All Submission Categories

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: 3 Pages.

Besides addressable color centers in diamond and lonsdaleite, operating as qubits for quantum computation or magnet field sensors, we address attention to the 3D lattice match between hexagonal lonsdaleite and tetragonal stishovite, both obtainable via shock wave synthesis route, where stishovite seeding may lower the high-pressure and low temperature synthesis conditions for lonsdaleite. Material science as well as geochemistry may gain profite from the remarkable three-dimensional lattice coincidence between stishovite and lonsdaleite.
Category: Condensed Matter

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

Authors: Óscar 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: 5 Pages.

This article attempts to dispel the biggest myth that has surrounded Newton's law of gravitation for over 300 years. It reveals the history of the term "law of universal gravitation" in physics and the consequences of its unjustified application to the law of two-body gravity. The role of Robert Hooke and his astonishing intuition, which paved the way for him to formulate the actual law of universal gravitation, is revealed in a new light.
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

The Observer-Invariant End of Spacetime at the Horizon of Schwarzschild Black Holes

Authors: René Friedrich
Comments: 5 Pages.

Time is not absolute but observer-dependent: This groundbreaking revelation by Einstein in 1905 has the potential to settle the famous disagreement between external observer Bob and infalling observer Alice:At first sight, the fact that Alice reaches the event horizon of a black hole within finite proper time prevails over Bob's "mere" observation that infalling matter does never reach the horizon.However, all external observers (even including Alice who is an external observer, too, before reaching the horizon) agree without exception that the horizon represents the ultimate simultaneity line (t = infinity), the temporal border of the spacetime manifold. Since spacetime is observer-dependent and thereby mere observation, this "concordant observation shared by all observers of the universe of spacetime" is perfectly adapted for determining extent and boundaries of spacetime.As a result, there can't exist any spacetime beyond observation, and when Alice - subject to infinite gravitational time dilation - reaches the event horizon, she is leaving our observer-dependent spacetime manifold. Consequently, the mass of a Schwarzschild black hole is not located inside the horizon but outside, in the approximate form of a nearby membrane, avoiding the unsolvable dilemma of a central spacetime singularity.
Category: Relativity and Cosmology

A Univariate Quartic Equation for Three-Generation of Charged Lepton G-2

Authors: Shan-Hong Li
Comments: 8 Pages. (Note by viXra Admin: Further repetition may not be accepted; the listing of references are not standard/complete)

We find a univariate quartic equation , and the coefficients of the equation are composed of the Schwinger term and the neutron spin g factor. The solution of the equation is two real roots and two conjugate imaginary roots. These roots are not arbitrary numbers, but are related to the anomalous magnetic moments of the three- generation of charged lepton. One of the two real roots is the anomalous magnetic moment of the electron, and the other is about one tenth of the anomalous magnetic moment of the τ lepton; the real number of the imaginary part of the two conjugate imaginary roots is about 10 times the anomalous magnetic moment of the muon. For this strange proportional relation, we have discussed it and found out the exact relation that they may exist. Finally, we believe that the anomalous magnetic moments of the three-generation of charged lepton is derived from the same high-order algebraic equation, indicating that algebra can also study quantum physics, and also suggesting that there may be algebraic relations and potential deep physical laws.
Category: High Energy Particle Physics

Nested Sampling: A Critical and Comprehensive Theoretical Guide

Authors: L. Martino, F. Llorente
Comments: 28 Pages.

The Nested Sampling (NS) technique has gained widespread attention, particularly in cosmology and astronomy, due to its ability to efficiently explore high-likelihood regions a feature akin to an implicit likelihood optimization that underlies its success. While the full theoretical derivation of NS is complex and involves several approximations, the central challenge lies in sampling from the likelihood-constrained priors, which is crucial for its performance. This work provides a comprehensive and detailed exposition of NS, clarifying both its theoretical foundations and practical challenges. We provide a thorough description of the NS procedure, emphasizing both its strengths and potential limitations. In doing so, this work seeks to deepen understanding of the method and to foster the development of future enhancements, novel variants, and more efficient implementations across a wide range of scientific applications.
Category: Statistics

Temperature Characteristics of Mems Pressure Sensor with High Proof Pressure for Adjustable Ranges up to 10u202660 Kpa

Authors: Mikhail Basov
Comments: 17 Pages.

Temperature characteristics are one of the key parameters determining the performance of MEMS pressure sensor under specific operating conditions. This research demonstrates a new microassembly design for differential pressure sensor with adjustable ranges of 10u202660 kPa. A distinguishing feature of this microassembly is its use of stops for pressure sensor chip with sensitivity of S = 0.61 ± 0.15 mV/V/kPa. Developed pressure sensor enables to withstand proof pressure of 1.5 MPa. The proposed base geometry of the microassembly significantly reduces the impact of residual mechanical stresses (RMS) by all mechanical connections of pressure sensor chip, which is subjected to thermocycling and barocycling. It has been proven that it is possible to achieve errors in the temperature hysteresis of the zero signal (THZ) of less than 0.25%/FS in a wide temperature range from -65u2070С to +85u2070С, despite the requirement of top mechanical stop (as an additional source of RMS), which is only necessary to increase the overload capacity of pressure sensor.
Category: Classical Physics

A Statement of the Cosmological Constant Problem and an Effect of the Reducing of Vacuum by Matter Based on Uncertainty Relations

Authors: Nekrasov Grigory Yu
Comments: Pages.

A problem of the connection of cosmology with elementary particle physics is shown on thelevel of uncertainty relations. At the scales about 10^-2 m the contribution of one single typevirtual elementary particles in the lower boundary of vacuum energy is considered. The observedvalue of vacuum energy or energy density on the large scale of the Universe corresponds only tothis scale. This is the energy about 3.34 GeV per each one cubic meter. The minimal high energyphysics scale achieved by experiments at present is considered. The lower boundary of theenergy is generated by the quantum vacuum of empty space and the quantum vacuum limited bymatter in the Universe mainly at scales down to 10^-15 m and more much are not in agreementwith the observed value, as that is established. These lower limits for the energies of the vacuumare considered in the model of estimating where they generate by the presence of virtual particlesin free space and the virtual particles which are limited by matter and exist together with matterin the Universe. The numerical values of the boundary energies are obtained using the computeralgorithm.
Category: Astrophysics

Hierarchical Geometric Cosmology with Einstein-Cartan Torsion: A Unified Framework for the Cosmological Constant, Dark Matter, and Baryogenesis

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

We present Hierarchical Geometric Cosmology (HGC v3.0), a framework that addresses three of the deepest open problems in modern cosmology — the cosmological constant, the nature of dark matter, and the matter-antimatter asymmetry — within a single geometric structure. The central premise is that our universe occupies the interior of a rotating black hole embedded in a parent spacetime governed by Einstein-Cartan theory, which extends General Relativity by incorporating the spin of matter as a source of spacetime torsion.The cosmological constant emerges naturally from the Israel-Darmois junction conditions at the black hole horizon, yielding Λ_eff = ζ Φ(a*)/R_s², where R_s is the Schwarzschild radius of the parent black hole and Φ(a*) encodes the spin contribution. With R_s ~ R_H, this produces the observed order of magnitude (10u207bu2075² mu207b²) without fine-tuning, resolving the 120-order-of-magnitude discrepancy with quantum field theory predictions.The Big Bang singularity is replaced by a torsion-induced bounce at the Cartan density ρ_C ~ 10u2076u2076 kg/m³, a semiclassical process some 30 orders of magnitude below the Planck scale. Numerical integration of the Kantowski-Sachs equations through the bounce shows that all initial conditions converge to an isotropic FRW universe with f_iso = 0.334 ± 0.001, independent of initial anisotropy.Dark matter halos are replaced by a gravitational shadowing potential derived from Regge-Wheeler/Zerilli tidal perturbation theory, with a characteristic scale r_c = R_s/√6. This model is tested against the full SPARC database of 175 disk galaxies, achieving a median χ²_r = 2.09 and performing comparably to NFW dark matter profiles (49 vs. 59 galaxy wins). The critical observational distinction lies at scales R > 2 Mpc, where shadowing predicts 5—15% suppression in weak gravitational lensing relative to NFW — a test accessible to Euclid and LSST.For baryogenesis, the cubic torsion term in the Dirac equation produces a mass asymmetry between matter and antimatter. We report honestly that the perturbative calculation yields δM/m ~ 10u207bu2076u2076 at the electroweak scale, far below the observed baryon asymmetry η_B ~ 6 × 10u207b¹u2070. We identify non-perturbative torsion effects near the Cartan density as the most plausible resolution and retain this as an open theoretical direction.Three primary open problems are explicitly identified: (i) the parameter ζ is geometrically motivated but not fully derived from first principles, with a factor ~4 gap remaining; (ii) HGC and NFW are statistically indistinguishable at the rotation curve level; (iii) torsion baryogenesis is qualitatively valid but quantitatively insufficient in the perturbative regime. Seven falsifiable predictions are presented for LiteBIRD, Euclid, LISA, DESI, CERN ALPHA/BASE, and JWST.
Category: Relativity and Cosmology

A Unifying View of Multiple-Try Metropolis and Particle Metropolis-Hastings Algorithms

Authors: L. Martino
Comments: 27 Pages.

Markov chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) methods are cornerstone techniques for Bayesian inference and stochastic optimization. The multiple-try Metropolis (MTM) algorithm generalizes the Metropolis-Hastings (MH) scheme by selecting the next state from a set of weighted candidates, improving exploration of the state space. Particle Metropolis-Hastings (PMH) integrates MCMC and SMC ideas to efficiently tackle high-dimensional targets with sequentially factorized structures, embedding a particle filter within an MH framework. While both approaches have been extensively studied, particularly for state-space models, their relationship has not been fully explored. In this work, we examine the connections and distinctions between MTM and PMH schemes, which motivates the design of novel, highly efficient algorithms for filtering and smoothing. Among these, we introduce a particle multiple-try Metropolis (P-MTM) method, which demonstrates excellent performance across a range of numerical experiments.
Category: Statistics

Emergence of Diverse Structures and Functions from a State Transition Function Facilitating Origin and Replication of the Byl Replicator.

Authors: Perry W. Swanborough
Comments: 14 Pages.

Replication of the John Byl cellular automata replicator (1989) and its origin from an isolated cell in an oriented state requires 183 explicit state transition rules, but the corresponding state-set inclusive of eight unoriented active states {1,2,3,4,5,L,E,F} and one oriented active state (^) prescribes a vast number of possible state transition functions supporting the replicator and its origin, each containing 93,262 explicit state transition rules. The particular state transition function supporting the origin and replication of the replicator, with all other 93,262 — 183 = 93,079 rules prescribing state conservation (neighbourhood CNESW - - > C’ = C), supports the appearance and development of structures and functions additional to appearance and replication of replicator structures. For abiogenesis, these results suggest the possibility that interactions of ancient replicators with each other and with coexisting persistent structures enabled the subsequent appearance of containment (membranes), metabolism and information processing, thereby founding ancestral life.
Category: Physics of Biology

On the Arbitrary Reduction of Energy and Energy-Momentum Tensor Components in the Alcubierre Warp Drive with Possibile Positive Values

Authors: Gianluca Perniciano
Comments: 7 Pages.

Building on the article [16] where a superluminal Alcubierre propulsion system with positiveenergy density and positive energy is possible as an improvement over [16], we make a smallmodification to have a positive energy and positive energy density for a curvature bubble of genericradius R and introduce a way to arbitrarily reduce such energy sources in the warped region,.Wecompute the remaining components of the Einstein tensor, which we find to be only partiallypositive; however, the WEC, DEC, and apparently the SEC are satisfied [20]. This implies a warpdrive without exotic matter, but rather a form of asymmetric ordinary matter.
Category: Relativity and Cosmology

The Relativistic Effect of Kinetic Star

Authors: Yi-Liu Fan
Comments: 2 Pages.

The paper discusses the "time paradox" and the "mass paradox" of "twin paradox" in the Special Theory of Relativity. The research discovers that the space-time outside a kinetic star shows a double structure.
Category: Relativity and Cosmology

WIMPs and Glowing Dark Matter

Authors: Clark M. Thomas
Comments: 5 Pages.

It may seem odd to talk about glowing dark matter. Something dark is typically hypothesized to be hiding in the darkness of space, along with dark energy. All favored dark models are simply outdated. This essay will explain how the brilliance of so-called dark matter has always been there. Dark phenomena clearly belong within any Theory of Everything.
Category: Astrophysics

Inversion Identity for The $k$-th Root Function

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

In this paper, we derive a formal inversion identity from the Taylor expansion of $sqrt[k]{x}$ to get $x$ as an infinite series of the function. Along with the derivation, we also give a proof of the identity by justifying some crucial mathematically rigorous statements regarding analyticity, validity of Cauchy's convolution, and the convergence, and also derive a trivial infinite series for $pi$, $e$ (Euler's constant) and a formal infinite series identity of $gamma$ (Euler-Mascheroni constant) in terms of their $k$-th roots.
Category: Number Theory

Study On Delta Between Primes And Triangular Numbers

Authors: Marcin Barylski
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

There are several interesting properties of triangular numbers and research work devoted to them. One of the them is correlation between them and primes- there is hypothesis that between every two different triangular numbers >1 there is always a prime number. This paper is focused on detailed examination of difference, mainly between triangular numbers and their closest (smaller or greater) primes (this difference is called in this work delta, δTP), including its extreme values, also in spirit of finding effective test to search for prime numbers.
Category: Number Theory

On the Origin of Cosmic Microwave Background Radiation

Authors: Sergey G. Fedosin
Comments: 41 Pages.

The alternative mechanism of the emergence of cosmic microwave background radiation (CMB), associated with the thermal radiation of primordial gas-dust clouds in the early Universe, is considered. The emergence of such clouds in the theory of infinite hierarchical nesting of matter is a natural stage in matter evolution. The mass, radius, and spatial concentration of typical primordial gas-dust clouds, the distance between neighboring clouds, and the power of CMB energy generation per unit volume and per nucleon of the early Universe were calculated. The masses and radii of these clouds correspond to the masses and radii of the observed Bok globules. The presented mechanism is consistent with the cluster model describing the appearance of angular multipoles in the CMB power spectrum. In addition to CMB radiation, cosmic infrared background (CIB) radiation and cosmic optical background (COB) radiation are also considered. According to the presented model, the sources of CIB are primordial protoplanetary clouds. As for the COB radiation, it is associated with the radiation of the first protostars. During evolution, each primordial cloud, with a mass of about 31 solar masses, first generates CMB radiation, and then CIB and COB radiations. Since protostars give rise to neutron stars, the concentration of primordial gas-dust clouds is also the concentration of observed neutron stars. In the course of the calculations, a new definition of the radiation intensity is used, which is based on the vector of the surface energy flux density and accounts for the angles of incidence of radiation on a flat receiver from all sides of the hemisphere. According to Poynting's theorem, the relationship between the intensity and energy density of black body radiation is derived from the concept of photons.
Category: Astrophysics

Comparison of Tensor and Vector Theories of Gravitation

Authors: Sergey G. Fedosin
Comments: 57 Pages.

Physical quantities in continuously distributed matter in curved spacetime, and equations for matter and fields are considered both from the point of view of tensor theory of gravitation and on the basis of vector theory of gravitation. An example in the first case is the general theory of relativity (GTR), which uses a scalar pressure field and a scalar acceleration field. In the second case, relativistic vector fields are taken into account, including the covariant theory of gravitation, the pressure vector field and the acceleration vector field. To analyze and compare the results in each approach, formulas derived from the principle of least action and from the corresponding Lagrangian are used. The problem of correlating scalar pressure with the principle of least action in the general relativity is considered. The conclusion is drawn that results of the general relativity, when scalar pressure is taken into account, are valid for relativistic uniform systems, but in a more general case, they require correction. Three versions of general relativity were analyzed: GTR1, GTR2 and GTRm. The GTR1 version is the closest version to the standard general theory of relativity, the GTR2 version follows exactly the principle of least action, and the GTRm version is characterized by the fact that the acceleration field and pressure field are represented not as scalar fields but as vector fields. Equations for metric, equations of motion, equations for fields, formulas for the energy and momentum, which follow from the Lagrangian formalism, are presented for all versions of general relativity. An explanation is given of where dark energy comes from and what it is within general relativity.
Category: Relativity and Cosmology

The Klein-Gordon-Fock Equation Correct for the Description of the Particle

Authors: Nekrasov Grigory Yu
Comments: 9 Pages. In Russian

The quantum mechanical Klein-Gordon-Fock equation in the space-time curved by the background sources is considered. The comparative analysis of this equation and the same equation in the flat Minkowski space-time is made. The Klein-Gordon-Fock equation is solved for the case of the microscopic Schwarzschild black hole near the event horizon. It is supposed that in curved spaces of special type this equation will describe the quantum particle, as then the fourth component of the probability current density — to within a multiplier the probability density will be strictly positive. The solution provided in this article is applicable for the description of a massive particle with spin 0 near the event horizon of the microscopic black holes. The main text of the article will be provided in Russian.
Category: Quantum Physics

On Uniqueness of Row Normal Form

Authors: Sanjeev Saxena
Comments: 4 Pages.

