proof of Pi's Irrationality

Authors: Timothy Jones
Comments: 5 Pages. A reader provided some corrections.

We give a proof of pi's irrationality that references principles of set theory and cardinality within the context of basic geometric properties of a circle.
Category: Number Theory

Dark Matter - Is it Needed?

Authors: Hartmut Schwab
Comments: 15 Pages.

The idea that the outer stars of a galaxy resp. the outer galaxies of a cluster of galaxies are rotating too fast around their gravitational centres and therefore can not be kept on their orbits by the gravitation of the baryonic matter leads to the assumption of extra matter, the u201edark matter", whose gravitation is suited to keep the outer stars/galaxies on their orbits.It is demonstrated that if the gravitational acceleration as a function of the distance from the gravitational centre is determined correctly, the observed orbital velocities can be explained solely by the baryonic matter and Newton’s laws and therefore u201edark matter" is not needed. It is not even necessary To explain the other effects ascribed to it, u201edark matter" is not neeeded either as the higher gravitational accelerations necessary for these effects are delivered solely by the baryonic matter.
Category: Astrophysics

A Gravitating Dark-Energy Hypothesis to Replace Mond

Authors: Remi Cornwall
Comments: 18 Pages. Dimesional consistency of eqns. 9 and 12 corrected. Epsilon constant now halved in eqn. 9. Grammar and presentational changes (size of images etc.) (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This paper completes a series of earlier papers on the Cosmological Constant as compressible fluid-like zeropoint energy, which acts as a 2nd order perturbation in the stress-energy tensor; this preceded an earlier sketch for a means for Dark-Energy to gravitate. We suggest a replacement to ad-hoc MOND type theories with the theory developed herein, in the light of recent DESI and JWIST discoveries, with a fully covariant version of our earlier gravitating dark-energy model: we show that dark-energy can gravitate if it is considered to be in compression by tidal effects; also it sets a natural cusp-like size limit to galaxies and clusters. A relation is suggested between the slope of the galactic rotation curve and the slope of the dark-energy/matter zone is found too. An earlier proof by the author that MOND type theories are not Heisenberg Uncertainty Principle compatible is cited here. Finally we ask if an extra-repulsive form of dark-energy may prevent collapse to singularity in Black hole solutions. All-in-all, this paper is a semi-classical treatment of gravity with vacuum corrections that might manifest on the large-scale.Keywords: Action, Cosmological Constant, Covariance, Dark-Matter, DESI, Gravitating Dark-Energy, Uncertainty Principle, JWIST, ΛCDM, MOND, Renormalisation Group, Rotation Curves, Singularities
Category: Relativity and Cosmology

On the Justification for Using a Dynamic Gravitational Field in an Expanding Universe

Authors: Pavlo Danylchenko
Comments: 4 Pages.

The feasibility of using the equations of the dynamic gravitational field to reflect the expanding Universe has been proven. Only thanks to this, the Hubble velocity of matter will not exceed the velocity of light at the event horizon of the Universe. The identity of the parameter b of the Schwarzschild solution of the equations of the gravitational field to the square of the relative frequency of the electromagnetic interaction of microobjects of matter and the equivalence of the limit velocity of matter to this parameter in its state of immobility has been justified. The falsehood of associating the parameter b with the coordinate velocity of light of the General Relativity (GR) has been proven.
Category: Relativity and Cosmology

Gravimetry and The Era of Development

Authors: Abdelmajid Ben Hadj Salem
Comments: 31 Pages. In French

This [article] represents the national report about the gravimetry activity in Tunisia, of the year 1970.
Category: Geophysics

On the Onset of Ultraviolet Turbulence in Quantum Chromodynamics

Authors: Ervin Goldfain
Comments: 15 Pages.

We derive the Renormalization Group (RG) behavior of non Abelian gauge theory under the assumption that spacetime has a scale dependent effective dimensional deviation in the deep ultraviolet (UV) and approaching zero in the infrared (IR). We derive the RG equation with dimensional fluctuations, cast it as a stochastic differential equation, and solve the associated Fokker—Planck equation. We show that the usual asymptotically free model of Quantum Chromodynamics (QCD) fails in the deep UV; instead, the coupling exhibits intermittent behavior, stemming from the multiplicative noise induced by dimensional fluctuations. We find that this turbulent regime leads naturally to a multifractal attractor, consistent with a Cantor—Dust spacetime structure and in line with our work on fractal spacetime description of ultraviolet physics.
Category: High Energy Particle Physics

A Derivation of the Alternating Power Sum Formula

Authors: Samuel Datu Castelo
Comments: 3 Pages.

This paper presents a derivation of the closed-form expression for the alternatingpower sum without employing Euler polynomials, which are traditionally used in suchformulations. Instead, the derivation utilizes the shifted series method together withdifferentiation techniques to construct a polynomial representation for the sum. Theresulting expression is then connected to classical number-theoretic constants throughthe Riemann zeta function and Bernoulli numbers. This approach provides an alterna-tive and more elementary framework for obtaining the alternating power sum formulawhile avoiding the formal machinery of Euler polynomials.
Category: Number Theory

Pure Primes: A Digit Restricted Prime Class Defined by Partition Primality

Authors: Christoper Mututu
Comments: 13 Pages. (Note by viXra Admin: Please cite listed scientific references)

We introduce and rigorously define a new class of prime numbers, which we term as pure primes. A natural number is classified as a pure prime if, for every contiguous partition of its decimal representation into equal length segments, each segment is itself a prime number. Restricting the digits to the prime set {2, 3, 5, 7}, we enumerate pure primes across all digit lengths. Specifically, our results establish the following counts: There are 4 total 1-digit pure primes. There are 4 total 2-digit pure primes. There are 15 total 3-digit pure primes. There is only 1 4-digit pure prime. There are 128 total 5-digit pure primes. There are 0 total 6-digit pure primes. There are 1325 total 7-digit pure primes. There are 0 total 8-digit pure primes. There are 469 total 9-digit pure primes. There are 0 total 10-digit pure primes. There are 214432 total 11-digit pure primes. There are 0 total 12-digit pure primes. There are 2884201 total 13-digit pure primes. There are 10 total 14-digit pure primes. There are 236 total 15-digit pure primes. As a result, the total pure primes across 1-15 digits are 3100825. These numbers span a huge range from 1-digit primes 2, 3, 5, 7 up to 15-digit primes in the quadrillion fully enumerated by our exhaustive computation. While a formal proof of absolute finiteness remains open, this extensive enumeration demonstrates that within this range, the class of pure primes is tightly constrained and structurally self-limiting. The partition invariance requirement imposes increasingly restrictive combinatorial conditions as the digit length grows. Empirically, this leads to long stretches of digit lengths admitting no pure primes. While these observations do not constitute a proof of finiteness, they indicate strong structural sparsity within this digit constrained prime class. This could hint at deep links between digit partition invariance and finiteness properties analogous to those conjectured in other constrained number sets. Our exhaustive computational investigation reveals strong structural constraints. Pure primes exist only for certain digit lengths with multiple lengths producing no examples. Beyond its combinatorial elegance, the discovery of pure primes opens new avenues for research into digit partition invariance, prime density constraints and the structure of prime subsets in discrete number spaces. These findings suggest a previously unrecognized form of order in the prime landscape, providing both a novel mathematical object and a framework for exploring finiteness within prime number theory.
Category: Number Theory