In this note, it is shown that the Row Echelon Normal Form (RENF), also called the Reduced Row Echelon Form (RREF) of a matrix, is unique. The result is proved from first principles.
Category: Algebra

Beals Conjuncture

Authors: P. Murugesha
Comments: 2 Pages.

In this paper i going to compare matrix determinants and Beals conjuncture.. In this paper I consider equation (az^2-by^2)x^2-(bx^2-cz^2)y^2=(ax^2-cy^2)z^2.Next taken bracket containing things in above equation as x, y, z and proved equation 5 as no solution.... So,I concluded in this paper( if we replace (az^l-by^m)x^n-(bx^n-cz^l)y^m=(ax^n-cy^m)z^l then again taken this equation az^l-by^m=x and bx^n-cz^l=y and ax^n-cy^m=z(note l, m, n are >=2). Then also get no solution (it get same as equation 5 only changes in x, y, z powers) it still holds the properties of relatively prime properties.) So we finally got answer for beal conjuncture. When powers greater than 2 has no solution.
Category: Number Theory

Error [Exploration] On Around-The-World Atomic Clocks Experiments

Authors: ZhengMing Yin
Comments: 7 Pages.

The Around- The- World Atomic Clocks experiment paper published in the 1972 American Science Journal contains many conceptual errors. Thank you very much for the brave experiments of 2 American scientists J.C. Hafele and Richard E. Keating, because their experiments brought new scientific thinking. Einstein proposed the Relativity time dilation, He should first define the concept and definition of time, But there is not these contents in the theory of elativity.。Thesis of losing conceptual and definitional constraints，it's just a science fiction, not the theoretical system. The concept and definition of time have been usefully explored by the author and will be presented in separate articles. This paper deals only with issues related to the Around- The- World Atomic Clocks experiment by American scientists J.C. Hafele and Richard E. Keating.
Category: Relativity and Cosmology

A New Method for Determining the Direction of Gyroscopic Torque as Part of Nonlinear Dynamics

Authors: Bojidar Djordjev
Comments: 20 Pages.

A new method for determining the direction of the gyroscopic torque is proposed based on the inertial potential of the changed direction of the orbital velocity in the plane of precession. It is shown that it is a theoretical prerequisite for the application of the Laws of Nonlinear Dynamics to the gyroscope. The Nonlinear Dynamics and the New Method for Calculating the Gyroscopic Torque are discussed. Theses and comments.
Category: Classical Physics

Unifying Trends in Physics, Chemistry and Biology, and the Emergence of Life

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

We present a unified theoretical framework (UPCB) that integrates trends in Physics, Chemistry, and Biology into a single Quantum Network Model. Departing from the point-particle paradigm of the Standard Model, we propose that fundamental entities are nodes in a quantum information processing network, emerging from a "Quantum Computing" substrate (2+2 → 3+1 Hopf fibration). Key unifications include: (1) The Wilsonization of the Gauge Theory, identifying the Electron as a time-like "T-quark" and unifying the lepton and quark sectors; (2) The topological unification of bonds, where chemical (covalent/ionic) and nuclear (Masonic) bonds are shown to be isomorphic graph connections distinguished only by the genus of the interaction (Grothendieck’s Dessins d’Enfant); and (3) The identification of Chirality Induced Spin Selectivity (CISS) as the macroscopicmanifestation of the fundamental gauge chirality. We argue that the emergence of Life is not accidental but mandated by the "diode-like" rectification of quantum information flow through chiral centers, challenging the traditional thermodynamic view of entropy in self-organizing systems.
Category: Quantum Physics

On the Limits of Prompt Repetition: A Multi-Asset Evaluation of LLM Inference Heuristics for Financial Time-Series Prediction

Authors: Avinash Chaurasiya
Comments: 20 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

Prompt repetition has recently been proposed as a simple inference-time modificationcapable of improving the performance of non-reasoning large language models(LLMs). By duplicating the input prompt, the technique aims to improve attentionutilization without incurring additional computational cost. While empirical gainshave been reported on deterministic language benchmarks, it remains unclearwhether such improvements generalize to stochastic prediction domains whereuncertainty originates from external information rather than prompt structure.In this work we conduct a systematic, multi-asset evaluation of prompt repeti-tion in financial time-series forecasting, spanning four representative instruments:GOOGL, MSFT, NVDA, and GLD. We compare a logistic-regression baselineagainst LLM predictions under both standard prompting and prompt repetition,assessing directional accuracy, Brier score, bootstrap confidence intervals, McNemarsignificance tests, and calibration reliability diagrams. Across all assets and allmetrics we find no statistically meaningful improvement attributable to promptrepetition. We further provide an information-theoretic proof showing that anytransformation preserving input entropy cannot increase predictive mutual infor-mation in noise-dominated environments. Our findings establish a clear boundarycondition for prompt-engineering techniques and underscore the necessity of domain-aware evaluation before deploying LLM inference strategies beyond natural languageprocessing.
Category: Artificial Intelligence

Directional Density of Gases Under FitzGerald-Lorentz Contraction: Predictions for Optical Interferometry

Authors: Alvydas Jakeliunas
Comments: 5 Pages.

We show that the directional density of an ideal gas remains isotropic under FitzGerald-Lorentz contraction of its container, while that of a bound solid becomes anisotropic. Com-bined with the invariance of the optical phase delay per molecule, this yields parameter-free predictions for gas-lled interferometers under Lorentzian Relativity. For a single gas-lled arm measured against a direction-independent reference, the signal on rotation is∆N= k(L/λ)(n−1)(v/c)2, where k depends on the interferometric conguration. Thesignal arises from the orientation-dependent molecule count in the contracted gas path. Wepredict null results for solid-dielectric interferometers regardless of refractive index.
Category: Classical Physics

Formulae for the Nth Derivative of Exponentiated Polynomials Via Infinite Matrix Operations

Authors: Habeeb Mohammed
Comments: 14 Pages.

We investigate how the sequence $f^{(n)}(0)$ acts for a specific class of functions: exponentiated polynomials, $e^{p(x)}$, of which we first look at $e{-x^2}$. This leads us into an textit{infinite dimensional matrix}, which can be analysed via tools from Lie algebra. To generalise this to all polynomials $p(x)$, we define a correspondence between the space of derivatives of $e^{p(x)}$, $cal{F}$, and a general vector space of polynomials, $cal{T}$; we find that the derivative in $cal{F}$ also corresponds to an operator in $cal{T}$. We then utilise Zassenhaus' formula to find how this operator iterates, hence giving us a general formula for the $n$th derivative of $e^{p(x)}$.
Category: Algebra

Transformative Integration of Agentic Generative AI in Food Safety Systems: Policy Framework, Implementation Guidelines, and Economic Impact Analysis

Authors: Satyadhar Joshi
Comments: 14 Pages.

This paper presents a comprehensive review of artificial intelligence (AI) applications in food safety and quality control, focusing on emerging technologies including generative AI, agentic AI systems, and automated compliance solutions. This review synthesizes current research and industry applications, highlighting how AI-driven systems are transforming food safety protocols, enhancing regulatory compliance, and improving overall food quality management. We examine various AI implementations, from optical imaging for bacterial detection to intelligent compliance agents and generative AI for supply chain optimization. This paper synthesizes current research and industry applications across multiple domains: automated visual inspection systems that detect contaminants with precision exceeding human capabilities; predictive quality analytics that forecast potential safety issues before they manifest; AI-driven regulatory compliance systems that continuously monitor and interpret complex regulatory requirements; and autonomous agentic systems that make real-time decisions without human intervention. The review also addresses significant technological innovations, including the FDA's development of AI tools for regulatory operations, generative AI applications for scenario planning and documentation, and cloud-based AI architectures deployed across major platforms. Critical challenges are examined, including data quality requirements, regulatory validation frameworks, system integration complexities, and ethical considerations. The paper concludes with policy recommendations for government implementation, proposing structured approaches to AI validation, data sharing incentives, regulatory modernization, research support, and ethical oversight. A dedicated response to FSIS-2025-0145 demonstrates how AI technologies can address overtime inspection fee challenges through remote monitoring and automated compliance.
Category: Quantitative Biology

Pi's Irrationality Using a TI84 Calculator

Authors: Timothy Jones
Comments: 2 Pages.

We use a TI-84 program to show pi's rationality implies that a radius at 90 degrees must have a defined slope, a contradiction.
Category: Number Theory

Invariant Dispersion and Emergent Quantum Wave Dynamics from Bounded Vacuum Capacity

Authors: Tanuj Kumar, Vandana [Doe]
Comments: 15 Pages.

In Ref. [1], a bounded vacuum-capacity framework was developed in which gravitation arises from static, localized deficits of a scalar vacuum-potential field normalized by the equilibrium condition Φ→c^2 in asymptotically flat spacetime. The same capacity principle was shown to imply a universal maximum force and a signal-speed bound v≤c.In the present work, we develop the dynamical sector of this framework. Linearized fluctuations about a localized vacuum-deficit configuration are shown to satisfy a Klein—Gordon—type equation, with the mass parameter determined by the intrinsic oscillation frequency associated with curvature of the vacuum response. The resulting plane-wave solutions yield an invariant quadratic dispersion relation of the form E^2=p^2 c^2+m^2 c^4, arising directly from the bounded stiffness-to-inertia ratio of the vacuum medium. The velocity-dependent inertial response m(v)=m/√(1-v^2/c^2 ) follows from this dispersion structure. In the low-momentum regime, the Schrödinger equation emerges as the nonrelativistic limit.The mass parameter entering both static and dynamical sectors is identified with the asymptotic vacuum-loading parameter defined in Ref. [1], relating gravitational and inertial mass to a single localized vacuum configuration. The results establish a unified dynamical description in which relativistic and nonrelativistic quantum wave equations arise from bounded vacuum capacity.
Category: Quantum Gravity and String Theory

Derivation of Great-Circle Distance Formula (Minimum Distance Between Any Two Points on Globe Given Latitude and Longitude)

Authors: Harish Chandra Rajpoot
Comments: 5 Pages, 3 Figures, Original Research

In this work, the great-circle distance formula is derived using hcr's inverse cosine formula. An analytic and the most generalized formula has been derived to accurately compute the minimum distance or great circle distance between any two arbitrary points on a sphere of finite radius which is equally applicable in the geometry of sphere. This formula is extremely useful to calculate the geographical distance between any two points on the globe for the given latitudes & longitudes. This formula is the most power tool which is applicable for all the distances on the tiny sphere as well as the large sphere like giant planet assuming them the perfect spheres.
Category: Geometry

Arrangement of Identical Touching Circles on a Spherical Surface Associated with Platonic Solids

Authors: Harish Chandra Rajpoot
Comments: 13 Pages, 10 Figures, Original Research

All the articles discussed and analyzed in this work are related to the five Platonic solids. A geometrical problem involving a finite number of identical circles mutually touching one another on the entire surface of a sphere of given radius is considered. Using elementary geometric relations together with tabulated parameters corresponding to the five Platonic solids, all important quantities, including the flat radius and arc radius of each circle, the total surface area covered by the circles, and the percentage of spherical surface coverage, are systematically evaluated. The derived parameters are useful for accurately drawing identical circles on a spherical surface and for the design and modeling of the five Platonic solids with identical flat circular faces.
Category: Geometry

Solid Angle Subtended by a Beam with Rectangular Profile Given Horizontal and Vertical Beam Angles

Authors: Harish Chandra Rajpoot
Comments: 6 Pages, 3 Figures, Original Research

In this work, previously reported analytical formulations are systematically discussed and analyzed to evaluate key beam parameters, including the solid angle subtended by a beam at a point source, the total area intercepted by the beam on a spherical surface, and the cone angle of an equivalent beam with a circular cross section. These formulations enable the transformation of a beam with a rectangular profile into an equivalent circular profile, and vice versa, while preserving the total radiation energy or luminous flux associated with the original beam emitted by a uniformly radiating point source. The presented analysis is particularly useful for radiometric applications involving the evaluation of radiation energy and directional intensity of uniform point sources, as well as for photometric applications concerned with luminous flux and luminous intensity in specified directions. Furthermore, the results facilitate the replacement of rectangular apertures with circular apertures, and conversely, without altering the total transmitted radiation energy or luminous flux. Consequently, the discussed formulations provide a valuable theoretical framework for the analysis and design of optical and beam-emitting devices, such as laser systems, based on uniformly radiating point sources.
Category: Geometry

Solid Angles Subtended by Archimedean Solids at Their Vertices

Authors: Harish Chandra Rajpoot
Comments: 3 Pages,

This paper presents a comprehensive tabulation of the solid angles subtended at the vertices of all thirteen Archimedean solids (convex uniform polyhedra). The solid angles are analytically evaluated using the standard solid angle formula in conjunction with tetrahedral decomposition. The resulting values constitute a consistent and complete set of reference data for the vertex geometry of the Archimedean solids, including the truncated tetrahedron, truncated hexahedron (cube), truncated octahedron, truncated dodecahedron, truncated icosahedron, cuboctahedron, icosidodecahedron, small rhombicuboctahedron, small rhombicosidodecahedron, snub cube, snub dodecahedron, great rhombicuboctahedron, and great rhombicosidodecahedron. These results provide useful quantitative tools for the geometric analysis and comparative study of uniform polyhedra.
Category: Geometry

Analytical Solutions for a Non-Uniform Tetradecahedron with Two Regular Hexagonal Faces and 12 Trapezoidal Faces

Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 4 Figures

This paper presents an analytical derivation of the fundamental geometric parameters of a non-uniform tetradecahedron composed of two congruent regular hexagonal faces, twelve congruent trapezoidal faces, and eighteen vertices lying on a common circumscribed sphere. Using HCR’s Theory of Polygon, explicit closed-form expressions are obtained for the solid angles subtended at the center by the hexagonal and trapezoidal faces, as well as for the corresponding normal distances of these faces from the center. From these results, exact formulas for the inradius, circumradius, mean radius, total surface area, and enclosed volume of the polyhedron are systematically derived. The analytical framework developed herein provides a general and rigorous method for the geometric characterization of non-uniform polyhedra.
Category: Geometry

Mathematical Analysis of a Tetrahedron: Dihedral Angles Between Consecutive Faces and Solid Angle at Its Vertex Given the Apex Angles

Authors: Harish Chandra Rajpoot
Comments: 12 Pages, 3 Figures, Original Research

In this paper, analytical formulas are derived using HCR’s Inverse Cosine Formula in conjunction with HCR’s Theory of Polygon. These formulas provide a simple and practical method for computing the internal (dihedral) angles between consecutive lateral faces of an arbitrary tetrahedron at any of its four vertices, as well as the solid angle subtended by the tetrahedron at a vertex when the apex angles between consecutive lateral edges meeting at that vertex are known. The resulting expressions are fully generalized and can also be applied to configurations in which three faces meet at a vertex of various regular and uniform polyhedra, enabling the calculation of the solid angle subtended by the polyhedron at that vertex.
Category: Geometry

Solid Angles Subtended by the Platonic Solids (Regular Polyhedrons) at Their Vertices

Authors: Harish Chandra Rajpoot
Comments: 5 Pages, 5 Figures

In this paper, the solid angles subtended at the vertices by all five platonic solids (regular polyhedrons) have been calculated by the author using the standard formula of solid angle. These are the standard values of solid angles for all five platonic solids i.e. regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron useful for the analysis of platonic solids.
Category: Geometry

Mathematical Analysis of Spherical Rectangle

Authors: Harish Chandra Rajpoot
Comments: 6 Pages, 2 Figures

In this work, all the articles have been derived using simple geometry & trigonometry. All the formulas are very practical & simple to apply in the case of any spherical rectangle to calculate all its important parameters, such as solid angle, surface area covered, interior angles, etc. & also useful for calculating all the parameters of the corresponding plane rectangle obtained by joining all the vertices of a spherical rectangle by the straight lines. These formulas can also be used to calculate all the parameters of the right pyramid obtained by joining all the vertices of a spherical rectangle to the center of the sphere, such as normal height, angle between the consecutive lateral edges, area of the rectangular base, etc.
Category: Geometry

Emergence of the Electroweak Scale from Continuous Dimensions

Authors: Ervin Goldfain
Comments: 11 Pages.