The Set of Real Numbers as a Function of the Set of Natural Numbers

Authors: Ayoub Zaroual
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references)

In this article, we prove that the number of elements in the set of positive real numbers Card (R+) can be expressed as a function of the number of elements in the set of natural numbers Card(N) using the formula: R+=N+0.9N(N-1)^2
Category: Set Theory and Logic

What Lies Behind It? an Attempt from Matter Field to Quantum Gravity

Authors: Kozo Koizumi
Comments: 19 Pages.

We present an algebraic framework demonstrating that the spacetime manifold is an emergent structure from the field module rather than a pre-existing background. By employing a Clifford-valued gauge construction, we show that Minkowski spacetime is rigorously identified through the algebraic consistency of the field module network. In this framework, the spin connection is integrated as a dynamical messenger field on the same footing as the standard gauge bosons. Furthermore, this construction naturally leads to a Yang-Mills type gravity sector, which is known to be 1-loop renormalizable and asymptotically free, thereby providing a consistent UV completion. This framework offers a structural resolution to the cosmological constant problem by ensuring that vacuum energy does not contribute to the field equations, providing a consistent foundation for integrating quantum gravity with the Standard Model without an a priori manifold.
Category: High Energy Particle Physics

Real Atomic Effect of the Reducing of the Vacuum in the Hydrogen Atom by the Heavy Proton (Nuclear Matter)

Authors: Nekrasov Grigory Yu
Comments: 33 Pages.

The computer program, when it will be executed, then gives a possibility of computation andgaining of real effect of the reducing of vacuum by matter based on uncertainty relations, i.e. theeffect, appearing in the electron-positron vacuum near heavy nuclear matter, is provided, as apotential solution of the vacuum energy calculation problem.
Category: Nuclear and Atomic Physics

The Riemann Hypothesis and its Generalization

Authors: Leckan M. Sibanda
Comments: 16 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

We present a complete, rigorous proof of the Riemann Hypothesis for the Riemann zeta function and its generalization to all primitive Dirichlet L-functions. The proof is based on a fundamental property of the zeros: for each zero (the imaginary part of a nontrivial zero) there exists a positive integer (called the optimal modulus) such that the zero times this integer divided by pi is exceptionally close to an integer. This Diophantine approximation leads to an exact phase-locking recurrence derived from the logarithmic identity. Using a decomposition of the Dirichlet series into residue classes, we obtain representations of the zeta function and its symmetric counterpart in terms of real, positive, strictly decreasing amplitudes and a fixed set of roots of unity. The vanishing of a certain anti-symmetric combination of the zeta function at a zero forces a simple trigonometric condition whose only solution is that the real part equals one-half. The argument is elementary, self-contained, and extends naturally to all Dirichlet L-functions. Numerical verification confirms the existence of the optimal modulus and the phase-locking identity to extraordinary precision, but the proof itself is purely analytic.
Category: Number Theory

Examining Toroidal Geometry in Terms of a 3-Dimensional "Tokamak", and the Use of a Toroidal Universe

Authors: Andrew Beckwith
Comments: 13 Pages. Accepted for publication in JHEPGC

We begin with a review of Friedman geometry in order to get a relationship between vacuum energy density and Hubble expansion parameter, Furthermore, in line with Utpal Sarkar, we can write the Hubble parameter as proportional to the square of background temperature, which is important in our derivational work. Furthermore, in line with work presented by Guth and Vilikin we can obtain g_00 = 1 for a time component in the Toroidal universe by setting a three dimensional line element as embedded in the square of a scale factor a (t) at times the "line element" given by R. Murdzek (which has a 3 dimensional presentation of a Ricci scale factor ). Incorporating the Guth and Vilikin "line element" trick, for a Toroidal universe allows 9_00 =1 for obtaining a non-zero stress energy tensor we can write as T_00 as a non-zero value even if the Ricci component R_00 in our construction is zero.
Category: Relativity and Cosmology

On the Observer-Relative Nature of Infinite Velocity in Euclidean Spacetime

Authors: Dong-Wook Kim
Comments: 14 Pages.

We propose Observer-Relative Infinity Theory (ORI), a mathematically rigorous extension of Special Relativity built on three independently established results. First, the constant c appears in Maxwell's equations as a structural ratio of electromagnetic constants, prior to any reference to light, establishing it as a geometric parameter of spacetime rather than a velocity law. Second, Ignatowski (1910) and Lee and Kalotas (1975) proved that the Lorentz transformation follows from spatial isotropy and the principle of relativity alone, without assuming a light-speed postulate, leaving c as a free parameter to be fixed by experiment. Third, Robinson's non-standard analysis (1966) provides a logically consistent number field containing infinite elements as genuine algebraic objects, within which infinite velocity is well-defined rather than a divergent limit.Building on these foundations, we introduce a projection operator that maps an unbounded true coordinate space onto a finite observed interval determined by the observer's resolution scale, denoted Omega. This operator is proven to be strictly monotone, strictly concave, sub-additive, and invertible. The resolution scale Omega is derived from the observer's characteristic energy and rest mass, and reduces exactly to c for photon-based detectors. True spacetime is taken to be Euclidean with a positive-definite metric; the Minkowski pseudo-Riemannian structure of standard relativity emerges from the projection to leading order. Newtonian mechanics is recovered exactly in the low-velocity limit, and Special Relativity is recovered approximately when velocities approach Omega. Four falsifiable, quantitatively distinct predictions are derived: energy saturation at ultra-relativistic collider energies, linear scaling of muon lifetime at extreme Lorentz factors, a measurable bound on quantum entanglement propagation speed, and a natural resolution of the Hubble tension through a time-varying resolution scale.
Category: Relativity and Cosmology

Dimensions and Coloring of a Graph

Authors: Volker Thürey
Comments: 7 Pages.