In this sequel to [6], we show that the electroweak vacuum v emerges as infrared scale generated by dimensional deviation ϵ≪1. The mechanism detailed here mirrors the mass-gap generation mechanism of non-Abelian gauge theory.
Category: High Energy Particle Physics

Geometric Analysis of a Spherical Triangle

Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 3 Figures

In this paper, the principal geometric parameters of a spherical triangle are derived using elementary geometry and trigonometry. The resulting formulae are practical and straightforward to apply for computing key quantities such as the solid angle subtended at the center, the covered spherical surface area, and the interior angles. The analysis is further extended to the corresponding plane triangle obtained by joining the vertices of the spherical triangle with straight line segments, allowing for the evaluation of its geometric parameters. In addition, the derived relations are applied to the right pyramid formed by connecting the vertices of a spherical triangle to the center of the sphere, enabling analytical computation of parameters such as the normal height, angles between consecutive lateral edges, and the area of the base.
Category: Geometry

The Rope of Time (In Italian)

Authors: Marco De Lorenzo
Comments: 43 Pages. (Note by viXra Admin: An abstract in the article is required and please cite and list scientific references)

This work presents a geometric re-foundation of physics, starting from a single postulate: the representation space is Euclidean, rigid, with fixed points and invariant distances. Time is not a coordinate, but an external parameter.From this postulate and the coherence condition (constant vector velocity of the representation system), we deduce the existence of a complex velocity, where is a geometric constant and a real phase.
Category: Classical Physics

Advancing U.S. Public Health Surveillance: Leveraging Generative AI and Agentic Systems for the National HIV Behavioral Surveillance Program

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

This comprehensive technical paper responds to the Centers for Disease Control and Prevention's Federal Register notice (Docket No. CDC-2025-0753) concerning the revision of the National HIV Behavioral Surveillance System (NHBS). We propose an integrated framework leveraging Generative AI (GenAI) and agentic systems to enhance the NHBS data collection methodology across 21 Metropolitan Statistical Areas (MSAs). Our approach addresses all five evaluation criteria specified by the Office of Management and Budget: (1) necessity and practical utility, (2) accuracy of burden estimates, (3) enhancement of data quality, utility, and clarity, (4) minimization of respondent burden through technology, and (5) assessment of information collection costs. Drawing on recent research in AI-assisted surveying, we demonstrate how Large Language Models (LLMs), adaptive interviewing systems, and human-AI hybrid frameworks can transform NHBS from a periodic cross-sectional survey into a dynamic, real-time surveillance tool while reducing the estimated 3,398-hour annual burden. We provide detailed implementation recommendations for the proposed three-year cycle, addressing ethical considerations, validation requirements, and quality assurance protocols for deployment in public health settings. This expanded framework includes comprehensive technical specifications, cost-benefit analyses, and risk mitigation strategies to support evidence-based decision-making for CDC leadership.
Category: Artificial Intelligence

The Geometry of Information: Unifying Gravity and Gauge Fields via the Mixed-State Quantum Geometric Tensor

Authors: Lee Jin Woo
Comments: 10 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org!)

Building upon the Relational-Informational Model (RIM), where spacetime geometry emerges from the Quantum Fisher Information of an observer’s reduced density matrix, I extend the framework to derive all four fundamental interactions from a single information-geometric object: the mixed-state Quantum Geometric Tensor (msQGT).The observer, defined as a subsystem within a globally static universe (Hˆtot|Ψ⟩ = 0), is generically described by a mixed state ρObs = TrSys|Ψ⟩⟨Ψ|. The real symmetric part of the msQGT—the Quantum Fisher Information Metric—reproduces the spacetime metric and gravitational dynamics as established in the companion paper. The imaginary antisymmetric part—the Uhlmann curvature—encodes gauge field strengths. Crucially, the gauge group is determined by the degeneracy structure of ρObs: non-degenerate spectra yield U(1) electromagnetism, two-fold degeneracies produce SU(2) weak interactions, and three-fold degeneracies generate SU(3) color symmetry. I derive the Einstein—Hilbert and Yang—Mills actions from a unified information-theoretic variational principle, interpret coupling constants as geometric rigidities tied to the eigenvalue structure of ρObs, and present explicit toy-model calculations. The frame-work identifies open problems including fermion generations, mass hierarchy, and the cosmological constant. These results suggest that all fundamental interactions arecomplementary manifestations of one underlying quantum information geometry.
Category: Quantum Gravity and String Theory

Non-Archimedean Functional Analysis Over Non-Archimedean Field ^{∗}ℝ_{c}^{}. Applications to Constructive Quantum Field Theory.part II.essential #-Self Adjointness of Hamiltonian H_{0}+V.

Authors: Jaykov Foukzon
Comments: 271 Pages.

Functional analysis works with TVS (Topological Vector Spaces), classically over archimedean fields like ℝ and ℂ.Canonical non-Archimedeanfunctional analysis, where alternative but equally valid number systems such as p-adic numbers ℚ_{p} etc. are fundamental, is a fast-growing discipline.This paper deals with TVS over non-classical non-Archimedean fields ^{∗}ℝ_{c}^{} ,^{∗}ℝ_{c}^{} and^{∗}ℂ_{c}^{}, ^{∗}ℂ_{c}^{}. Definitions and theorems related to non-Archimedean functional analysis onnon-Archemedean field ┊^{∗}ℝ_{c}^{}┊ and on complex field ┊^{∗}ℂ_{c}^{}┊=┊^{∗}ℝ_{c}^{}┊+i┊^{∗}ℝ_{c}^{}┊are considered. Applications to constructive quantum field theory also are considered
Category: Mathematical Physics

On Continuous Dimensions and Infrared Mass Generation

Authors: Ervin Goldfain
Comments: 33 Pages.

We show that dimensional regularization of a generic self-interacting field theory induces an infrared mass scale, when the running spacetime dimension approaches the critical value D=4. The mechanism is universal and does not rely on spontaneous symmetry breaking alone. Instead, it arises from logarithmic corrections generated by renormalization group (RG) flow near the bifurcation point at ϵ≡4-D→0. We explicitly indicate how dimensional transmutation converts marginal couplings into dynamically generated mass scales. This framework unifies the origin of the Higgs vacuum expectation value and electroweak boson masses with the Yang—Mills mass gap and the QCD scale Λ_QCD. Step-by-step derivations are provided, without appealing to nonperturbative assumptions beyond RG consistency. The paper focuses on the regime bordering relativistic quantum field theory and complex dynamics.
Category: Mathematical Physics

The Phenomenon of the Golden Ratio as a Necessary Condition for the Effectiveness of the Double Step Running System

Authors: Andrey V. Voron
Comments: 6 Pages.

The article substantiates the hypothesis that the unity of the object-system "Double step running" is due to the "golden" proportional relations of the object-system to its subsystems and between the subsystems themselves.
Category: Physics of Biology

Intrinsic Dynamical Equations of the Three-Body System and the Gravitational Superposition Theorem

Authors: Jinyong Liu
Comments: 13 Pages. (Note by viXra Admin: For the last time, please cite and list scientific references!)

This paper addresses a classic core challenge in celestial mechanics—the three-body problem. Based on the fundamental axioms of Newtonian mechanics, we rigorously derive and prove a universal "Gravitational Superposition Theorem." This theorem states that the total gravitational field produced at any point in the external space by a finite set of point masses is completely equivalent to the gravitational field produced by a single point mass located at the system's center of mass, with a mass equal to the sum of the individual masses. Using this theorem as a cornerstone, the classical three-body problem can be precisely reduced to three strictly analytically solvable two-body relative motion problems.This research not only provides a theoretically self-consistent and mathematically complete framework for the long-standing three-body problem but also fundamentally reconstructs the theoretical paradigm and logical starting point for modeling multi-body gravitational systems.
Category: Mathematical Physics

Identification of a Continuous Microhertz Signal Consistent with Gravitational Radiation from Zeta Phoenicis

Authors: Herbert Weidner
Comments: 8 Pages.

We investigate whether longu2011term atmospheric pressure measurements contain a coherent, frequencyu2011stable signal consistent with the expected gravitationalu2011wave (GW) emission from the eclipsing binary Zeta Phoenicis. The system's orbital period of 1.6697739 days implies a GW frequency of 13.863 $mu$Hz, a regime inaccessible to conventional interferometric detectors. Using 20 years of hourly pressure data from more than 100 stations, we apply a communicationsu2011engineering approach combining coherent integration, superheterodyne frequency shifting, and iterative phaseu2011demodulation to isolate weak, structured oscillations. After compensating for frequency drift and multiple phase modulations, we recover a narrow, persistent spectral feature at the predicted frequency. One modulation matches the annual Doppler signature expected from Earth’s orbital motion, while additional lowu2011frequency sidebands may reflect longeru2011period dynamical influences within the source system. We further derive a longu2011term decrease in the GW frequency, corresponding to a secular increase in the orbital period, and provide a prediction for the period in 2026. These results demonstrate that phaseu2011sensitive demodulation techniques can extract ultrau2011lowu2011frequency, coherent signals from noisy geophysical data and may offer a complementary pathway for probing continuous GW sources in the microhertz regime.
Category: Astrophysics

Empirical and Structural Analysis of Twin Prime Sub-classifications via Modular Quadruplet Intersections

Authors: Christoper Muoki Mututu
Comments: 13 Pages. (Note by viXra Admin: For the last time, Please cite and list scientific references!)

We investigate a structural subclassification of twin prime pairs based on intersections between two modular quadruplet configurations, an admissible (2, 4, 2) prime pattern and a complimentary forbidden quadruplet pattern eliminated modulo 3. We define an overlap counting function ��(��) measuring the number of twin primes up to �� arising from such structural intersections and compare it to the total twin prime count ��(��). Computational data up to ��=3×1011 shows that the ratio ��(��)=��(��) ��(��) increases from approximately 0.4 at 103 to approximately 0.6568 at 3×1011. We prove that the structural configurations underlying the overlap occur infinitely often as arithmetic patterns and that ��(��)→∞ ���� ��→∞. We do not prove infinitude of twin primes nor do we establish a limiting value of ��(��). However, the data suggests that the overlap subclass forms a substantial and stable proportion of observed twin primes at large computational scales. This work provides an empirical decomposition of twin primes that they may compliment probabilistic models such as the Hardy-Littlewood heuristic.
Category: Number Theory

A Novel Math: L-Gebra Applied to Fermat’s Extended Last Theorem, Riemann Hypothesis & Beyond

Authors: Arthur V. Shevenyonov
Comments: 4 Pages.

An introduction to L-gebra, a promising algebraic apparatus spanning areas as diverse as calculus & number theory to name but a few, and bridging the otherwise distinct if disparate operations & operators, should suffice for a potent yet succinct treatment of Fermat’s augmented last proposition & Riemann’s hypothesis.
Category: Number Theory

Sunqm-8: {N,n} QM for the Condensed Matter Structure And/or the Bio-QM Structure (That Formed Under Either the Residue e-Force or the Residue G-Force)

Authors: Yi Cao
Comments: 23 Pages.

In the past 10 years (since 2016), I discovered that not only the Solar system, but also the whole universe (from the Virgo Super Cluster down to a quark) can be described by a brand new {N,n//6} quantum mechanics. The {N,n//q} QM structures of the Solar planets (Jupiter, Saturn, Neptune, Earth, etc.) further showed that, while the gravitational force strength decreasing (by mass decreasing), the q values also decreased from 6 to 5, 4, 3, and 2. In the current paper, I extended the {N,n//q} QM description to the condensed matter structure and the bio-structure (both of them governed by either the residue E-force, or the residue G-force). 1) I used a one-dimensional {N,n//q} QM with 1 < q < 2 to describe the condensed matter structure in a 1D-space, and then used a set of one-dimensional {N,n//q} QM (with a set of variable q(s), 1 < q < 2, and a set of anisotropic but concentric r1(s)) to describe the condensed matter structure in a 3D space. 2) Furthermore, I used the {N,n//q} QM to describe a bio-structure (e.g., a human body) with 1 < q < 2, and with a set of anisotropic and un-concentric r1(s). 3) The originally isotropic distributed force-line (from a point-centered E-force), is redistributed anisotropic (in a residue E-force), and the density is greatly increased in the direction of the residue E-force’s exertion direction (in 1D-space), and thus greatly increased the force strength in that 1D direction (out of the rθφ-3D space), and formed a strong "σ-bond" in that 1D direction. 4) The same explanation for the residue G-force explained the quasi "σ-bond" in the two arms of a galaxy, and thus contributed to the "Dark matter" effect. Thus, the matter in the two arms of a galaxy can also be treated as the condensed matter (that governed by the residue G-force) in a spiral-line 1D-space (along this galaxy arm’s 1D-space). Similarly, the multi-arm structure (that extended from any one node of the cosmic web) can also be treated as the condensed matter structure that governed by the residue G-force in a multiple (curved-line) 1D-space. 5) If the (newly defined) "primary force strength density" equals to a constant value for all three primary forces (G-, E-, and S-), then we may can estimate the size of our universe.
Category: Quantum Physics

Mathematical Analysis of a Great Rhombicuboctahedron

Authors: Harish Chandra Rajpoot
Comments: 15 Pages. 5 Figures (Note by viXra Admin: An abstract in the article is required!)

In this work, the principal geometric parameters of the great rhombicuboctahedron, an Archimedean solid, are analytically derived. This polyhedron consists of 12 congruent square faces, 8 regular hexagonal faces, and 6 congruent regular octagonal faces of equal edge length, with 72 edges and 48 vertices lying on a circumscribed spherical surface. By applying HCR’s Theory of Polygon, explicit expressions are obtained for the solid angles subtended by each square, hexagonal, and octagonal face, as well as their corresponding normal distances from the center of the great rhombicuboctahedron. The formulation further yields the dihedral angles between adjacent faces, the inscribed radius, circumscribed radius, mean radius, surface area, and volume. The derived formulas are useful for the geometric analysis, design, and modeling of uniform (convex or non-convex) polyhedra.
Category: Geometry

Mathematical Analysis of a Great Rhombicosidodecahedron

Authors: Harish Chandra Rajpoot
Comments: 16 Pages. 5 Figures (Note by viXra Admin: An abstract in the article is required!)

In this paper, the principal geometric parameters of the great rhombicosidodecahedron, the largest Archimedean solid, are analytically derived. This polyhedron consists of 30 congruent square faces, 20 regular hexagonal faces, and 12 congruent regular decagonal faces, all of equal edge length, with 180 edges and 120 vertices lying on a circumscribed sphere. By applying HCR’s Theory of Polygon, explicit expressions are obtained for the solid angles subtended by each square, hexagonal, and decagonal face, along with their corresponding normal distances from the center of the great rhombicosidodecahedron. The derived formulation further yields the dihedral angles between adjacent faces, the inscribed radius, circumscribed radius, mean radius, surface area, and volume. The resulting formulas are useful for the geometric analysis, design, and modeling of uniform polyhedra.
Category: Geometry

Deriving Rotational Quaternions Using Pivot Vectors

Authors: Russell P. Patera
Comments: 8 Pages.

Pivot Vectors are used to derive the orthonormal triad that forms the basis vectors of rotational quaternions. Pivot Vectors are also used to derive Hamilton’s rules for quaternion algebra, which forms the foundation of quaternion parameterization of attitude. The concepts of simultaneous rotations and sequential rotations are used with Hamilton’s rules to derive the quaternion composition rule for rotations. The quaternion derivation of the rotation composition rule is compared to the Pivot Vector derivation to clarify the respective attitude parameterizations.
Category: Classical Physics

On the Fixed Degree Scholz-Brauer Problem

Authors: Theophilus Agama
Comments: 8 Pages.

Denote the minimal length of a fixed degree d>1 addition chain that leads to n by l^d(n). We introduce the concept of a strong Brauer number of rank d>1 and show that all numbers belonging to this class satisfy the inequality l^d(d^n-1)1 framework.
Category: Number Theory

A Family of Analytic π-Series via Modulus-q Expansions

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

We present a new analytic framework for generating infinite series representations of pi. Using the digamma reflection formula and a modulus-q expansion strategy, we construct families of convergent pi-series. In particular, we derive a novel modulus-5 weighted series (Khan's pi-series), which converges absolutely and exhibits faster convergence than the classical Leibniz series. The approach generalizes naturally to arbitrary moduli q >= 3, yielding entire families of pi-series with adjustable convergence behaviour.
Category: General Mathematics

A Critical View of Quantum Physics

Authors: Jorma Jormakka
Comments: 26 Pages.

The article explains issues that I see as problems in quantum mechanics and quantum field theory. The first section briefly reminds of the main errors in the Relativity Theory that I noticed in earlier papers. Sections 2-5 discuss why I consider wave-particle dualism and wavefunction as incorrect concepts. Section 6 looks at some problems in QED and explains why the theory is not verified by measurements of anomalous magnetic moments and the fine structure constant. Section 7 gives a few comments about the Standard Model and addresses the question why false theories have been accepted in theoretical physics
Category: Quantum Physics

Il Dio Onnipotente (The Almight God)

Authors: Rosario D'Amico
Comments: 15 Pages. In Italian

This paper aims to reflect and investigate what is meant strictly speaking by the term "Omnipotence", that is, what it means to suppose the existence of a real Omnipotent being, who cannot therefore suffer what he does not want or, in other words, can make everything he wants without incurring any contradiction, since what is contradictory cannot be real.The approach will be strictly philosophical, eschewing commonplace drifts, devotional intents, or moral judgments in favor of conceptual clarity.Based on this analysis, we will finally cover each of the following issues:can there be an Almighty being who is also good and understandable? If so, could He abolish or sacrifice evil in the world?****Questo contributo si propone di riflettere e indagare su cosa debba intendersi stricto sensu con il termine "Onnipotenza", cioè cosa comporti supporre l’esistenza di un essere reale Onnipotente, il quale non può quindi subire ciò che non vuole o, il che è lo stesso, può realizzare qualsiasi cosa di logicamente possibile voglia, non potendo essere reale ciò che è contraddittorio.L’approccio sarà strettamente filosofico, evitando così derive banali, intenti devozionali o giudizi morali a favore della chiarezza concettuale. Sulla base di questa analisi, tratteremo infine ciascuna delle seguenti questioni: è possibile che un essere Onnipotente sia anche buono e comprensibile? Se sì, potrebbe abolire o sacrificare il male nel mondo?
Category: Religion and Spiritualism

Stepwise Emergence of Spacetime Dimensions in the Big Bang Framework

Authors: Grigol Keshelava
Comments: 19 Pages.

This study proposes a refined hypothesis on the stepwise emergence of spacetime dimensions from a zero-dimensional (0D) singularity within the Big Bang framework, integrating insights from emergent cosmology, quantum gravity, and stochastic dynamics. Building on prior theoretical developments such as loop quantum gravity and matrix theory, the model posits a sequential dimensional activation: 0D (singularity) → one-dimensional (1D; time, enabling causality) → two-dimensional (2D; light, activating electromagnetic fields) → three-dimensional (3D; space, facilitating cosmic expansion). This phenomenological approach offers a novel pathway toward reconciling quantum mechanics and general relativity, circumventing classical singularities through emergent, temperature-dependent transitions governed by effective Lagrangians and stochastic relaxation. By examining the pre-geometric 0D phase, the 1D temporal scaffold, the 2D wave-like behavior of light, and the full 3D spatial regime, the hypothesis provides a comprehensive framework for early-Universe evolution. Its implications extend to black holes, Big Crunch scenarios, and observational tests via cosmic microwave background (CMB), gravitational wave (GW), and Big Bang nucleosynthesis data, challenging unified 4D spacetime paradigms while aligning with holographic dualities and entanglement-based geometries.
Category: Relativity and Cosmology

Conceptual Design of a Near-Future Laser ICF Driver "Pulsar"

Authors: Juchi Ye
Comments: 17 Pages. License: CC BY-NC-ND

The use of Fusion as the energy source of a rocket propulsion system has great potential to significantly reduce travel times to the outer planets [3]. Research into this has been focused on more far-future and exotic fusion drives [9], while recent research efforts into near future fusion drives focus mainly on Zeta-Pinch or direct fusion [10-12]. However, Inertial-confinement fusion (ICF) could also be a viable option [3]. This paper is a proposal of a conceptual fusion drive - "Pulsar". Studies include the selection of fuel options (D-T and D-3He), concept, thermal-mechanical properties, theoretical performance, and applications compared to other options. A combination of Deuterium-Tritium (D-T) fuel, reflected fiber pulse laser system, spin-aligned frozen fusion pellets, and a Tantalum-Hafnium Carbide (TaHfC) heat shield shows great promise for a viable near-future ICF propulsion system.
Category: Nuclear and Atomic Physics

General Relativity is the Global Solution

Authors: Christopher R. Brown
Comments: 5 Pages.

General relativity is a constrained theory built upon a handful of postulates including general covariance, diffeomorphism invariance, and the relational character of spacetime. These postulates are not optional features. They are the structural commitments that give general relativity physical meaning. When these commitments are enforced exactly, without supplementary assumptions, the theory selects a unique static geometry governed by a single measured scale. This geometry reproduces the linear Hubble law, the nonlinear curvature of the Type~Ia supernova distance modulus, and the CMB monopole temperature, without a cosmological constant and without dark energy. The geometry predicts zero redshift drift. Any expanding model predicts nonzero drift. The ELT-ANDES measurement is a clean, binary test.
Category: Relativity and Cosmology

Emergent Gravitation and Quantum Wave Dynamics from a Bounded Vacuum

Authors: Tanuj Kumar, Vandana [Doe]
Comments: 23 Pages.

We present a framework in which gravitation, inertia, and wave dynamics emerge from the response of avacuum endowed with finite potential capacity. The theory is formulated in terms of a scalar vacuumpotentialfield whose absolute normalization is fixed by relativistic considerations, such that the equilibrium value at infinity equals c. Static relaxation of localized vacuum-potential deficits reproduces Newtonian gravity in the coarse-grained limit, while time-dependent redistribution generates propagating disturbances governed by a universal wave equation. Finite vacuum capacity implies intrinsic upper bounds on transmissible force and signal speed, yielding F_max = C^4/G and v ≤ c without invoking spacetime geometry or independent kinematic postulates. Vacuum microstructure further leads to a universal lattice dispersion relation with Planck-suppressed corrections, consistent with current astrophysical and gravitational-wave constraints. Gravitational redshift, lensing, horizons, and quantum correlations arise as energetic consequences of bounded vacuum response. The vacuum is modelled as a Dynamical Planck Network (DPN), providing a conservative and internally consistent bridge between relativistic gravitation and quantum wave phenomena.
Category: Quantum Gravity and String Theory

Proof of Beal Conjecture by Reductio ad Fermat

Authors: Francesco Aquilante
Comments: 6 Pages.

Beal Conjecture, which asserts that for $a^k + b^m = c^n$ with $k, m, n > 2$, the bases $a, b,$ and $c$ must share a common prime factor. We prove it to be true with an approach that utilizes a sequence of rational perturbations $delta={delta_i}:{delta}_{i in mathbb{N}} subset mathbb{Q}$ , $delta_i > 0$ and $lim_{i to infty} delta_i = 0$ to treat such Diophantine equation as the critical limit-state of a geometrically constrained configuration. By defining a sequence of non-degenerate triangles $mathcal{T}_delta$ with rational side lengths ${a^k, b^m, c^n - delta_i}$, we establish a continuous mapping to the moduli stack of elliptic curves $mathcal{M}_{1,1}$.We demonstrate that the requirement for {rationality of the configuration} (the existence of a rational altitude $h_delta$) induces a sequence of Frey-Hellegouarch curves $E_delta$ that converge algebraically to the limit-state $E_{Beal}$. For signatures where $min(k,m,n) geq 3$, we invoke Ribet’s Level-Lowering Theorem to show that the associated Galois representation $ho_{E,n}$ is necessitated to reside within the {em empty space} of weight-2 cuspidal modular forms $S_2(Gamma_0(2))$.Simply speaking, our proof follows the often anticipated path of reasoning by which if Beal Conjecture were trueit must ultimately stand on the foundationthat underpins the validity of Fermat's Last Theorem.Furthermore, we provide a formal textit{Parity Lemma} to delineate the bifurcation at $n=2$, explaining why the modular sieve permits coprime solutions in Fermat-Catalan and Pythagorean signatures. This topological and arithmetic framework confirms that for strictly hyperbolic signatures, a solution exists if and only if $gcd(a, b, c) > 1$.
Category: Number Theory

Dimensionality as Regulation of the Ultraviolet-Infrared Mixing

Authors: Alaya Kouki
Comments: 47 Pages.

From Restraint Relativity it is possible to consider a corpuscle as a packet of strings. The variation of the length of this packet is equal to the phase speed of the corpuscle as a packet of waves times an universal constant. In a system of units whereħ=c=a=1 the string vector becomes equal to the wave vector.From quantum mechanics we had deduce that a Planck oscillator can emit or absorb power only by quanta of power multiple integer of hν^2. This allows us to divide space-time in modular cells of action, momentum, energyu2026etc in the phase space geometry of the oscillator and resolve the problem of vacuum energy density. Planck system of units serves only for black holes topology. Space-time is a four open dimensions. More than four the dimension should be curled. Resolving gravitation field by computer machine with model using modular cells will be without singularities . Einstein equations of gravitational field are available only in a quasi static asymptotically flat Universe. Constants G & Λ of General Relativity are proportional to the inverse square of the Universe radius. We propose a new model for the hydrogen atom based on vacuum energy. Vacuum energy its self depends on the dimensionality of space. For weak gravitational field the Infrared cut off depends on real particle (the electron) and for strong gravitational field the Ultraviolet cut off depends on fictive particle ( Planck atom). The holographic principle depends largely on the dimensionality of the space. Also the principle of uncertainty depends in which gravitational field considered weak or strong: we introduce for this Planck fine structure, Planck charge, Planck electric permittivity, Planck electric impedenceu2026etc.
Category: Quantum Gravity and String Theory

Relativistic Action at a Distance Formulation of the Strong Interaction

Authors: Juan R. González-Álvarez
Comments: 12 Pages.

We demonstrate that the strong interaction can also be formulated as an action-at-a-distance theory. Starting from the nonlinear gluon wave equation, we derive an exact momentum-space propagator for the gluon with a dynamical mass determined by the color current. To make its momentum dependence analytically manageable, we also obtain an exact discrete spectral representation of the propagator, whose lower-order truncations reproduce the Refined and Very Refined Gribov-Zwanziger propagators used in confinement studies, but now with a dynamical origin for their mass scales. Ordinary Yang-Mills theory and the free gluon propagator are recovered in the ultraviolet regime, where confinement becomes irrelevant.
Category: Nuclear and Atomic Physics

A Pair of Straight Lines and the Condition to Meet Each Other or to Stay Parallel

Authors: Binay Krishna Maity
Comments: 3 Pages.

We have two straight lines in a graph. We need to determine if these straight lines will meet each other or remain parallel if we extend these lines. This paper helps us to determine this question. If we square a certain type of polynomial (x^pn - x^p(n-1) -....... -x^p1-1) and take the coefficient of x of this square on the x axis and the power of x on the y axis and if we make the graph, the spectra that will be created, consider the initial and final part of the spectra as two straight lines, then those two straight lines will meet each other or be parallel, it will depend on the n and p of this polynomial. That is, on the length and step of the polynomial.
Category: General Mathematics

Why Was that Difficult to Prove the Goldbach Conjecture?

Authors: Óscar E. Chamizo Sánchez
Comments: 6 Pages.

An ancient conjecture, named after its discoverer as Goldbach conjecture [1][2], that is to say, every even number greater than 2 can be represented by the sum of two primes, is a simple and intractable statement that has been torturing mathematicians for more than 250 years. We wonder if the divide et impera method, so useful in programming and algorithmics, could provide some service here. The goal is simplify and separate the whole problem into three independent and fairly manegeable subproblems. An approach that, as far as I know, has not been tested before
Category: Number Theory

On the Origin of the Entropy of Black Holes

Authors: Farid Abrari
Comments: 6 Pages.

Using the c-SRQM framework, a quantum description of the black hole singularity was introduced previously. According to the theory, a black hole singularity is made of a series of concentric spherical shells of a constant diametral step size A. Each shell was shown to be occupied with a fixed set of Primordial Stem Particles (PSP) of rest mass m_bar = h/Ac. The PSP appears to be defining a regime where the point-like and field-like particle physics meet. The innermost shell with diameter A, if it were to be found standing alone by itself, would be the smallest possible black hole in nature with mass of M_1 = Ac^2/4G, named the Unit Black Hole (UBH). In this article we have shown that the Beckenstein-Hawking entropy of black holes can be traced back to the available microstate arrangements of the PSPs on the singularity shells. The Page time and total evaporation time obtained from this theory are shown to be consistent with the existing theoretical predictions, a non-trivial hint to the consistency of the proposed theory.
Category: Quantum Physics

Diamane Fibonacci Nanotubules for Quantum Computing

Authors: Hans Hermann Otto
Comments: 7 Pages.

We propose 13-protofilament Fibonacci nanotubules of diamane for quantum computing. Generated color centers acting as qubits may have coherence times greater than a minute and a processing temperature much higher than 1 K enabling the construction of energy-efficient computers. The proposal paws the way to develop purely organic hollow nano-fibers for quantum computing operating around room temperature, where qubits are better protected against noise decoherence. However, 2D diamond structures are superior with regard to thermal stability and will therefore retain lasting importance in computational and sensor technology.
Category: Condensed Matter

Khasi Words and Similar Words from Languages Around the Globe

Authors: Anindya Kumar Biswas
Comments: 17 Pages.

In this article, we describe words similar, in spelling, from languages around the globe to a set of words of the Khasi language. The languages are Malay, Indonesian, Vietnamese, Japanese, Turkish from Asia; Hawaiian from the pacific;Hausa, Bemba from Africa; Portugese( from Portugal, from Brazil), Latin, Early English, Icelandic, Danish, Swedish andNorwegian. We use the dictionaries of the languages written in Roman script.
Category: Linguistics

Proof of Goldbach’s Conjecture and the Twin Prime Conjecture

Authors: Ryujin Choi
Comments: 3 Pages. (Note by viXra Admin: Please cite and list scientific references and submit article written with AI assistance to ai.viXra.org)

We study the distribution of integers obtained by removing fixed residue classes moduloprimes. Using an explicit upper-bound sieve argument, we show that admissible integers cannotoccupy arbitrarily long contiguous intervals. In the case of two arithmetic progressions, thisleads to the existence of simultaneous prime values. As a consequence, Goldbach’s conjectureand the twin prime conjecture follow.
Category: Number Theory

Construction Of Pythagorean Triple

Authors: Binay Krishna Maity
Comments: 4 Pages. (Note by viXra Admin: Please cite and list scientific references!)

A very common term in mathematics is a^2 + b^2 = c^2. This equation has been discussed since many years ago. This equation is called the Pythagorean Theorem. And for a, b and c as positive inintegers a, b and c are called Pythagorean triples. For example, 3, 4, 5 is a Pythagorean triple. Because, 3^2+ 4^2 = 5^2. Some more examples are (5,12,13), (9,12,15), and (12,16,20) etc. Any number has more than one Pythagorean triple. For example, with 12 numbers (5,12,13), (9,12,15), (12,16,20) and (12, 35, 37) these four Pythagorean triples are obtained. Now I am given a positive integer number and I have to show how many Pythagorean triples can be found with that number? There may have some solutions in the mathematics. This paper provides an alternative solution to evaluate these Pythagorean triplets for any given number.
Category: General Mathematics

Solving Ultimate Nth-Degree Algebraic Equations: Evading the Abel-Ruffini Theorem?

Authors: Arthur Shevenyonov
Comments: 6 Pages.

The present paper proposes a most parsimonious scheme to arrive at solutions for polynomial algebraic (or ODE) equations of an arbitrary degree (order). The former option treats the polynomial as an implied characteristic of a difference/recurrent equation (dubbed an AlD, or algebraic-to-difference path). The latter (i.e. ODE) domain, rather than building on the more trivial differential-difference parallelism, embarks on first reducing ODE to algebraic (while making use of, say, Mikusinski-style E-operators) then applying the above procedure (what amounts to a DARF/DARE, or differential to algebraic to recurrent/functional equation path).
Category: General Mathematics

Para-complex Numbers and Their Applications in Hyperbolic Systems From Einstein’s Relativity to Supersonic Fluid Dynamics

Authors: Ahcene Ait Saadi
Comments: 4 Pages. (Note by viXra Admin: Please cite listed scientific references!)