In the first part, we introduce several notions of graph dimensions. These concepts are inspired by the classical idea of a `dimension' which was introduced around 60 years ago. We provide simple examples to illustrate them. In the second part, we define new types of graph colorings, derived from the Chromatic Number and the Chromatic Index. We apply coloring to both the vertices and the the edges. Finally, we generalize the concept of the `Chromatic Number of the Plane'.
Category: Combinatorics and Graph Theory

Triality-Resolved Spectral Update Theory: A Minimal Transition-Operator Route to Standard-Model Structure and Gravity

Authors: Casey McGrath
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

We present a minimal theorem stack in which low-energy physics emerges from a single causaltransition operator acting on a microscopic configuration Hilbert space. Proper time is defined as accumulated transition information along causal histories. An exact order-three invariant sector of the microscopic dynamics yields three fermion generations. In the infrared, the theory is described by an almost-commutative spectral triple whose visible finite algebra reproduces the Standard Model gauge group. The finite Dirac operator acts on a 96-dimensional fermionic Hilbert space and admitsa triality-circulant Yukawa core diagonalized by a discrete Fourier basis, producing exact zeroth-order harmonic flavor states. Minimal non-circulant defects generate CKM and PMNS mixing, while a compatible singlet Majorana block closes the neutrino sector by the seesaw mechanism. The bosonic spectral action yields Einstein—Hilbert, Yang—Mills, and Higgs terms from a single operator expansion. The fine-structure sector reduces to one microscopic spectral invariant. We isolate the remaining genuinely downstream inputs to explicit microscopic Yukawa coefficients, defect blocks,and the computation of the relevant spectral invariant.
Category: Relativity and Cosmology

Angular Homothety and a Geometric Characterization of Twin Primes

Authors: Youssef Ayyad
Comments: 14 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This article establishes a fundamental connection between number representation in different bases and the geometry of regular polygons. We demonstrate that every positive integer $N$ admits a unique decomposition $N = b^m - R$ where $b^m$ is the smallest power of the base $b$ exceeding $N$, and $R$ is expressed in base $b$. This arithmetic fact translates geometrically into an angular homothety mapping the regular $b^m$-gon to the $N$-gon. Through this geometric lens, we obtain a natural classification of numbers: primes appear as elements whose associated fractions are irreducible regardless of the base. Our main result is a striking geometric characterization of twin primes: for a prime $p$, the pair $(p, p+2)$ consists of twin primes if and only if for every base $b$ with $2 leq b < p$, the ratio of homothety factors $lambda_b(p+2)/lambda_b(p)$ equals $p/(p+2)$. We explore connections between this characterization and the twin prime conjecture, offering a new geometric perspective on this ancient problem.
Category: Number Theory

The Structural Irreversibility of the Thermodynamic Arrow of Time: Why Time-Reversed Thermodynamics Breaks Down

Authors: Richard Kaufman
Comments: 4 Pages.

Among the laws of physics, only the second law of thermodynamics distinguishes a preferred direction in time. This thermodynamic arrow is commonly expressed in terms of entropy increase toward the future. An equivalent formulation is Clausius’s statement that heat energy flows spontaneously from hotter to colder bodies. It is tempting to suppose that reversing time would simply reverse this process, such as watching a film backward, with heat energy flowing from cold to hot and thermodynamic evolution retracing earlier states. In this paper, we examine that intuition within classical thermodynamics. Drawing on earlier pedagogical and technical analyses, we argue that reversing the direction of spontaneous heat flow does not produce a reversed thermodynamic evolution. Instead, temperature differences amplify, equilibrium ceases to function as a stable endpoint, and thermodynamic description breaks down. In this sense, the thermodynamic arrow of time is structurally irreversible. This paper is aimed at the undergraduate level.
Category: Thermodynamics and Energy

Errors in the Quantum Mechanical Calculation of the Fine Structure Spectrum of the Hydrogen Atom

Authors: Jorma Jormakka
Comments: 7 Pages.

The article explains why the Quantum Mechanical derivation of the fine structure spectrum is incorrect. The main error is that in this derivation the substitution of the momentumto partial derivatives of space coordinates is not the same as in the non-relativistic case andtherefore the energy levels that are obtained are not energy levels for the relativistic Hamiltonian,they are energy levels for the non-relativistic Hamiltonian and different. If the error is corrected,the fine structure spectrum formula is not obtained correctly. There are also two other errors inthe Quantum Mechanical derivation.
Category: Quantum Physics

No Relation Between the Fine Structure Constant and Electron Anomalous Magnetic Moment

Authors: Jorma Jormakka
Comments: 7 Pages.

The article shows that there is no such relation between the fine-structure constantand the anomalous magnetic moment of an electron that is claimed by Quantum Electrodynamics. Indeed, there is no clear mathematical relation between these constants. The anomalous magneticmoment is not a real property: it is an apparent magnetic moment caused by force weakening whenacting on a fast spinning electron. The magnetic moment of the electron in its rest frame isthe Bohr magneton.
Category: Quantum Physics

A Further Specification of the Cosmological Constant Problem by Account the Two Fermions in the Standard Model and an Effect of the Reducing of Vacuum by Matter Based on Uncertainty Relations

Authors: Nekrasov Grigory Yu
Comments: 90 Pages.

A Cosmological constant problem has been considered in the light of the Standard model ofelementary particles (SM) by account two fundamental fermions in the SM, i.e. the u quark and the electron and their antiparticles. All the movings of these virtual pairs and their orientations in space have been taken into account in the offered model of vacuum. This led to the more precise and accurate estimation of vacuum energy of the Universe on the discovered and examined minimal scale ~1.5×10^-15 m on uncertainty relations, than in the previous work. Comparison of this estimation with the calculation realized by the method and the model used in the previous work was carried out. The current calculation is the full computation, i.e. for the full scale range, which begins from 1.5×10^-15 m and can think terminated for the effect on the maximal linear size2.00008×10^-11 m for the experimental data and the theoretical data on hydrogen and heliumatoms correspondingly. These atoms are considered due to that they are the most common in the entire Universe. Thus, one can say that the vacuum energy near matter in the special effect of the reducing of vacuum by matter has been assessed more precisely as it is in the real world.
Category: Astrophysics

Fundamental Rebound Theories Justify Generalizing Newton's 340-Year-Old Third Law and Its Implications.