This paper explores the properties of para-complex numbers ({R}[j], j^2=1, j ) as the fundamental framework for hyperbolic physical systems. By introducing the zero-divisor identity (1-j)(1+j)=0, we demonstrate a major simplification of Lorentz transformations and a formal unification between the relativistic light cone and the Mach cone in fluid mechanics. This approach aligns with recent research archived on viXra.org, aiming to renew the study of applied hyperbolic algebra.
Category: Algebra

Approximation of pi

Authors: Mieczyslaw Szyszkowicz
Comments: 5 Pages.

Archimedes used the perimeter of inscribed and circumscribed regular polygons to obtain lower and upper bounds of the number pi. He started with two regular hexagons and he doubled their sides from 6 to 12, 24, 48, until 96. Applying the perimeters of 96 side regular polygons, Archimedes obtained the bounds for the number pi: 3+10/71<pi<3+1/7. His algorithm can be executed as a recurrence formula called the Borchardt-Pfaff-Schwab method. Dörrie proposed an improvement of this algorithm to produce narrower interval which encapsulates pi. Here a linear combination of the bounds is realized to obtain an improved accuracy. Many other linear combinations are presented to approximate this mathematical constant.
Category: General Mathematics

Theory of Complex Momentum in Phase Spacetime

Authors: Jarosław Grabiec
Comments: 32 Pages. In Polish (Note by viXra Admin: Please cite listed scientific references)

This paper introduces a macro-scale complex form of mechanical momentum, revealing its dual nature and emphasizing its vector phase of motion ��(��) in spacetime. It offers practical diagnostic tools and clearly describes the dynamical relationship between rotational and translational motion. It proposes a proven mathematical framework using complex numbers, used by engineers in electrical engineering, which can also be applied in orbital mechanics, simplifying calculations. A non-standard approach in a non-inertial frame explains, using an example, the astrophysical energy mechanism seen from within the Coriolis effect. It focuses the reader's attention on energy transfer in the energy channel of the Coriolis effect as a looped (implicit) coupling mechanism between the active and reactive energy channels.
Category: Astrophysics

The 3.998D Manifold Framework: Assessment of Geometric Unification and the Resolution of Galactic Rotation Anomalies

Authors: Charles Opoku
Comments: 16 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This paper presents the results of the application of the 3.998D Manifold Framework, a geometric theory that posits all physical phenomena emerge from a scalar field (ϕ-Field) within a space of spectral dimension d_s=3.998 and a dimensional deficit of δ=0.002. Unlike standard cosmological models that require Dark Matter and Dark Energy to resolve dynamic anomalies, this framework introduces a single density-dependent mechanism governed by a manifold stiffness constant (β_eff ≈ 5.01). The proposed Framework further demonstrate that this constant is the geometric reciprocal of a 13.4% metric compaction, providing a structural bridge that unifies subatomic bond tension with galactic dynamics. Analysis shows that this geometric constraint successfully unifies physics across subatomic, galactic, and cosmic scales. Statistical modelling of galactic rotation curves for the Milky Way, Andromeda (M31), Triangulum (M33), UGC 128, and NGC 2403, demonstrates that the framework’s saturation mechanism replaces the need for invisible mass, achieving statistically significant improvements over Newtonian predictions while recovering standard gravity in high-density solar systems.
Category: Relativity and Cosmology

Max Bank: a Saturating Magnetic Marx Generator

Authors: Max Artusy
Comments: 2 Pages.

A unique method is described to make the Marx circuit usable with saturating magneticswitching. This also enables high repetition rate operation. The properties of near indestructibleswitching, combined with self grading voltage distribution, offer unique scalable high voltage pulser design. Additionally the leading edge is pulse compressed.
Category: High Energy Particle Physics

Integrating Environmental Sensors and Real-Time Feedback for Enhanced Study Productivity

Authors: Shanmuka Gottimukkala
Comments: 4 Pages.

High school students often experience reduced productivity during study sessions due to environmental distractions (noise, poor lighting, temperature) and unmanaged emotional states. Commercial affective computing systems are typically expensive and inaccessible. This study develops and evaluates a low-cost Raspberry Pi-based IoT desk assistant that monitors ambient conditions and delivers real-time visual feedback to enhance emotional awareness and focus.
Category: Digital Signal Processing

Unified Substrate Hydrodynamics: A Deterministic Framework for Spacetime Theory

Authors: Yogan Rameg
Comments: 5 Pages. (Note by viXra Admin: Please cite listed scientific references and submit article written with AI assistance to ai.viXra.org) Copyright © 2026 by the author(s), all rights reserved.

I present Spacetime Theory, a unified framework that reinterprets the vacuum not as a staticgeometric manifold, but as a dynamic Superfluid Active Nematic Lattice. By adopting a"Spacetime-First" approach, we demonstrate that General Relativity and Quantum Mechanicsemerge as macroscopic and microscopic limits of substrate hydrodynamics. Gravity is derivedas the convective acceleration of a superfluid inflow, while matter is identified as localized,phase-locked standing waves ( -resonances). We provide a first-principles derivation of thegravitational constant , the speed of light , and a mechanical solution to the Michelson-Morley null result through wave-structure deformation. This deterministic model replaces probabilistic "dice-throwing" with geometric resonance within a lattice under 10^35 Pa of hydrostatic pressure.
Category: Relativity and Cosmology

Foundations for Gravitational Potential Energy Evaluation

Authors: Nicolas Poupart
Comments: 14 Pages.

We investigate the evaluation of the effective gravitational binding energy in heterogeneous stellar systems by comparing a permutation-invariant estimator with heuristic formulations based on sequential aggregation. While the exact method provides a physically consistent reference independent of population ordering, the heuristic estimator introduces a path dependence that strongly influences the reconstructed dark-to-baryonic mass ratios.A systematic exploration of permutations quantifies this sensitivity and shows that an optimized ordering substantially reduces the systematic bias, producing an error distribution that is approximately Gaussian and centered near zero. When applied to galactic rotation data, the heuristic approach achieves significantly higher resolution rates than the exact estimator.These results highlight a fundamental trade-off between physical invariance and algorithmic sensitivity, and suggest that the trajectory through population space is an intrinsic component of sequential gravitational energy models.
Category: Astrophysics

The Röntgen Interaction and the ‘Resolution of the Abraham—Minkowski Dilemma’

Authors: Andrew T. Hamilton
Comments: 3 Pages.

The Röntgen interaction between a point electric dipole and the electromagnetic field does not carry over to a continuous medium that is polarized. Consequently, the kinetic momentum density of light in a medium is not, as Barnett contends in his ‘Resolution of the Abraham—Minkowski Dilemma’, the Abraham form (although the canonical momentum density is the Minkowski form). As a result, the resolution is flawed, albeit not fatally.
Category: Classical Physics

On the Inconsistency of the Standard Model in Exactly Four Dimensions

Authors: Ervin Goldfain
Comments: 22 Pages.

The triviality of the Higgs sector in exactly four spacetime dimensions is a serious obstruction to the ultraviolet completion of the Standard Model for Particle Physics (SM). In contrast with the Asymptotic Safety program of Quantum Gravity, this brief report argues that minimal spacetime fractality above the SM scale, a) is a natural extension of Renormalization Group dynamics and b), it can successfully explain away the triviality problem.
Category: High Energy Particle Physics

On the Irrationality of Some Notable Constants

Authors: Francesco Aquilante
Comments: 7 Pages.

We present the quantum Riemann sum ($Q$-sum) operator framework and use it to prove the irrationality of the Riemann-$zeta$ function at odd integers, the Dirichlet-$beta$ function atat all positive integers $n geq 2$, as well as that of the Euler-Mascheroni constant ($gamma$). By establishing a recursive functional hierarchy, we circumvent the classical ``parity barrier'' that has traditionally isolated even and odd zeta-type constants. We utilize the $p$-adic Newton Polygon to demonstrate that the arithmetic complexity of the operator kernel is an invariant of the functional hierarchy. Therefore, the irrationality of the transcendental anchors $zeta(2)$ and $beta(1)$ necessitates the irrationality of the entire chain. This line of reasoning can be extended to incorporate $gamma$, thereby substantiating its long-held irrationality.
Category: Number Theory

Structure Formation by Application of the Theory of Variable Speed of Light

Authors: Markus Schönlinner
Comments: 9 Pages.

One of the greatest challenges of the Standard Model is the question of the structure formation in the early universe. From the currently widely accepted way of temporal evolution ofthe scale factor follows, that the structure formation by gravitational forces seems not possible without additional assumptions and only with the help of Dark Matter. The theory of variable speed of light, however, suggests a linear progression of the scale factor. This corresponds to the model of the "empty universe". A linear evolution of the scale factor is compatible with the observations. It is investigated, which consequences a linear evolution of the scale factor has on the structure formation and this is compared to the classical derivation.
Category: Relativity and Cosmology

[Corpuscular-Simple Theory Gravity-Antigravity:] Non-reactive Accelerated Movement

Authors: V. F. Chibisov, I.V. Chibisov
Comments: 11 Pages. 22 figures.

This article is devoted to the presentation of the consequences of the models of the Corpuscular-Simple Theory (CST) [1], revealing the phenomena of gravity, antigravity, and non-reactive accelerated movement.
Category: Quantum Gravity and String Theory

Letter: Spatial Space as a Time-Averaged Quantum Clock Caused by an Orthogonal State-System — an Alternative Explanation to LQG and CDT

Authors: Holger Döring
Comments: 4 Pages.

A model is proposed in which space itself is not a fixed background object, but arises from ultrafast rotating of orthonormal quantum states. The fundamental frequency is (Planck time). Observable spatial and temporal quantities emerge as averages over many Planck cycles. The model provides natural explanations for: Lorentz invariance as effective symmetry, gravitation as a statistical mean of rotational dynamics and cosmic fluctuations as frozen rotational phases. Furthermore, experimentally accessible signatures are predicted, including transverse correlations in the holometer, time noise in atomic clocks, and quadratic Planck corrections for the speed of light.
Category: Quantum Gravity and String Theory

The Essence of Relativity: The Theory of Composite Motion under Impulse Duality

Authors: Petr Kunčík
Comments: 5 Pages. (Note by viXra Admin: Please cite and list scientific references)

This work presents an alternative interpretation of kinematic phenomena currently attributed to time dilation and spacetime curvature. Based on mechanical experiments involving impulse duality (Primary and Secondary), it demonstrates that the addition of the speed of light to the speed of its source in absolute space is not only logically possible but essential for the correct interpretation of physical reality.
Category: Relativity and Cosmology

The Proton as a Phase Resonator: Quarter-Wave Origin of the Compton Wavelength

Authors: Najib Chairi
Comments: 7 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

In a previous work [1], we highlighted a kinematic relation linking the measured charge radius of the proton Rp to a frequency f, whose numerical value coincides remarkably well with the Compton frequency of the proton: f=c/{(pi/2)Rp} This relation can also be rearranged into a compact geometric form: the non-reduced Compton wavelength of the proton is equal to one quarter of the equatorial perimeter associated with Rp, namely lambda_c,p = (pi/2)Rp. The present article continues this analysis by proposing a wave interpretation of the appearance of the factor pi/2. Relying on standard results from wave physics (standing waves, mixed boundary conditions, resonators), we show that in a wave-based reading this factor is neither arbitrary nor fitted, but constitutes the universal signature of a fundamental quarter-wave mode associated with a central phase constraint. Within this framework, the nucleus is no longer interpreted as a rigid boundary, but as a phase tuner. The same structure is immediately extended to the neutron by simple substitution, providing a coherent length scale associated with its Compton frequency.
Category: Nuclear and Atomic Physics

The Hubble Tension

Authors: Yvan Raverdy
Comments: 3 Pages.

This title expresses an incompatibility, within the standard model of cosmology, between measurements of the Hubble constant from the CMB and those from the Red Shift. We had previously proposed a new concept called the" Intrinsic Space" [1]; a fictitious Universe whose energy is only one of the void, which exclude masses and radiations. In this article, we compare the evolution over time of the radius of the Universe, as analyzed in [2], between the real Space and this fictitious Space to calculate the Hubble constant in both cases. Thus we propose a solution to the problem .
Category: Relativity and Cosmology

From the Kinematics of Elliptical Motion to the Law of Gravitation

Authors: Viktor Strohm
Comments: 4 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)

The law of force governing the motion of a material point along an elliptical orbit is derived using a purely kinematic approach. Starting from the differential equations of motion in a Cartesian coordinate system, an angular equation of motion is obtained that directly leads to Kepler’s second law. It is shown that the acceleration is directed toward the focus of the ellipse and is inversely proportional to the square of the distance. The derived expressions are applied to the Earth—Moon system. Forces calculated using Newton’s second law are compared with those given by Newton’s law of universal gravitation. The relative discrepancy does not exceed 1.1%, confirming their equivalence.
Category: Mathematical Physics

Idempotents in Rings with Unity

Authors: Lamarr Widmer
Comments: 14 Pages. (Note by viXra Admin: Please cite and list scientific references)

If * is a binary operation on a set S, an element a is an idempotent for * if a*a=a . In this paper, we provide an alternative equivalent definition for idempotents in a ring with unity. This definition facilitates the calculations in several theorems characterizing the idempotents in rings of the form Z_n . This material was developed by the author while teaching a one semester undergraduate class in modern algebra. Some of this material was presented to students in that class and we believe all of it is suitable for students after a basic introduction to rings. We include numerous theorems determining the number of idempotents in Z_n for various factorizations of n . These theorems, along with specific examples and calculations, lead to the eventual general theorem which shows how the number of idempotents in Z_n is determined by the number of prime divisors of n.
Category: Algebra

Unified Resolution of Cosmological Tensions from Phase Accessibility Dynamics

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

Recent observations from DESI and other surveys reveal two seemingly independent cosmologicaltensions. Dark energy exhibits late-time deviation from a cosmological constant (w0 = −0.70±0.09),while structure growth shows low-redshift suppression relative to CMB-inferred expectations.We show that both effects can be consistently interpreted within a single phenomenological framework based on loss of phase accessibility in an expanding universe. From an effective late-timedescription, we derive specific, testable functional forms for w(z) and σ8(z), governed by a singleeffective coupling parameter γ0 ≃ 0.16—0.20, achieving quantitative agreement with current data.The framework predicts a scaling correlation between dark energy evolution and structure growthsuppression that is not reproduced by ΛCDM or standard extensions. These results suggest thatcosmic acceleration and late-time structure formation may be understood as coupled manifestationsof effective phase dynamics, rather than independent phenomena requiring separate explanations.
Category: Astrophysics

An Infinite Sum in Vol.1 "Integrals and Series-Elementary Functions" (1986) by Prudnikov, Brychkov and Marychev

Authors: Edgar Valdebenito
Comments: 9 Pages.

In this note, we study the sum S=(1/2)+(1/10)+(1/30)+(1/68)+...
Category: Number Theory

Cyclic Cosmology with Informational Inheritance (CCHI): A Framework for Perturbative Memory Across Quantum Bounces

Authors: Samuel Krieger Bonini
Comments: 15 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This paper introduces the Cyclic Cosmology with Informational Inheritance (CCHI), a theoretical framework proposing that the universe undergoes repeated cycles of expansion and contraction mediated by a quantum bounce, during which statistical information about the pre-bounce matter distribution survives into subsequent cycles, encoded in the post-bounce perturbation spectrum. Built upon Loop Quantum Cosmology (LQC), the framework presents three original contributions: (i) a master equation governing the cyclic evolution of the primordial power spectrum via Bogoliubov transfer coefficients, which admits a stabilized fixed point whose resonant peaks nucleate Primordial Black Holes (PBHs); (ii) a derivation of Entropic Time Dynamics (ETD) from the effective LQC metric, identifying the chrono-entropic coupling Γ(ρ) = 1 − ρ/ρ_c = g_00^(eff) with no free parameters; and (iii) a unified mechanism connecting gravitational baryogenesis, PBH dark matter in the asteroidal mass window, and the seeding of supermassive black holes observed by JWST at z > 10. The framework yields falsifiable predictions for CMB-S4, LiteBIRD, the Roman Space Telescope, and LVK O4/O5, and addresses the cyclic entropy problem through the distinction between thermodynamic entropy (diluted each cycle) and entanglement entropy (transferred as low-entropy correlations).
Category: Relativity and Cosmology

Preliminary Investigation into the Construction of a 10-Dimensional Discrete Spacetime Basis Based on Ramanujan Modular Formulas

Authors: Shengliang Dong
Comments: 15 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

Aiming at the "continuous-discrete" contradiction between general relativity and quantum theory at the Planck scale, as well as the defects of existing unified theories, this paper takes Ramanujan modular formulas as the mathematical core to construct a 10-dimensional discrete spacetime basis with a fractal fiber topological network. The basis is formed by the topological tensor product coupling of 3-simplices, elliptically deformed Calabi-Yau-like manifolds and fractal fibers, and the spacetime evolution is regulated by modular parameters. This study clarifies the intrinsic origin mechanism of time, initially constructs a correlation framework for 60 typical topological real particles, provides topological interpretations for the Higgs signal and the origin of photons, proposes a cyclic universe model, and predicts candidate particles for dark matter. This research offers a new research path for the unification of quantum and classical theories, and also provides a basic framework for the interdisciplinary research of high-dimensional spacetime and number-theoretic geometry.
Category: High Energy Particle Physics

Hydrodynamic Model of Gravity as a Secondary Effect of Interaction of Electromagnetic Dipoles and the Hypothesis of Superfluid Space-Time.

Authors: Pavlo Kandyba
Comments: 13 Pages. (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

A hypothesis based on expriment is proposed according to which the fabric of space-time isa superfluid subatomic medium in which matter exists as a process of vortex collapse ofcaverns (cavitation). Vortices are considered as thermodynamic phenomena associated withthermal motion and release of free energy of the medium, which is confirmed experimentallyin agreement with E = mc ^ 2, and are quantized due to superfluidity. Gravity is interpretedas a secondary effect of interaction of EM dipoles, which is supported by the hydrodynamicmodel.
Category: Quantum Gravity and String Theory

Evidence for Planet Nine with a 3,600-Year Period and 1 AU Perihelion as the Cause of Recurring Apocalypses

Authors: Pavlo Kandyba
Comments: 12 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org) https://doi.org/10.5281/zenodo.18524556

A model of Planet Nine as a massive body (3—4 M⊕) in a highly eccentric retrograde orbit (e ≈ 0.994—0.996, q ≈ 1 AU,Q ≈ 400—500 AU, T ≈ 3600 yr), presumably captured from interstellar space. The model relies on the clustering of ETNOorbits by dynamic traps and chaotic diffusion, the instability of their configurations, and the millennial scale. It also shows quasi-periodicity in climatic and cultural-historical markers (the Younger Dryas event, the Piora oscillation) consistent with a periodof T ≈ 3600 yr. A historically confirmed possible observation of Planet Nine’s perihelion passage of 1 AU has been established,supported by the exploratory LLM reconstruction of the comet’s orbit observed in CE 60 for 180 days and documented in Chinesechronicles.
Category: Astrophysics

Idiostasis, The Mechanism of Identity

Authors: Chance Moore
Comments: 28 Pages.

Despite radical shifts in medical, technological, and scientific capability, the phenomenon of consciousness has remained stubbornly elusive, even a hundred years after we mapped Time and Space. While many theories have been proposed, the barrier between the physical and the mental remains unreconciled. In Idiostasis, I propose both an identity model and an ontologically mathematical framework as a mechanistic solution to the ‘hard problem of consciousness,’ by considering the Universe as its own Observer, through a process of continually shifting perspective through orders of complexity. By this method, I propose a singular contextualization of the currently accepted mathematics of quantum physics through quantized self-measurement, conceptually reproducing both particles and the strong force, mathematical constants, and quantum activity, as well as demonstrating inherent mechanisms for quantum entanglement, quantized gravity, dark energy, dark matter, and how both the Riemann �� (Zeta) Function and Euler’s Identity may have foundational applications.
Category: Quantum Gravity and String Theory

Singularities in 3D: the Mathematical Challenge of Euler and Navier-Stokes

Authors: Diego Cordoba Gazolaz, Luis Martinez Zoroa, Abdelmajid Ben Hadj Salem
Comments: 19 Pages. In French, the article was translated from spanish.

The Euler and Navier-Stokes equations describe the motion of fluids, but it is still unknown whether their smooth solutions (with $C^{infty}$ regularity) can develop singularities in finite time.This enigma, one of the «Millennium Prize Problems», poses fundamental mathematical and physical challenges. In this article, we review what singularities are, the main mechanisms proposed for their formation (selfsimilar solutions and vorticity cascades) as well as recent advances that shed new light on this problem in three dimensions.
Category: Mathematical Physics

Efficient PII Removal with Zero-Shot NER and Dual Encoders

Authors: Rutvik Acharya, Nitin Agarwal
Comments: 5 Pages.

Personally Identifiable Information (PII) removal is a critical task in data privacy and security, requiring the identification and redaction of sensitive entities such as names, addresses, and social security numbers from unstructured text. Traditional Named Entity Recognition (NER) models used for PII removal are limited to predefined entity types, necessitating retraining for each new PII category. This paper presents zero-shot NER architectures that enable the efficient removal of any type of PII without extensive retraining.We leverage two advanced architectures for zero-shot NER in the context of PII removal: bi-encoder and poly-encoder models. The bi-encoder architecture separates the encoding of input text and PII entity types into distinct transformer models, allowing for efficient and scalable processing. PII entity type encodings can be pre-computed and reused across different input texts, reducing computational overhead. The poly-encoder architecture enhances the bi-encoder approach by incorporating a post-fusion step to model interactions between input text and PII entity representations explicitly, addressing the lack of inter-entity understanding in standalone bi-encoder models.To evaluate the effectiveness of these architectures for PII removal, we conduct experiments using a diverse, high-quality dataset containing various types of PII. We compare the performance of our proposed models with existing zero-shot NER approaches, such as GLiNER, in terms of precision, recall, and F1 score. The results demonstrate that our bi-encoder model outperforms GLiNER in identifying and removing PII entities, setting a new benchmark for zero-shot NER in the context of data privacy and security.These architectures offer several advantages for PII removal, including the ability to recognize an unlimited number of PII entities simultaneously, faster inference with preprocessed PII entity embeddings, and better generalization to unseen PII categories. These advancements enable the development of efficient and scalable PII removal systems capable of handling diverse and evolving PII requirements, ensuring compliance with data privacy regulations and protecting sensitive information.In this paper, we present an adaptive approach to PII detection that dynamically selects between GLINER and Presidio models based on contextual analysis. Our methodology first analyzes input text for regional markers, script patterns, and format variations to determine the most suitable model for PII detection. GLINER is prioritized for Western contexts and standardized formats, while Presidio handles region-specific and non-standard patterns. This context-aware selection is complemented by a robust validation framework that includes both primary and secondary validation layers, confidence scoring, and enhanced processing for ambiguous cases. Experimental results demonstrate an 12%-14% improvement in overall accuracy compared to single-model approaches, with particularly strong performance in handling diverse regional formats and multi-script environments, while maintaining acceptable processing overhead.
Category: Artificial Intelligence

Structural Analysis of the Generalized Collatz Tree (k-Tree)

Authors: Ammar Hamdous
Comments: 18 Pages. Creative Commons Attribution 4.0 International

In earlier work [1], we introduced a refined and more structurally representative Collatz tree, within which we identified a singularity. A subsequent preprint [2] established a methodological generalization of the Collatz sequences that preserves this singularity by extending it to a generalized singularity. In the present paper, we investigate the structure of the generalized Collatz tree—referred to as the k-Tree—arising from this transformation. Our analysis focuses on the ordering, propagation, and interaction of branch beginnings across ranks, with particular attention to the structural sets Bk and Ak. This study aims to elucidate the internal architecture of the generalized tree and to clarify the extent to which the geometric and dynamical features of the classical Collatz tree persist under the generalization.
Category: Number Theory

Unified Proof of Sendov’s Conjecture via the Potential Stability Paradigm: Topological Rigidity and Field Monotonicity

Authors: Francisco Petitti
Comments: 43 Pages.

Sendov’s Conjecture (1962) asserts that if all zeros {z1, ..., zn} of a polynomial P(z) of degree n >= 2 lie within the closed unit disk D, then each zero zk is at a distance no greater than 1 from at least one critical point of P(z). While recent advancements have confirmed the conjecture for sufficiently large n, a unified proof for intermediate degrees has remained an open challenge due to the complex discrete interactions of root configurations.This paper provides a complete resolution of Sendov’s Conjecture for all n >= 2 by reformulating the problem within the framework of Logarithmic Potential Dynamics. We treat the critical points as equilibrium positions in a complex force field generated by the zeros of the polynomial. By isolating a fixed root z1 = a in [0, 1], the total force F(z) = P'(z)/P(z) is decomposed into a Local Attraction Force and a Collective Cloud Repulsion Force.By employing a Laurent series expansion of the cloud force, we derive a universal upper bound for the remainder Rn(z) based on the majorization of the spectral moments (power sums) of the roots constrained within D. We prove that the repulsion magnitude is strictly dominated by the geometric decay of the local force at the boundary |z - a| = 1.0. Furthermore, by analyzing the Hessian of the potential, we establish the strict monotonicity of the radial force field in the region r >= 1.0, thereby precluding the existence of equilibrium points beyond the unit radius of any given zero. This analytical framework bridges the gap between asymptotic analysis and discrete root geometry, confirming the conjecture for the general case.
Category: Functions and Analysis

Foundations of Mathematics and Quantum Theory

Authors: Felix M Lev
Comments: 17 Pages.

As shown by Gödel and other mathematicians, foundational problems of classical mathematics (CM) arise because this theory involves the entire infinite set of natural numbers. Therefore, CM must be modified in some way. A problem discussed in a wide literature is how mathematics should be treated: (1) as a purely abstract discipline, independent of nature; or (2) as a discipline that must ultimately describe nature. Most physicists accept only viewpoint (2), while many mathematicians and philosophers adopt viewpoint (1). However, currently approach (1) did not solve the problem of how CM should be modified, and quantum theory (QT) is considered to be the most general theory for describing nature. Therefore, CM must be modified so that it correctly describes QT. As shown in our publications, finite mathematics (FM) satisfies this condition. It involves a finite ring $R_p=(0, 1, ...p-1)$ where addition, subtraction, and multiplication are performed modulo $p$. FM does not contain any foundational problems and is a more general theory than CM: the latter is a degenerate special case of the former in the limit $ptoinfty$. The purpose of this paper is to provide a brief overview of our results to make them understandable to a wide audience of mathematicians and physicists.
Category: General Mathematics

Recurrence, Alignment, and Temporal Structure in the Sun—Earth—Moon System

Authors: Pedro A. Kubitschek Homem de Carvalho
Comments: 30 Pages. (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org!)

The classical three—body problem is traditionally formulated as the predictionof complete spatial trajectories of three interacting masses under gravitation, a taskknown to be generally non—integrable and chaotic. In this work, we adopt a complementary perspective focused on the Sun—Earth—Moon system, where the most stable and observable features arise not from translational motion but from rotational recurrence and angular phase closure. We introduce an angular—toroidal phaseformalism in which the three bodies are represented by periodic phase variablesassociated with Earth rotation, Earth orbital motion, and lunar orbital motion. These phases naturally define a three—torus T3, within which the system evolves as a helical flow. Observable cycles such as the solar day, the synodic month, and the year emerge as alignment events corresponding to phase closure conditions. An alignment operator is proposed to characterize the temporal coherence of these events. The approach does not aim to recover full three—body trajectories, but in stead provides an analytic and geometrically transparent description of recurrence and temporal structure in the restricted three—body problem.
Category: Mathematical Physics

Spatial Chess: Complete Rules and Specifications

Authors: Vyacheslav Potapov
Comments: 14 Pages. English translation of the original 2011 Russian publication. Physical model specifications included.

This paper presents the formal rules and structural design of "Spatial Chess," a three-dimensional extension of the classical game played on an 8 × 8 × 8 grid. The authorprovides a detailed specification of the physical board consisting of 512 cubic fields and 64transparent levels. While the movement logic is partially based on the FIDE Istanbul 2000standards, it is significantly expanded to account for 3D vectors, including the introductionof the "Lion" (8-direction corner-to-corner piece) and the 24-direction Knight. This modelserves not only as a complex strategy game but also as a specialized tool for developing3D spatial orientation and cognitive skills essential for pilots, astronauts, and engineers.
Category: Combinatorics and Graph Theory

A Bottom-Up Unification of Quantum Mechanics, the Standard Model, Gravity and Cosmology from a Discrete Hypercubic Spacetime Lattice

Authors: Bertrand Jarry
Comments: 31 Pages. Creative Commons Attribution 4.0 International (CC-BY 4.0) (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org)

This paper presents a bottom-up unification of quantum mechanics, the Stan-dard Model, gravity, cosmology, and quantum gravity derived from a single minimal structure: a four-dimensional hypercubic spacetime lattice with local unitary nearest-neighbor evolution. All known physical laws emerge as continuum and long-wavelength limits of the discrete dynamics, while ultraviolet divergences and classical singularities are regulated by the lattice scale.
Category: Quantum Gravity and String Theory

Compatibility Between the NFW Method and DMbQG Theory in M33

Authors: Manuel Abarca Hernandez
Comments: 11 Pages. In Spanish (Note by viXra Admin: Author name is required in the article after the article title)

This work aims to contrast the validity of the DMbQG theory in M33, as well as the compatibility of the virial mass results calculated by this theory and the NFW method. The data sources used are the papers by Kam and Carignan [1] and López Fune and Salucci [2]. The DMbQG theory was initially developed for the MW and M31 and was successfully verified with results published in Abarca, M. [4]. In Abarca, M. [3], it was demonstrated that this theory offered virial mass results for the MW and M31 similar to those calculated using the NFW method. In Abarca, M. [6], the compatibility of both methods was also demonstrated in two galaxies outside the Local Group.By applying the DMbQG theory using the rotation curve data published by Kam and Carignan [1] for M33, a virial mass of 1.3 E11 MSUN is obtained. However, Kam and Carignan obtain a virial mass of 5 E11 MSUN using the NFW method, which is more than half of the estimated virial mass for the MW. This result is inconsistent, given that M33 contains only one-tenth of the baryonic mass of the MW.Since the DMbQG theory is based on the hypothesis that dark matter is a product of the propagation of the gravitational interaction according to a quantum gravity mechanism, it follows that this theory supports the fact that the ratio of baryonic mass to virial mass should be very similar in all galaxies, at approximately 10%. However, with the results from Kam and Carignan, this ratio for M33 drops to 2%.Given that the DMbQG theory can only be applied in the halo region where the baryonic mass is residual, a specific region of the rotation curve was selected for the application of the theory based on this criterion. Furthermore, two points were selected to calculate the two parameters associated with the NFW method. The selection criterion for these points ensured that both theories used the same data to calculate the virial mass. The results have been excellent, as the virial mass calculated with NFW has been reduced to 1.1 E11 MSUN, and the mass calculated with the Direct Mass method is 1.3 E11 MSUN; these results differ by only 15%. Moreover, with these results, the baryonic mass to virial mass ratio is 9%, which is very similar to the typical ratio observed in most galaxies.
Category: Astrophysics

A New Polyhedron Obtained by Truncation of Rhombic Dodecahedron

Authors: Harish Chandra Rajpoot
Comments: 19 Pages, 15 Figures

In this paper, a new convex polyhedron is introduced, obtained by systematically truncating all 24 edges of a rhombic dodecahedron such that the newly generated 24 congruent vertices lie exactly on a common spherical surface. The resulting truncated rhombic dodecahedron is a non-uniform convex polyhedron composed of 12 congruent rectangular faces, 6 congruent square faces, and 8 congruent equilateral triangular faces, with a total of 48 edges and 24 identical vertices. Using HCR’s Theory of Polygon, closed-form analytical expressions are derived for the radius of the circumscribed sphere passing through all vertices, the normal distances of the rectangular, square, and equilateral triangular faces from the center of the polyhedron, as well as its total surface area and enclosed volume. In addition, analytical formulae are obtained for the solid angles subtended at the center by each type of face, the dihedral angles between any two faces meeting at each of the 24 vertices, and the solid angle subtended by the truncated rhombic dodecahedron at each of its vertices.
Category: Geometry

Mathematical Analysis of Rhombic Dodecahedron Using Theory of Polygon

Authors: Harish Chandra Rajpoot
Comments: 11 Pages, 9 Figures

In this paper, a comprehensive mathematical analysis of the rhombic dodecahedron is presented, and closed-form analytical expressions are derived for a polyhedron consisting of 12 congruent rhombic faces, 24 edges, and 14 vertices lying on a common circumscribed sphere. Using HCR’s Theory of Polygon, generalized formulae are obtained for the face angles and diagonals of the rhombic faces, as well as for the radii of the circumscribed sphere, inscribed sphere, and midsphere. Analytical expressions are further derived for the total surface area and enclosed volume in terms of the edge length. In addition, the solid angles subtended at the vertices and the dihedral angles between adjacent faces are evaluated. It is also shown that this convex polyhedron can be constructed by assembling twelve congruent right pyramids with rhombic bases and a specific normal height.
Category: Geometry