Authors: Ajay Sharma
Comments: 45 Pages. (Note by viXra Admin: Further repetition will not be accepted)

Consider compiling a set of one hundred observations of diverse rebounding bodies on a common surface. These bodies may have identical mass and composition but different shapes, such as spheres, cones, needles flat sheets, and irregular forms, and they rebound to different heights. These experiments are classified into realistic physical systems. In existing physics, motion of bodies is explained by the coefficient of restitution method (CORM), the equations are based Newton’s third law of motion (NTLM) and kinematical equations, which explains phenomena qualitatively. Hertz contact method also explains motion of bodies in very limited range. Under suitable conditions, a spherical ball may rebound nearly to its original height, retracing its path, and the rebound height may be treated as an indicator of reaction force. In contrast, a flat body rebounds to a minimal height whereas irregular bodies rebound along trajectories at varying angles. The typical motions of sphere, flat and irregular bodies may be explained with a single equation quantitatively. Even recoil velocity of gun based on the conservation of momentum, is not quantitatively confirmed at macroscopic level. Thus, NTLM may be used independently to explain the behavior of rebounding bodies, in realistic system NTLM is expressed as Reaction (FBA) = − [Kshape × Kcomposition × Ktarget × Kother] Action (FAB). Accordingly for recoil of gun we get, vgun = - mbullet .vbullet /Zmgun. The proposed generalization may be examined through quantitative experiments involving diverse parameters, where �� and Z represent experimentally determined coefficients. This law is specifically tailored for rebounding bodies in real-world scenarios, providing a phenomenological framework to describe experimental observations. Distinct theoretical treatments may be required for rebounding bodies in idealized versus real-world systems. The results motivate consideration of an extended interaction framework that incorporates geometric and surface-dependent factors within a Newtonian context.
Category: Classical Physics

Sequences of Consecutive Primes and Benford's Order

Authors: René-Louis Clerc
Comments: 15 Pages. In French

Prime numbers are essential in various areas of cybersecurity, from encrypted messages to secure payments, HTTPS websites, and RSA cryptography techniques.Numerous computational and theoretical studies on these "unbreakable" numbers in arithmetic ((+)) are regularly published. Here we continue the article [15] by considering sequences of consecutive primes of the type considered then, primes with a certain initial digit, primes without a certain digit, primes with a certain digit, primes with a certain last digit, primes with a certain digit in a certain position, etc. We will calculate the maximum number of consecutive elements in such sequences, as well as the probabilities of obtaining a certain consecutive number from one of these primes. We will also seek to determine whether there is a possible order for the maximum lengths of sequences of such primes with reference to these various digits, and in which cases this may be a Benford-type order (inverse natural order of integers).We will conclude with some results concerning the maximum differences between any consecutive primes or those possessing a certain property.
Category: Number Theory

Emergent Quantumcorrelations as Global Fixpoint-Structure in Topotronics-Theory.

Authors: Holger Döring
Comments: 27 Pages.

Reconstructed are the fundamental basic ideas in a minimal superdeterminism-model. The global fixpoint structure codates measurements and apparates simultaneous, while an selector-operator generates stability and coherence via a minimal-principle. Discrete functions of measuring and on reproduce exactly the quantum correlations . The CHSH- correlations appear at the emergent quantum area , whereby the Bell-unequation appears as an emergent, not as a fundamental physical law. The minimal measurable deviation is . The model offers a consistent, logical structure, connecting superdeterminism and emergent effective quantum logic. Classical quantum structure appears not on a fundamental level but only as an emergent overlaying structure like in the demand of Einstein. Particularly in the connection with a theory of quantum gravity it is necessary to explore new ways of formulating fundamental spacetime-structures like quantum states to get greater significance in the clearness of a consistent underlying description.
Category: Quantum Gravity and String Theory

Deep Learning Reveals Breakthroughs in Fluid Dynamics Equations

Authors: Abdelmajid Ben Hadj Salem
Comments: 5 Pages. In French

A new approach finds solutions to fluid dynamics equations and could be applied to solve one of the Millennium Prize Problems.
Category: General Mathematics

Resolving the Fourth Spatial Dimension via Motion in Spacetime: Toward a 5D Geometry of Force and Beyond

Authors: Stuart Hood
Comments: 20 Pages. (Note by viXra Admin: Author name is required in the article after the article title)

This paper re-examines the concept of the fourth dimension, proposing the natural next step beyond static three-dimensional forms to be the motion of those forms through both space and time. This reading eliminates the need for an abstract fourth orthogonal axis and unites two previously competing traditions — Charles Hinton’s recursive four-dimensional geometry and the spacetime framework of Hermann Minkowski and Albert Einstein.The mathematics and logic of Hinton’s tesseract have long demanded a fourth spatial axis, while relativity established time as the fourth dimension. By regarding the "four-dimensional edges" of the tesseract (and other polytopes) not as representing an additional orthogonal direction through space but instead as displacement vectors tracing paths of change through both space and time, all currently-established representations of four-dimensional forms can be conceived as worldlines — precisely the four-dimensional histories of three-dimensional objects that Minkowski described.Treating the fourth dimension as both the spatial and temporal capacities of motion (the animated, ever-moving world we live in) opens a natural extension: a fifth dimension of force. This geometric interpretation aligns with historical attempts in Kaluza—Klein theory and its modern successor, Space-Time-Matter theory. The five-dimensional penteract is shown to mirror the structure of particle interactions and cosmological dynamics. The framework then continues beyond physical space toward consciousness, proposing a sixth dimension of possibility (observation, knowledge, awareness) and a seventh of intelligence (reason/logic, choice/will).
Category: History and Philosophy of Physics

On the Recent Breakthrough on Dynamical Higgs Mechanism for DRGT Gravity: How Finally it is a Framework Permitting to Resolve the Strong Coupling Problem

Authors: Emmanuel Kanambaye
Comments: 17 Pages.

In a recent paper, we claimed to propose a solution to the strong coupling break-down problem of massive gravity by making the theory massless above some energy-scale.Although interesting, it seems in light of the valuable work of Bonifacio, Hinterbichler and Rosen ({it Constraints on a Gravitational Higgs Mechanism}), that our proposed mehanism fails in its aim.At first glance, this may seem to spell the end for our model; fortunately things are rather more subtle because our model has indeed all the necessary tools to solve the strong coupling problem of massive gravity.Indeed, we are going to demonstrate in this paper, that for some appropriate expressions of the free tensor $mathcal{S}_{muu}(g,Psi,varphi^{ho},A)$ of our recently proposed model; it becomes possible to cleverly raise up the strong coupling scale of dRGT massive gravity without spoiling its ghost-free and local character.More precisely, through a concrete example, we will prove in the present paper that our recently proposed framework of massive gravity is well and truly a framework permitting to resolve both the strong coupling break-down problem as well as the van Dam-Veltman-Zakharov (vDVZ) discontinuity of dRGT massive gravity; we achieve this breakthrough by highlighting the first rather simple and improved four-dimensional dRGT massive gravity capable of overcoming both strong coupling problem and vDVZ discontinuity.
Category: Relativity and Cosmology

Deduction of the Electron Charge

Authors: Antonio Jose Mejias Marquina
Comments: 2 Pages.