Analytical Study of the Rhombicuboctahedron Using the Theory of Polygon

Authors: Harish Chandra Rajpoot
Comments: 18 Pages, 11 Figures

In this paper, the circumscribed radius of a rhombicuboctahedron is derived using an alternative geometrical approach based on HCR’s Theory of Polygon. An explicit analytical expression is obtained for the radius of the circumscribed sphere passing through all 24 congruent vertices of a rhombicuboctahedron with a given edge length. Using the same theoretical framework, closed-form formulae are subsequently derived for the normal distances of the equilateral triangular and square faces from the centre, the total surface area, and the enclosed volume. In addition, analytical expressions are presented for the solid angles subtended at the centre by each equilateral triangular face and each square face, the dihedral angles between any two faces meeting at a vertex, and the solid angle subtended by the rhombicuboctahedron at any of its 24 identical vertices.
Category: Geometry

Mathematical Analysis and Modeling of Pyramidal Containers, Right Pyramids, and Related Polyhedra

Authors: Harish Chandra Rajpoot
Comments: 32 Pages, 22 Figures

This paper presents generalized analytical formulas for computing key geometric parameters of pyramidal flat containers with regular polygonal bases, right pyramids, and related polyhedra. The derived expressions include the V-cut angle (obtained using HCR’s Theorem), edge length of the open end, lateral edge length, dihedral angle (derived using HCR’s Corollary), surface area, and volume. These formulas provide a unified framework for the systematic analysis and construction of such structures for arbitrary regular polygonal bases. The results are applied to pyramidal flat containers with square, regular pentagonal, hexagonal, heptagonal, and octagonal bases, as well as to right pyramids and bipyramidal polyhedra. The underlying geometric construction is based on the rotation or folding of two coplanar planes about their intersecting straight edges, as formulated in HCR’s Theorem, and is illustrated through practical paper models.
Category: Geometry

Two Mathematical Proofs of Bond Angle in a Regular Tetrahedral Structure

Authors: Harish Chandra Rajpoot
Comments: 9 Pages, 5 Figures

This paper presents a proof of the angle between any two bonds in a molecule possessing a tetrahedral structure, such as the methane molecule, in which all four σ-bonds (corresponding to four hydrogen atoms bonded to a central carbon atom) are equally inclined in three-dimensional space. The bond angle is derived using two independent approaches. The first method involves formulating the geometry of a right pyramid with a regular n-gonal base, while the second employs HCR’s formula for regular polyhedra. Both approaches lead to the same result, thereby providing a simple and rigorous geometric justification of the tetrahedral bond angle.
Category: Geometry

Mathematical Analysis of Non-Uniform Polyhedra with Two Regular N-Gonal Faces and 2n Trapezoidal Faces Inscribed in a Sphere

Authors: Harish Chandra Rajpoot
Comments: 12 Pages, 4 Figures

In this paper, all the important formulas have been generalized which are applicable to calculate the important parameters, of any non-uniform polyhedron having 2 congruent regular n-gonal faces, 2n congruent trapezoidal faces each with three equal sides, 5n edges & 3n vertices lying on a spherical surface, such as solid angle subtended by each face at the center, normal distance of each face from the center, inner radius, outer radius, mean radius, surface area & volume. These are useful for the analysis, designing & modeling of different non-uniform polyhedrons.
Category: Geometry

Mathematical Derivations of Inscribed & Circumscribed Radii for Three Externally Tangent Circles

Authors: Harish Chandra Rajpoot
Comments: 14 Pages, 6 Figures

In this paper, the generalized formula have been derived to analytically compute the radii of the circles inscribed and circumscribed by three mutually tangent circles of known radii. The formulation and analysis of three external tangent circles packed in the smallest rectangle and the intersection circles have also been done. . The analytic formula derived here can also be used in case of three tangent spheres in three dimensions. These formula are also used for calculating any of three radii if rest two are known & computing the length of common chord, angle of intersection & area of intersection of two intersecting circles. The generalized formula derived here can also be used to derive the recurrence relations for circle packing over a plane which can further be extended into 3 dimensions for sphere packings.
Category: Geometry

Mathematical Analysis of Trapezohedron/deltohedron Having 2n Congruent Right Kite Faces

Authors: Harish Chandra Rajpoot
Comments: 19 Pages, 8 Figures

The generalized formula, derived here, are equally applicable on any n-gonal trapezohedron having 2n congruent right kite faces, 4n edges & 2n+2 vertices lying on a spherical surface with a certain radius. These formula have been derived by the author Mr H.C. Rajpoot to analyse infinite no. of the trapezohedrons having congruent right kite faces simply by setting n=3,4,5,6,7,u2026u2026u2026u2026u2026u2026upto infinity, to analytically compute all the important parameters such as ratio of unequal edges, outer radius, inner radius, mean radius, surface area, volume, solid angles subtended by the polyhedron at its vertices, dihedral angles between the adjacent right kite faces etc. These formula are very useful for the analysis, modeling & designing of various n-gonal trapezohedrons/deltohedrons.
Category: Geometry

Mathematical Analysis of Spheres at Polyhedral Vertices, Face Filleting, and Sphere Packing in Right Pyramids and Platonic Solids

Authors: Harish Chandra Rajpoot
Comments: 28 Pages, 6 Figures

In this paper, the generalized formula has been derived that is applicable to locate any sphere, with a certain radius, resting in a vertex (corner) at which n no. of edges meet together at angle α between any two consecutive of them such as the vertex of platonic solids, any of two identical & diagonally opposite vertices of trapezohedron (uniform polyhedron with congruent right kite faces) & the vertex of right pyramid with regular n-gonal base. These are also useful for filleting the faces meeting at the vertex of the polyhedron to best fit the sphere in that vertex. These are used to determine the distance of the sphere from the vertex, the distance of the sphere from the edges, the fillet radius of the faces, etc. The formula has been generalized for packing the spheres in the vertices of right pyramids & all five platonic solids.
Category: Geometry

Fish" Paradox in Practice and Possible Reality of "Strange" Mathematical Solutions

Authors: Roman Vinokur
Comments: 3 Pages. (Note by viXra Admin: Please cite listed scientific references)

A feasible scenario to the famous "minus two fish" paradox is shown in this paper. Not rarely theoretical models produce ‘strange’ mathematical solutions which look incredible to common sense but describe real phenomena. Such examples teach engineers and scientists not to dismiss ‘strange’ mathematical results without verifying their feasibility. In particular, the prediction of the positron by Paul Dirac confirms this pattern.
Category: History and Philosophy of Physics

Mathematical Analysis of Disphenoid (Isosceles Tetrahedron)

Authors: Harish Chandra Rajpoot
Comments: 12 Pages. 4 Figures

In the present work, general analytical expressions are derived for the fundamental geometric properties of a disphenoid using three-dimensional coordinate geometry. Specifically, closed-form formulae are obtained for the volume and total surface area, as well as for the radii of the inscribed and circumscribed spheres, and solid angle subtended by disphenoid at its vertex. In addition, explicit coordinates of the four vertices are determined, together with the coordinates of the in-center, circum-center, and centroid of the disphenoid. The proof for in-center, circum-center and centroid to be coincident is presented and mathematical equation governing all disphenoids is also derived in a closed form.
Category: Geometry

The Result of the Product of Two or More Prime Numbers is Always Equal to the Result of the Sum of Only Two or Three Prime Numbers

Authors: Silvana di Savino
Comments: 5 Pages.

The even number 2n, which is the product of two or more prime numbers with 2, is always equal to the sum of only two prime numbers equidistant from half their sum; the odd number 2n+1, which is the product of two or more prime numbers with 2 + the odd number 1, is always equal to the sum of two prime numbers equidistant from half their sum, which is an even number + a prime number, 1+2n, which is the difference between the two equidistant primes.
Category: Number Theory

Geometric Unification of Fermion Masses and Mixing Angles from Clifford Torus Topology

Authors: Dmitry Makaryev, Victor Shcherb
Comments: 5 Pages. (Note by viXra Admin: For the last time, please submit article written with AI assistance to ai.viXra.org)

The Standard Model accommodates fermion masses and mixing through nineteen free Yukawa andmixing parameters, offering no structural explanation for their values. Here we derive the complete flavor structure from two principles: the quadratic (Born-rule) form of the mass Hamiltonian and a topological constraint from the Clifford torus in S3 that fixes the interference amplitude to A = √2. We prove that the number of generations N = 3 is the unique solution to the identity Q × N = 2, where Q = 2/3 is the Koide ratio and 2 = χ(S2) is the Euler characteristic of the Bloch sphere.A single geometric phase δ = 2/9 rad determines the entire lepton sector: the charged-lepton mass spectrum (predicting mτ = 1776.97 MeV versus the experimental 1776.86 ± 0.12 MeV), all three PMNS mixing angles via the exact relations θ23 = π/4, θ13 = 2δ/3, θ12 = π/4 − δ (derived from Z3 representation theory), and the Wolfenstein parameter λ = sin δ. The neutrino sector follows from a complementary phase shift Δδ = 3π/4, predicting absolute masses withP mν = 0.059 eV. The universal mass curve evaluated at θ = π yields a dark-sector state at 53.85 MeV whose π-harmonic at 17.14 MeV matches the Atomki X17 anomaly to 0.8% with zero free parameters. The Clifford torus geometry further predicts maximal CP violation (δCP = −π/2) and exact μ—τ reflection symmetry. In total, the model yields twenty experimentally testable predictions from two input masses and onefundamental constant.
Category: Quantum Physics

The Implications of the Time Dilation Factor in Classical Theories

Authors: Guanghui Xie
Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific references written by other scholars))

The time dilation factor in special relativity is an algebraic expression derived through mathematical deduction based on the principle of the constancy of the speed of light and the principle of special relativity. This expression quantitatively describes the functional relationship between the velocity of motion and the change in time. In this paper, Building upon classical physical theories and taking as a premise that the speed of information propagation equals the speed of light, through theoretical derivation, The algebraic expression of the time dilation factor in special relativity is also derived. And within the framework of classical physics, it clarifies the physical meaning of this time dilation factor, theoretically explains why it lacks a contraction function, and provides a new pathway for further research into the theory of temporal relativity.
Category: Relativity and Cosmology

Two Relativistic Derivations of the Photon Energy-Frequency Relation and Their Pedagogical Significance

Authors: Zilu Liu
Comments: 7 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

This paper presents two novel methods for deriving the photon energy formula. Both methods are strictly rooted in the framework of special relativity, without presupposing quantum theory. Method One rigorously established the proportional relationship between energy and frequency by analyzing the complementarity between the relativistic Doppler effect and the conservation of energy and momentum in Compton scattering, and subsequently determined Planck's constant h through experiments. Method Two derives the photon energy formula naturally by examining the limiting behavior of a massive particle as its velocity approaches the speed of light, incorporating the frequency ratio relationship. These derivations compensate for the deficiency in traditional pedagogy where this formula is introduced merely as a prescriptive conjecture fitting experimental results, providing a profound and unified perspective for understanding this fundamental formula and highlighting the role of relativity as its theoretical foundation, thus holding significant value for physics education.
Category: Quantum Physics

Expansion of the Number π Into a Series of Trigonometric Functions and Its Connection with the Riemann Zeta Function

Authors: Bohdan Dykyi
Comments: 8 Pages. (Note by viXra Admin: Please cite and list scientific references)

In this work, series expansions for negative and positive integer powers of the number π are derived using Viète’s infinite product and differentiation techniques. A representation of these powers in terms of trigonometric series involving tangent functions is obtained. Furthermore, a connection between these expansions and the values of the Riemann zeta function at even arguments is established. Explicit formulas for the reciprocal values of the zeta function are presented. Several illustrative examples are provided.
Category: Functions and Analysis

Counting Fixed Degree D>2 Addition Chains

Authors: Theophilus Agama
Comments: 19 Pages.

We denote the length of an addition chain with fixed degree d>2 leading to n by l^d(n). We study the counting function F_d(m,r):=#{nin [d^m,d^{m+1})~:~ l^d(n)<m+r} establishing upper and lower bounds, which generalizes previous classical investigations of De Koninck, Doyon, and Verreault.
Category: Number Theory

Neural Networks with {-1,0,1} Weights Playing Atari Space Invaders (1) Trained by Evolution Strategy

Authors: Hidehiko Okada
Comments: 8 Pages.

In prior work, discrete-weight neural networks trained via evolutionary algorithms have been investigated, demonstrating the feasibility of binary-weight models on reinforcement learning tasks including Atari Space Invaders. In this study, we extend this line of research by evaluating ternary-weight neural networks with weights in {-1,0,1} and comparing their performance with binary-weight counterparts {-1,1}. Using Evolution Strategy to train multilayer perceptron controllers for the Atari Space Invaders task, the author analyzes the effects of weight representation and evolutionary hyperparameters. Experimental results show that ternary-weight networks achieved higher average performance than binary-weight networks with identical architectures, although the difference was not statistically significant. Additionally, a larger population size combined with fewer generations was found to be more effective than smaller populations with longer training durations, consistent with prior findings. These results suggest that population size plays a critical role in compensating for the limited global search capability of ES.
Category: Artificial Intelligence

Quantum Mechanic Results of Bell-test Experiments Explained Classically, Intuitively and Comprehensible

Authors: Gerard van der Ham
Comments: 12 Pages. 1 diagram, 4 figures

The fact that particles of an entangled pair are detected from different viewpoints in Bell-test experiments is the reason why no conclusions can be drawn from the comparison of the outcomes of the detections. Bell-test experiments are about correlations between probabilities for certain combinations of (spin-) outcomes and relative angles of settings of the detectors at which the spin of the particles is measured. Quantum Mechanics predicts those correlations correctly. Because repeated experiments show equal results there must be conditions for a pair of entangled particles to show a certain combination of results. This article explains how to find these conditions and the probabilities with which the certain combinations of outcomes occur. The implication of this explanation is that ‘entanglement’ turns out not to be the Quantum Mechanic phenomenon as is usually assumed.
Category: Quantum Physics

Dimensional Regularization as Gateway to Complex Dynamics

Authors: Ervin Goldfain
Comments: 31 Pages.

A while ago, a couple of readers asked me to write a user-friendly essay on the path leading from Dimensional Regularization (DR) of Quantum Field Theory to the far-from-equilibrium regime of ultraviolet phenomena. Below is my answer to their request. The takeaway point of this essay is that DR provides a natural route to fractal spacetime and complex behavior, without introducing unnecessary assumptions or extra degrees of freedom.
Category: High Energy Particle Physics

De Broglie's Phase Waves of Electrons on Cyclotron Orbits

Authors: Martin Kraus
Comments: 3 Pages.

The discrete energy levels of electronic orbits in Bohr's model of hydrogen atoms may be derived from the requirement that de Broglie's electronic phase waves are continuous on closed orbits. This short work applies the same reasoning to electronic cyclotron orbits. The resulting discrete energy levels include the non-relativistic Landau levels and additional (half-integer) levels. It is hypothesized that transitions to these additional levels are usually strongly suppressed.
Category: Quantum Physics

A New Approach Towards Relativity and Time Dilation

Authors: Vipin Choudhary
Comments: 9 Pages.

Time Dilation is the difference in the amount of time two clocks measure in the same inertial frame. Many studies have explored the relativity of time dilation using various approaches. However, the scientific and mathematical explanation of time dilation of moving things and light pulse clocks still has limited research. Therefore, this article examines relativity by utilizing scientific and mathematical approaches, the experience of moving things and light pulse clock ticks have been examined. The study revealed that the time elapsed for the same process is different for the different observers. Here, it showed that the time can be expressed in the form of a wave. In addition, the relative distance changes between the observers, and the observing subject time flows differently for the observer relative to the observing subject.
Category: Relativity and Cosmology

Orbital Motions

Authors: Stefano Giardinelli
Comments: 34 Pages. In Italian (Note by viXra Admin: Author name, an abstract and scientific references are required in the article - Please conform!)

How is the trajectory of a space probe designed? What are the invisible laws that govern the flight of satellites and their encounters with distant worlds? In this treatise, The author guides the reader through a logical and mathematical journey that reveals the secrets of space navigation. Starting from Newton's fundamental principles and Kepler's laws, the work explores with clarity and rigor the pillars of orbital dynamics: u2022 The basics of motion: From circular orbits to the energy analysis of bound systems. u2022 Flight technology: The operation of rocket engines, the Tsiolkovsky equation, and propellant management. u2022 Interplanetary maneuvers: The Hohmann transfer, the study of spheres of influence, and the complex geometry of Lagrange points. u2022 The Mars mission: A detailed, step-by-step analysis for planning a trip to the Red Planet. u2022 The slingshot effect: How to use the gravity of planets (Gravity Assist) to navigate the Solar System without consuming propellant. Richly illustrated and accompanied by a precise mathematical framework, Orbital Motions is an essential work for physics students, space enthusiasts, and anyone who wishes to understand orbital mechanics.
Category: Classical Physics

On the Constant Real Part of the Non-Trivial Zeros: A Proof of the Riemann Hypothesis

Authors: Sriramadesikan Jagannathan
Comments: 11 Pages. (Note by viXra Admin: Please cite and list scientific references)

This paper presents a proof of the Riemann Hypothesis by examining the geometric and arithmetic properties of the Dirichlet eta function. By assuming the existence of zeros off the critical line, and analyzing the resulting alternating series in the complex plane, we establish a logical contradiction. The proof relies on insights into the structure of these series, demonstrating that all non-trivial zeros must possess a real part of exactly 1/2.
Category: Number Theory

Impossible Relativistic Rest on Inclined Planes

Authors: Antonio León Sánchez
Comments: 2 Pages.

An impossible relativistic rest on an inclined plane is used here to demonstrate in a very simple and concise way that the theory of special relativity is formally inconsistent, with the second of its fundamental principles being the cause of its formal inconsistency.
Category: Relativity and Cosmology

Dihedral Angles Between the Polygonal Faces in Various Regular & Uniform Polyhedra: Platonic and Archimedean Solids

Authors: Harish Chandra Rajpoot
Comments: 29 Pages, 8 Figures, Original Research

This paper presents a set of practically oriented analytical formulas and tables compiled from geometric data of various convex uniform polyhedra, listing the dihedral angles between two faces with or without sharing a common edge. These formulas and tables are specifically intended to support the physical and computational construction of convex uniform polyhedral shells composed of different regular polygonal faces. By using the tabulated dihedral angles, wire-frame and shell models of polyhedra can be constructed efficiently by successively joining adjacent planar faces at their common edges with the correct angular orientation. The presented data are particularly useful for applications in geometric modeling, structural design, educational model fabrication, and the development of computational algorithms for polyhedral assembly. Overall, the tables provide a convenient and reliable reference for the practical realization and analysis of convex uniform polyhedral structures.
Category: Geometry

Solid Angle of Torus on Normal Geometrical Axis Using [a] Method of Concentric Cones

Authors: Harish Chandra Rajpoot
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please don't name title, equation/formula etc after the author's name) 3 Figures.

An analytical method is developed for the exact evaluation of the solid angle subtended by a torus at a point lying on its axis of symmetry, i.e., on the line perpendicular to the mid-plane of the torus and passing through its centre. The method is based on a geometric enclosure of the torus between two coaxial right circular cones with a common apex at the observation point and with axes coincident with the torus axis. It is shown that the solid angle associated with the torus equals the algebraic sum i.e. the difference between the solid angles subtended by the outer and inner bounding cones at the apex [1,2,3]. This construction leads to closed-form expressions for the solid angle as a function of the torus radii and the axial distance of the observation point, without recourse to surface integration. The resulting formulation provides a concise geometric characterization of toroidal visibility and is well suited for applications in geometric modeling, where exact angular measures are required, and in photometry and radiative transfer, particularly for the evaluation of irradiance, flux, and angular response of axially symmetric toroidal sources and apertures.
Category: Geometry

An Approach to Identify Machine Learning and Deep Learning Methods and Data to Understand Common Time Variant Models That Can Forecast Both Crypto Currencies and Equities

Authors: Sanath Shenoy
Comments: 27 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

Cryptocurrencies and Equity markets have been the most attractive investment in the modernworld. While these are very attractive, they have been subject to a lot of volatility in their behaviour due to multiple reasons, such as macroeconomic conditions and regulation in various economic activities within countries and the world as a whole. There has been minimal research in this area to use Machine learning and deep learning approach to help predict the price of cryptocurrency and equities and other behaviours. There has also been very minimal research to understand the relationship on whether the change in the value of equities has an effect on the value of cryptocurrencies and vice versa. These effects are due to correlational causal or any other kind of relationship between the values of both the investment asset classes. This research aims to identify models based on Machine learning and deep learning that can predict the price or value of cryptocurrencies and equities.
Category: Artificial Intelligence

Modular Exponentiation Identities: Classification and Minimal Exponents

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

This paper investigates two specific modular exponentiation identities involving fixed integers. First, we determine the set of non-negative integers m satisfying a^N ≡ a^m (mod N) for a fixed N > 1 and all integers a, deriving the minimum such m. Second, we analyze the minimum positive integer n such that a^mn ≡ aⁿ (mod x) holds for a fixed x > 1 and all integers a, m. We provide explicit formulas for these minimal exponents in terms of the prime factorization exponents and the Carmichael function λ(·).
Category: Number Theory

Invariance of Ensemble Overlap in Stochastic Binary Processes: A Gauge Theory of Strategy

Authors: Goutham Netha Anagandula
Comments: 2 Pages. Pedagogical derivation connecting the Gambler's Fallacy to Gauge Invariance

The "Gambler’s Fallacy" is often treated as a cognitive bias, but it can be rigorously understood as a violation of translational invariance in probability space. In this paper, we define a "Global Correlation Functional" G(u20d7S) representing the expected overlap between a fixed control sequence (strategy)u20d7 S and a random Bernoulli target vector (nature)u20d7 X. By treating the strategyu20d7 S as a gauge degree of freedom, we demonstrate—both analytically and via Monte Carlo simulation—that the expectation value of the overlap is invariant under all local permutations ofu20d7 S. We conclude that in memoryless systems, the derivative of success with respect to strategy is identically zero(∇u20d7 S G = 0), implying that all strategies are microcanonically equivalent. This framework offers a pedagogical bridge between classical probability and the concept of gauge invariance in theoretical physics.
Category: Mathematical Physics

Quantum Vacuum as Relic of Primordial Spacetime

Authors: Ervin Goldfain
Comments: 18 Pages.

The vacuum of Quantum Field Theory (QFT) is a state containing an infinite amount of quantum entanglement between different regions of spacetime. This nonlocal behavior reflects a fundamental property of quantum fields: vacuum fluctuations are globally correlated and cannot be localized to finite regions of spacetime. Here we point out that this property echoes the behavior of primordial spacetime undergoing a sustained transition to classical chaos. Following the entropy framework of [6], we interpret vacuum entanglement as relic of primordial spacetime. Moreover, building off our previous contributions [7-10], results suggest that gravitational physics, the QFT vacuum and the complex behavior of ultraviolet phenomena share a common root in the dynamics of primordial spacetime.
Category: Quantum Physics

From Information to Geometry: A Relational Framework for Emergent Spacetime via Quantum Fisher Information

Authors: Lee Jin Woo
Comments: 4 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

I present a rigorous formulation of the Relational-Informational Model (RIM), proposingthat spacetime geometry is not a fundamental background but an emergent property arisingfrom the entanglement structure between an observer and the universe. By defining theobserver as a quantum reference frame within a globally static universe (Htot|Ψ⟩= 0), I de-rive the spatial metric directly from the distinguishability of quantum states. Utilizing the Taylor expansion of the Bures distance, I prove that the physical metric corresponds to the Quantum Fisher Information Matrix (QFIM) of the observer’s reduced density matrix. Furthermore, addressing the critique regarding zero net flux in static Rindler frames, I employthe First Law of Entanglement Entropy involving the Modular Hamiltonian to demonstratethat the thermodynamic evolution of this informational geometry naturally obeys the Ein-stein field equations. This framework provides a concrete mathematical pathway unifyingQuantum Information Theory with General Relativity.
Category: Quantum Gravity and String Theory

Geodesy and Numerical Analysis Exams

Authors: Abdelmajid Ben Hadj Salem
Comments: 20 Pages. In French

This booklet covers several topics from geodesy and numerical analysis exams given to engineering students studying Geomatics at ESAT school.
Category: Geophysics

Directional Density of Gases Under FitzGerald-Lorentz Contraction: Predictions for Optical Interferometry

Authors: Alvydas Jakeliunas
Comments: 6 Pages. Updated with the help of authors mentioned in the reference

We show that the directional density of an ideal gas remains isotropic under FitzGerald-Lorentz contraction of its container, while that of a bound solid becomes anisotropic. Combined with the invariance of the optical phase delay per molecule, this yields parameter-free predictions for gas-filled interferometers under Lorentzian Relativity. For a gas cell of physical length L measured against a direction-independent reference, the peak-to-peak fringe shift on a full rotation is ΔN = k(L/2λ)(n−1)(v/c)², where k is the number of passes through the gas (k = 1 for a single pass, k = 2 for a round trip). The signal arises from the orientation-dependent molecule count in the contracted gas path. We predict null results for solid-dielectric interferometers regardless of refractive index.
Category: Classical Physics

Emergence of the Electroweak Scale from Continuous Dimensions

Authors: Ervin Goldfain
Comments: 11 Pages.

In this sequel to [6], we show that the electroweak vacuum �� emerges as infrared scalegenerated by dimensional deviation �� ≪ 1. The mechanism detailed here mirrors themass-gap generation mechanism of non-Abelian gauge theory.
Category: High Energy Particle Physics

On Continuous Dimensions and Infrared Mass Generation

Authors: Ervin Goldfain
Comments: 34 Pages.

We show that dimensional regularization of a generic self-interacting field theory inducesan infrared mass scale, when the running spacetime dimension approaches the criticalvalue �� = 4. The mechanism is universal and does not rely on spontaneous symmetrybreaking alone. Instead, it arises from logarithmic corrections generated byrenormalization group (RG) flow near the bifurcation point at �� ≡ 4 − �� → 0. Weexplicitly indicate how dimensional transmutation converts marginal couplings intodynamically generated mass scales. This framework unifies the origin of the Higgsvacuum expectation value and electroweak boson masses with the Yang—Mills mass gapand the QCD scale ΛQCD. Step-by-step derivations are provided, without appealing tononperturbative assumptions beyond RG consistency. The paper focuses on the regimebordering relativistic quantum field theory and complex dynamics.
Category: Mathematical Physics

The Phenomenon of the Golden Ratio as a Necessary Condition for the Effectiveness of the Double Step Running System

Authors: Andrey V. Voron
Comments: 6 Pages.

The article substantiates the hypothesis that the unity of the object-system "Double step running" is due to the "golden" proportional relations of the object-system to its subsystems and between the subsystems themselves.
Category: Quantitative Biology

On the Fixed Degree Scholz-Brauer Problem

Authors: Theophilus Agama
Comments: 8 Pages.

Denote the minimal length of a fixed degree d>1 addition chain that leads to n by l^d(n). We introduce the concept of a strong Brauer number of rank d>1 and show that all numbers belonging to this class satisfy the inequality l^d(d^n-1)1 framework.
Category: Number Theory

Il Dio Onnipotente (The Almighty God)

Authors: Rosario D'Amico
Comments: 15 Pages. In Italian

This paper aims to reflect upon and investigate what is meant strictly speaking by the term "Omnipotence"; specifically, what it entails to posit the existence of a real Omnipotent being. Such a being, therefore, cannot be subject to what he does not will or, in other words, can bring about anything logically possible that he wants, since that which is contradictory can be neither real, nor imagined, nor, consequently, willed. The approach will be strictly philosophical, eschewing commonplace drifts, devotional intents, or moral judgments in favor of conceptual clarity. Based on this analysis, we will finally cover each of the following issues: can there be an Almighty being who is also good and understandable? If so, could He abolish or sacrifice evil in the world?

Questo contributo si propone di riflettere e indagare su cosa debba intendersi stricto sensu con il termine "Onnipotenza", cioè cosa comporti supporre l’esistenza di un essere reale Onnipotente, il quale non può quindi subire ciò che non vuole o, il che è lo stesso, può realizzare qualsiasi cosa di logicamente possibile voglia, non potendo essere reale né immaginato, né quindi voluto, ciò che è contraddittorio. L’approccio sarà strettamente filosofico, evitando così derive banali, intenti devozionali o giudizi morali a favore della chiarezza concettuale. Sulla base di questa analisi, tratteremo infine ciascuna delle seguenti questioni: è possibile che un essere Onnipotente sia anche buono e comprensibile? Se sì, potrebbe abolire o sacrificare il male nel mondo?
Category: Religion and Spiritualism

Emergent Gravitation and Quantum Wave Dynamics from a Bounded Vacuum

Authors: Tanuj Kumar, Vandana [Doe]
Comments: 25 Pages. Minor revision

We present a framework in which gravitation, inertia, and wave dynamics emerge from the response of a vacuum endowed with finite potential capacity. The theory is formulated in terms of a scalar vacuum-potential field whose absolute normalization is fixed by relativistic considerations, such that the equilibrium value at infinity equals c^2. Static relaxation of localized vacuum-potential deficits reproduces Newtonian gravity in the coarse-grained limit, while time-dependent redistribution generates propagating disturbances governed by a universal wave equation. Finite vacuum capacity implies intrinsic upper bounds on transmissible force and signal speed, yielding F_max=c^4/G and v≤c without invoking spacetime geometry or independent kinematic postulates. Vacuum microstructure further leads to a universal lattice dispersion relation with Planck-suppressed corrections,Δv/c≃-1/8 (E/E_P )^2,consistent with current astrophysical and gravitational-wave constraints. Gravitational redshift, lensing, horizons, and quantum correlations arise as energetic consequences of bounded vacuum response. The vacuum is modelled as a Dynamical Planck Network (DPN), providing a conservative and internally consistent bridge between relativistic gravitation and quantum wave phenomena.
Category: Quantum Gravity and String Theory

On the Inconsistency of the Standard Model in Exactly Four Dimensions

Authors: Ervin Goldfain
Comments: 23 Pages.

The triviality of the Higgs sector in exactly four spacetime dimensions is a seriousobstruction to the ultraviolet completion of the Standard Model for Particle Physics (SM).The conventional explanation of triviality is that SM represents a highly successful lowenergy description of nature, but it is incomplete and cannot be the final word on how theUniverse works. An intriguing resolution of the triviality problem is the AsymptoticSafety (AS) program developed in the framework of Quantum Gravity. In contrast withAS, this brief report argues that minimal spacetime fractality above the SM scale, a) is anatural extension of Renormalization Group dynamics and b), it can successfully explainaway the triviality problem.
Category: High Energy Particle Physics

Evidence for Planet Nine with a 3,600-Year Period and 1 AU Perihelion as the Cause of Recurring Apocalypses.

Authors: Pavlo Kandyba
Comments: 15 Pages.

A model of Planet Nine (P9) as a massive body (5 M⊕) in a highly eccentric retrograde orbit (e ≈ 0.994—0.996, q ≈ 1 AU, Q ≈ 400—500 AU, T ≈ 3600 yr), presumably captured from interstellar space. The model relies on the clustering of ETNO orbits by alignment with P9. It also shows quasi-periodicity in climatic and cultural-historical markers (the Younger Dryas event, the Piora oscillation) consistent with a period of T ≈ 3600 yr. A historically confirmed possible observation of Planet Nine's perihelion passage of 1 AU has been established, supported by the exploratory reconstruction of the comet's orbit observed in CE 60 for 180 days and documented in Chinese chronicles.Numerical modeling showed relative stability over a period of 1000,000 years.
Category: Astrophysics

Dimensional Regularization as Gateway to Complex Dynamics

Authors: Ervin Goldfain
Comments: 31 Pages.

A while ago, a couple of readers asked me to write a user-friendly essay on the pathleading from Dimensional Regularization (DR) of Quantum Field Theory to the far-fromequilibrium regime of ultraviolet phenomena. Below is my answer to their request. Thetakeaway point of this essay is that DR provides a natural route to fractal spacetime andcomplex behavior, without introducing unnecessary assumptions or extra degrees offreedom
Category: High Energy Particle Physics

On the Constant Real Part of Non-Trivial Zeros: A Proof of the Riemann Hypothesis

Authors: Sriramadesikan Jagannathan
Comments: 11 Pages. References appended

This paper presents a proof of the Riemann Hypothesis by examining the geometric and arithmetic properties of the Dirichlet eta function. By assuming the existence of zeros off the critical line, and analyzing the resulting alternating series in the complex plane, we establish a logical contradiction. The proof relies on insights into the structure of these series, demonstrating that all non-trivial zeros must possess a real part of exactly 12.
Category: Number Theory