In this work the charge of the electron will be deduced from Gauss’s flow law.
Category: Classical Physics

Polynomial Feature Engineering for Analytical Ridge Regression: A Case Study in Aerospace Anomaly Detection

Authors: Ansh Mathur, Atrishman Mukherjee, Supratik Dey
Comments: 11 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We investigate the effectiveness of polynomial feature engineering when combined with analytical ridge regression for multi-class classification tasks. Using the NASA Shuttle dataset as a case study, we demonstrate that degree-4 polynomial features enable closed-form solutions to achieve 99.43% test accuracy in 45 milliseconds of training time. This accuracy matches or exceeds previously reported results while offering substantial computational advantages through elimination of iterative optimization. Our systematic evaluation across six feature configurations reveals that test accuracy improves monotonically from 87.33% withlinear features to 99.43% with degree-4 polynomial interactions, representing a 12.10% absolute improvement. Generalization gaps remain below 0.3% across all tested configurations, indicating robust performance despite increased model capacity.These findings suggest that ex-plicit polynomial feature expansion, when properly regularized, provides a computationally efficient alternative to iterative learning methods for problems with polynomial structure. We discuss the applicability of this approach to safety-critical aerospace applications where deterministic training guarantees and rapid model updates are valued.
Category: Artificial Intelligence

Shaking up Einsteins Theory[:] the Anomalous Acceleration of the Pioneers

Authors: Henk Dorrestijn
Comments: 9 Pages.

If we assume the Lorentz contraction in Einstein's theory is based on a misconception, we can understand the unexpected acceleration found when measuring the orbital motion of Pioneers 10 and 11. Rejecting the Lorentz contraction leads to a better understanding of the concept of velocity. The actual velocity of an object, when accounting for the time delay, turns out to be slightly greater than the traditionally determined velocity. The time delay in a gravitational field decreases with distance so the actual velocity decreases compared to the traditionally determined velocity. This can be considered the anomalous acceleration measured for the Pioneers. Our calculations show that the actual velocity explains the anomalous acceleration very accurately.
Category: Relativity and Cosmology

Logarithmic-Spiral Tubular Wormholes in Einstein-Cartan-Maxwell Theory

Authors: Rodolfo Rossetti
Comments: 9 Pages. License: CC BY 4.0.

We announce the discovery of a non-vacuum, static, and dynamically stable traversable wormhole solution that fundamentally shatters the "exotic matter" requirement inherent in General Relativity. By embedding a scaled tubular neighborhood of a three-dimensional logarithmic spiral within the Einstein-Cartan-Maxwell framework, we derive a spacetime metric where the throat’s existence is dictated by the curve’s intrinsic Frenet-Serret curvature (κ) and torsion (τc). We rigorously prove the Rossetti-Geometric Autonomy Theorem: the fundamental existence condition, Eq(22), is structurally decoupled from the axial magnetic potential As(0). This result demonstrates that the wormhole is not a fragile artifact supported by external fields, but a self-sustaining topological necessity of the spiral geometry. The Null Energy Condition (NEC) is not merely violated; it is transmuted by the torsional pressure of the non-Riemannian connection, rendering the century-old search for "exotic fluids" obsolete. The crowning achievement of this work is the definitive solution to the stability problem. Through the derivation of the Radial Master Equation, we reveal an Effective Potential Veff(s) characterized by an insuperable torsional barrier at the throat (s→0). Unlike standard solutions prone to catastrophic collapse, the spiral-torsion coupling generates a positive-definite, divergent shield that immunizes the throat against radial fluctuations.Furthermore, we formalize the principles of Chronodynamics: establishing the exact Frame Dragging condition for bidirectional time travel and proving the Torsional Discharge Limit.We rigorously demonstrate that the quantum vacuum energy divergence at the Cauchy horizon is perfectly absorbed by the non-Riemannian spin-torsion coupling, acting as a geometric capacitor. This eliminates the radial collapse predicted by semiclassical gravity, formally defeating Hawking’s Chronology Protection Conjecture. This concludes the transition from speculative geometry to Torsiodynamics: the formal proof of a topologically eternal and dynamically invulnerable passage through spacetime. This work does not merely add to the literature; it rewrites the fundamental constraints of traversable topology and time travel.
Category: Relativity and Cosmology

Not All Weights Are Equal

Authors: Benjamin Cowherd
Comments: 9 Pages. github.com/orbits64/project-synapse

We test whether a dedicated ternary-weight reasoning component outperforms a homogeneous MLP baseline on multi-step logic problems requiring generalization to unseen entities. In two runs, the reasoning component outperformed the baseline by 12.2% and 19.6% respectively, while the baseline overfit and stalled. We propose a full architecture built on this result, with separate components for reasoning and language.
Category: Artificial Intelligence

Operational Simultaneity in Einstein’s Train—Lightning Thought Experiment

Authors: Richard Kaufman
Comments: 13 Pages.

Einstein’s train—lightning thought experiment is commonly used to illustrate the relativity of simultaneity, yet many textbook treatments leave implicit a modeling choice: observers typically analyze the scenario as though their own inertial frame were at rest. This assumption is not required by the postulates of special relativity, which permit an observer to adopt any inertial frame as at rest and itself as moving relative to it. This paper develops an operational method that makes this choice explicit by treating the rest frame of the light sources as the reference for assigning propagation distances, with the sources understood as lightning in ground air or as bulbs fixed to the track. Working in the train frame, and using only the train’s proper length, the relative velocity v, and the invariance of the speed of light c, the observer computes the propagation times and infers simultaneous emissions. The choice of source frame is an operational decision tied to the physical configuration of the emitters and does not introduce an absolute rest frame or imply any dependence of c on source or observer motion. The analysis does not replace the conventional interpretation but complements it by making explicit the operational choices involved in simultaneity judgments. Because the train—lightning experiment is almost universally presented as requiring the train observer to infer non-simultaneity, the analysis highlights how that conclusion depends on an implicit operational modeling choice while remaining consistent with special relativity. For completeness, Appendix A applies Selleri synchronization to Einstein’s original thought experiment, illustrating how admissible synchronization conventions affect simultaneity assignments.
Category: Relativity and Cosmology

Predictsol: A Prediction Tokenizer for the Solana Network

Authors: Lukas Saul, Robert Gludo, Jonathan Gludo
Comments: 4 Pages.

Predictsol demonstrates an architecture for a new generation of decentralized prediction markets which are transparent and modular. For the first time we separate the tokenization and associated redemption tasks of a prediction market into their own smart contract: the prediction tokenizer. A tokenizer contract offers the ability to create "events" such that outcomes are represented by tokens. These tokens are minted by participants in exchange for deposit, and the contract allows any users to redeem these tokens as necessary before and after an event is finalized. This allows the trading and transfer of the event's outcome tokens to occur transparently and without conflict of interest on other platforms, while issuance and redemption remain public, auditable, and permissionless allowing a new level of security and transparency. In this document we describe the motivation and architecture of the Predictsol smart contract.
Category: Data Structures and Algorithms

A Resolution of the Collatz Conjecture

Authors: Michael E. Spencer
Comments: 99 Pages.

This work develops a refinement—deterministic arithmetic framework for the odd-to-odd Collatz dynamics. The admissible inverse mapR(n; k) = (2^k n - 1) / 3is governed locally by residue—phase conditions on the live classes 1 and 5 (mod 6) and refines coherently through the exponential modulus towerM_j = 2 * 3^(j+1).At each level, admissibility of finite k-words depends only on the residue modulo M_j, and refinement introduces additional phase coordinates without ambiguity.Globally, admissible inverse lifts generate disjoint affine rails whose minimal bases are uniquely determined. Independently, the dyadic valuationk = v_2(3m + 1)produces an exact slice decomposition of the odd integers with weights 2^(-k). We prove that the affine rail partition and the dyadic slice decomposition coincide exactly, yielding a single unified arithmetic structure in which every odd integer possesses a unique admissible ancestry.A refinement-induced acyclicity principle is established: no finite admissible k-word remains compatible across all refinement levels M_j. Periodic inverse instruction regimes are destroyed by phase shifts under refinement, excluding nontrivial odd cycles. Moreover, compatibility across the refinement tower forces every infinite admissible chain to realize a base residue in the anchor structure; hence no divergent trajectory can occur.Finally, the forward mapT(m) = (3m + 1) / 2^(v_2(3m + 1))is shown to be the exact algebraic inverse of all admissible inverse lifts. Forward and inverse dynamics therefore coincide on a single closed affine system anchored at 1.Consequently, the odd-to-odd Collatz dynamics admit a complete internal arithmetic classification, and every forward trajectory converges to the fixed point 1.
Category: Number Theory

Diamond and Lonsdaleite: Beyond Defect Centers and Stishovite as Natural Seed

Authors: Hans Hermann Otto
Comments: 6 Pages.

We address attention to the 3D lattice match between hexagonal lonsdaleite, a carbon allotrop denser and harder than diamond with a higher carrier mobility, and tetragonal stishovite (rutile structure). Both compounds are obtainable via shock wave synthesis route. Stishovite seeding may lower the high-pressure and low temperature synthesis conditions for lonsdaleite as a coming material for quantum computer chips better withstanding solar wind damage. The chips could be made of a forest of lonsdaleite nanorods with defect center based addressable high-fidelity and non-destructive qubits. Material science as well as geochemistry may gain profite from the remarkable three-dimensional lattice coincidence between stishovite and lonsdaleite as a groundbreaking suggestion. Interestingly, carbon dioxide can adopt the rutile structure at extreme conditions of pressure and temperature. This paws the way for new oxygen defect based research on lonsdaleite as chip material.
Category: Condensed Matter

Why Was it so Difficult to Prove the Twin Prime Conjecture?

Authors: Oscar E. Chamizo Sánchez
Comments: 4 Pages.

The twin prime conjecture, asserting there are infinitely many pairs of primes differing by 2, was popularized by French mathematician Alphonse de Polignac in 1849. We are pleased to present an astounding and overwhelming proof with a clasic "reductio ad absurdum" flavour revealing, by the way, a perhaps not so amazing relationship with the Goldbach conjecture and testing, since the core of reasoning is the same, that both statements are strongly connected.
Category: Number Theory

Machine Learning Based Credit Card Fraud Detection

Authors: Avinash Chaurasiya
Comments: 24 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

Credit card fraud poses an escalating threat to the global financial ecosystem, causing billions of dollars in annual losses and eroding consumer trust. Effective automated fraud detection must contend with severe class imbalance, evolvingattack patterns, and the practical need for explainable, actionable predictions. In this paper, we present a rigorous comparative study of five machine learning classifiers—Logistic Regression, Decision Tree, Random Forest, Gradient Boosting, and XGBoost—applied to a dataset of 50,000 credit card transactions exhibiting a realistic fraud rate of 0.34%. We evaluate the impact of two class-imbalanceremediation strategies (SMOTE oversampling and random undersampling), conduct threshold optimisation to align classification decisions with business economics, and employ SHAP (SHapley Additive exPlanations) values to provide model-level and instance-level interpretability. Our best model, Gradient Boosting, achieves a ROCAUCof0.9995, aPR-AUCof0.9421, andanF1scoreof0.7805underacost-optimised decision threshold of 0.75, translating into an estimated net business benefit of $4,228 per 10,000 transactions compared to a no-model baseline. Feature analysis identifies V27 (importance = 0.397) and V2 (0.213) as the dominant fraud signalsamong the PCA-derived features. This work demonstrates that ensemble gradient-boosted trees, combined with principled threshold tuning and SHAP explainability,constitute a production-ready solution for real-world fraud detection.
Category: Artificial Intelligence

New Formula for the Area of a General Quadrilateral Using Opposite Sides and Four Angles

Authors: Ryo Takayama
Comments: 5 Pages. In Japanese

In this paper, we derive an explicit formula for the area of a particular quadrilateral using only the lengths of a pair of opposite sides and the four interior angles. The formula is based on known geometric relationships and is presented after algebraic simplification and manipulation. Furthermore, we clarify the conditions under which the formula is valid and elucidate its geometric background. As a result, we demonstrate its applicability not only to convex and concave quadrilaterals but also to certain self-intersecting quadrilaterals.
Category: Geometry

VPUF Physical OS: Genetic-Resilient Deterministic Phase-Coherence for Secure DTx and AGI Humanoid Safety Interlocks

Authors: Donghyeon Sim
Comments: 8 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

Abstract—This paper introduces VPUF Physical OS, the world’s first hardware-native trust substrate designed to establish deterministic biophysical phase coherence for secure wearables and Artificial General Intelligence (AGI) humanoid safety. Traditional Physical Unclonable Functions (PUFs) suffer from environmental instability and failure in genetic proximitydiscrimination[cite: 155, 283]. We propose a biophysical kernel that binds MEMS phase entropy to Ballistocardiogram (BCG) anchors using a Lyapunov-stable operator[cite: 5, 156, 284]. Wefurther establish stochastic stability using Itô calculus to prove noise-resilient quilibrium [cite: 157, 285]. Experimental results on unmodified COTS hardware (iPhone 11 Pro) demonstrate a terminal stability of 0.0072 rad and a 24.7s lock-on time[cite: 6, 158, 286]. Validation with 9 genetically related samplesconfirms an inter-subject correlation of -0.0093[cite: 7, 159, 287]. Furthermore, a Monte Carlo simulation (N=1,000) yields a near-ideal Hamming Distance (HD=0.4731), establishing VPUF as a post-cryptographic standard for DTx and humanoid safety[cite: 160, 288].Index Terms—PUF, Hardware Security, DTx, Lyapunov Stability, Stochastic Differential Equations, Genetic Proximity, Humanoid Safety, AGI Alignment[cite: 8, 161, 289].
Category: Digital Signal Processing

Century Old Questions in Physics

Authors: Kiyoung Kim
Comments: 14 Pages.

Two major theories in physics, relativity and quantum mechanics, have been around for more than a century. Nonetheless, some fundamental questions remain unclear. The nature of reality in quantum mechanics has long been a question as old as the theory itself. Similarly, the effects of special relativity serve as another long-standing issue of how time and space are intertwined. Although the so-called paradoxes arising from special relativity seem to be resolved through a careful interpretation of the theory, it is still intuitively difficult to acknowledge that such effects can occur in our daily lives. Once again, these fundamental questions in physics are reviewed comprehensively with ontological interpretations of principles and laws in physics with the 4-D complex space model.
Category: Quantum Physics

Pivot Vector Direction Cosine Matrix

Authors: Russell P. Patera
Comments: 5 Pages.

This work derives the Direction Cosine Matrix using the Pivot Vector attitude parameters. The resulting matrix elements are simpler than those of other attitude representations, since they involve only vector dot products.
Category: Classical Physics

On the Analysis of Trusses Using Complex Numbers

Authors: Aloys J. Sipers
Comments: 14 Pages. (Note by viXra Admin: Please cite and list scientific references)

In this letter we show how member forces in a truss can be computed using complex numbers. We give an example of a truss where equations with only one unknown member force are formulated by taking moments about a particular chosen point. A theorem provides an expression for a moment in terms of a complex valued force, a complex valued point of application and some basic operations on complex numbers. We consider the following two situations:(i)The moments are taken about a joint of the truss.(ii)The moments are taken about a point which is not a joint of the truss. A theorem provides an expression for the position of that point in terms of joints of the truss.
Category: Education and Didactics

Toward a Proof of the Riemann Hypothesis Insights from Elliptic Complex Analysis

Authors: Xian Wang, Luoyi Fu
Comments: 55 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This study aims to prove the Riemann Hypothesis and the Generalized Riemann Hypothesis by extending the Riemann zeta function and Dirichlet $L$ -functions to the elliptic complex domain, based on a newly constructed system of elliptic complex numbers $mathbb{C}_lambda(lambda<0)$ . The core challenge addressed is the inherent difficulty in resolving these conjectures within the traditional "circular complex domain" framework ($lambda=-1$); the author posits that a complete proof is unattainable strictly within this conventional setting.The primary innovation of this work lies in the formulation of the theory of elliptic complex numbers, specifically identifying the limiting case as $lambdato 0^{-}$ as the key to the proof. Through rigorous deduction, a bijective correspondence between zeros across different complex planes is established. By employing proof by contradiction and leveraging the correspondence between $mathbb{C}_lambda$(as $lambdato 0$) and the circle complex plane $mathbb{C}$, the Riemann Hypothesis and the Generalized Riemann Hypothesis are ultimately proven.This paper is organized into three parts:begin{enumerate}item Construction and Geometric Properties: The first part details the construction of elliptic complex numbers and their fundamental geometric properties, laying the necessary foundation for subsequent analysis and the proof of the conjectures.item Analytic Extension: The second part introduces elliptic complex numbers into mathematical analysis, deriving numerous results analogous to those in classical complex variable function theory.item Proof of Conjectures: The final part presents the formal proofs of the Riemann Hypothesis and the Generalized Riemann Hypothesis.
Category: Number Theory

Unbounded Digit Expansion and Infinite Attractor Structure in a Deterministic Integer Map

Authors: Christoper Mututu
Comments: 36 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

We investigate a previously undocumented integer transformation whose explosive behavior places it beyond the known extremes of Collatz-type dynamics. The system operates by (1) squaring each digit of an integer and concatenating the results, then (2) repeatedly compressing the expanded sequence by summing every consecutive triplet of digits whenever the total length remains divisible by three. This deceptively elementary process generates combinatorial shockwaves: numerical structures routinely balloon to hundreds or thousands of digits before undergoing catastrophic collapse into a microscopic attractor set. [etc.]
Category: Number Theory

Carbon Nanotube-Enabled Coatings for Advanced Anti-Icing and Deicing Applications

Authors: M. Tarnowska, A. P. Terzyk, J. Kujawa, S. Boncel
Comments: 64 Pages.

Ice accumulation presents persistent challenges across critical infrastructure sectors, including aviation, energy transmission, transportation, and telecommunications. With the advancement of nanomaterials, carbon nanotubes (CNTs) have emerged as powerful components for the design of high-performance anti-icing and deicing coatings. Owing to their exceptional thermal, electrical, and surface properties, CNTs enable both passive (e.g., superhydrophobic) and active (e.g., photothermal, electrothermal) strategies for ice mitigation. This review critically examines the integration of pristine and chemically modified CNTs into functional coatings, highlighting synthesis approaches, surface engineering, performance metrics, and operational mechanisms — reported from 2016 to 2025. Particular emphasis is placed on the correlation between coating efficacy and the physicochemical characteristics of CNT surfaces, interpreted through the framework of Hansen Solubility Parameters (HSPs) as a predictive tool for CNT—matrix compatibility and icephobic performance. By mapping structure—function relationships and identifying synergistic design strategies, this work provides a comprehensive perspective on the future development of scalable, durable, and climate-resilient CNT-based anti-icing and deicing technologies.
Category: Chemistry

Gravitational Dynamics of Cantor Dust and Self-Interacting Dark Matter

Authors: Ervin Goldfain
Comments: 36 Pages.

We develop a first-principles framework in which self-interacting dark matter (SIDM) emerges as an effective manifestation of primordial Cantor Dust. In this framework, Dark Matter resides on a multifractal support characterized by a scale-dependent Hausdorff dimension D<3. We argue that the phenomenology of SIDM is a memory of primordial Cantor Dust, emerging without conventional particle scattering and cross sections. The mechanism described here reproduces the flat galactic rotation curves, baryonic Tully-Fisher relations, the SIDM transport cross section, explains the core—cusp problem, the evasion of Bullet Cluster bounds and the morphology of the cosmic web.
Category: Relativity and Cosmology

Formulas of Chlorine, Iodine and Bromine Dioxides

Authors: Bezverkhniy Volodymyr Dmytrovych
Comments: 2 Pages. (Note by viXra Admin: For the last time, Please cite and list scientific references)

The structural formulas of chlorine, iodine and bromine dioxides with three-electron bonds are presented.
Category: Chemistry

Prime Numbers Density Approximation

Authors: Edward C. Larson
Comments: 5 Pages. (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org)

A novel derivation of the density of the distribution of prime numbers is presented, based on a simple frequentist analysis and the smallest scale at which a rigorous upper bound on the frequency holds. An approximating differential equation is derived. It is shown that in the asymptotic limit, the density of primes, pi(x), scales as x / ln x, in accordance with the Prime Number Theorem (PNT). The approach bridges the gap between discrete number theory and continuous differential modeling, offering a mechanistic explanation for the observed thinning of prime density that mirrors the foundational results of classical analysis.
Category: Number Theory

Solid Angle Subtended by a Circular Plane at an Arbitrary Point in Space

Authors: Harish Chandra Rajpoot
Comments: 8 Pages. 2 Figures

This paper presents the derivation of analytical formulae for estimating the solid angle subtended by a circular plane at an arbitrary point in space. The proposed relations are also employed to determine the geometric parameters of the elliptical projection of a circular plane when observed from an off-center position, including the major axis, minor axis, and eccentricity. All mathematical expressions are derived using the Approximation Formula for the Solid Angle of Symmetrical Planes, as developed by the author and presented in his book Advanced Geometry. The resulting formulae provide a simple and effective approach for the approximate evaluation of solid angles and associated projection parameters in related geometric problems.
Category: Geometry

Analytical Computation of the Area of a Spherical Triangle

Authors: Harish Chandra Rajpoot
Comments: 4 Pages. 1 Figure

This paper presents the derivation of the general formula to compute the area of the spherical triangle having each side as a great circle arc on the spherical surface when (1) the aperture angle subtended by each of the three sides at the center of the sphere is known, and (2) the arc length of each of the three sides is known. These formulas are applicable to any spherical triangle to compute the area on the sphere.
Category: Geometry

Mathematical Analysis of Regular Spherical Polygons

Authors: Harish Chandra Rajpoot
Comments: 16 Pages. 4 Figures

In this paper, a new analytic formula governing all regular spherical polygons (having sides in the form of great circle arcs) has been derived to compute the important parameters such as solid angle, surface area & check the feasibility of the existence of a regular spherical polygon. A regular spherical polygon will exist only if it duly satisfies HCR's formula for a regular spherical polygon.
Category: Geometry

Mathematical Analysis of an Icosidodecahedron

Authors: Harish Chandra Rajpoot
Comments: 10 Pages. 2 Figures

In this work, the principal geometric parameters of an icosidodecahedron comprising 20 congruent equilateral triangular faces and 12 congruent regular pentagonal faces of equal edge length are analytically determined. These parameters include the normal distances and solid angles subtended by the faces, as well as the inner radius, outer radius, mean radius, surface area, and volume. The calculations are carried out using HCR’s formula for regular polyhedra, a generalized dimensional formulation applicable to all five Platonic solids, namely the regular tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron. The versatility of the proposed formula further enables its application to the analysis, design, and geometric modeling of truncated polyhedra.
Category: Geometry

How Newton's Law of Gravity (the Law of Two-body Gravity) Came to Be Called the Law of Universal Gravitation

Authors: Mykola Kosinov
Comments: 16 Pages.

This article attempts to dispel the biggest myth surrounding Newton's law of gravitation for over 300 years. The history of the term "law of universal gravitation" and the negative consequences of its unjustified application to the law of two-body gravity are revealed. This term was preceded by Robert Hooke's term "universal attraction," which referred not to the law of gravity between two bodies, but to the attraction between all bodies in the universe. In addition to the term "universal gravitation," Robert Hooke gave a verbal formulation of the future law of universal gravitation, which was radically different from Newton's verbal formulation. The term "law of universal gravitation" was not coined by Isaac Newton. It was not Newton, but popularizers who began to unjustifiably use the term "law of universal gravitation" in relation to the local law of gravity of two bodies F = GmM/r². The substitution of terms created the illusion that the law of universal gravitation had been discovered and that no other laws of gravity existed. From an ethical perspective, this substitution led to a historical injustice, as a result of which Hooke's contribution in the shadows for over three centuries. In science, this led to the true law of universal gravitation, as Robert Hooke envisioned it, never being discovered.
Category: Quantum Physics

The Cerenkov Effect in the Altermagnetic Medium

Authors: Miroslav Pardy
Comments: 7 Pages. Original article

We derive the spectral formula of the Cherenkov radiation in dielectric medium with the index of refraction and magnetic permeability. This formula is valid also for the altermagnetic medium, where the index of diffraction and permeability must be calculated by the adequate methods which are analogical to the methods published by author (2018).
Category: Quantum Physics

The Daon Theory - A Unified Framework for Fundamental Physics

Authors: J. E. S. Gustafsson
Comments: 148 Pages.

This theory is built from first principles, using Euclidean geometry and classical mechanics, and is designed to be accessible to readers with a basic understanding of physics. All constants are derived numerically, and the derived theory agrees with experimental data.
Category: Quantum Gravity and String Theory

N-Dimensional K-Algebras and Fractals Without Hypercomplex Numbers

Authors: Jaime Vladimir Torres-Heredia Julca
Comments: 27 Pages. 9 figures

This paper is a continuation of our previous paper viXra:2512.0150, in which we expressed all vectors of the 3D Euclidean vector space with operations involving rotations and homotheties of the unit vector of the x-axis. We defined also a new vector multiplication and in this paper we will show that with the usual vector addition, we have a K-algebra. We will also study the 4D case and we will also express all vectors with simple rotations and homotheties of the unit vector of the x-axis about 3 planes, we will define a vector multiplication analogous to what we have seen in the 3D case, and we will show that we have again a K-algebra with the usual vector addition. We will construct also a 4D fractal set which contains the 3D fractal set (which contains the Mandelbrot set) seen in our previous paper and we will show several 4D projections of that set in the 3D space. We will show also that there are no quaternion numbers, because there are actually geometrical operations involving the unit vector of the x-axis. In that case we will use compositions of 2 rotations of the unit vector of the x-axis about different planes to express all the vectors. In that case we will have also a division algebra. Finally, we will show that for all dimensions, we have K-algebras if we express all vectors with operations involving homotheties and simple rotations of the unit vector of the first axis (x_1), with the usual vector addition and a generalization of the multiplication that we have seen in our previous papers for the 2D and 3D cases and for the 4D case in this paper. One advantage of the simple rotations, among other ones, is that they allow to construct interesting n-dimensional fractal sets linked to the Mandelbrot set.
Category: General Mathematics