All Submission Categories

Plumbing the Pudding Model of the Atom

Authors: Eric Stanley Reiter
Comments: 11 Pages.

We will explore in this sequence: 1) the experimental evidence initiating the nuclear model, 2) ER’s (author) experimental repeat, 3) threshold model vs quantization, 4) ER’s past experiments with gamma ℽ and alpha α rays, 5) ER’s recent tests splitting the α upon a crystal surface, and 6) conclusion. In addition to a serious challenge to quantization, this report reveals how so-called nucleons can diffract. We know of experiments that show what appears to be nucleon interference (1, 2). If you do not think this is a problem, please read up on the measurement problem. The short answer to this problem is: if it diffracts, it is a wave. We explain particle-like effects as thresholds.
Category: Quantum Physics

Charge Quantization, the Topology of the Universe, and the Hopeful Abolition of Monopoles

Authors: Warren D. Smith
Comments: 18 Pages. Paper on internet since 2000, now uplaoded to VIXRA for archival purposes.

This is the first of a 3-part series of papers on "the shape of the universe." Inspired by P.A.M.Dirac's discovery that the existence of even one "magnetic monopole" would logically force electric charge to be quantized (explaining why all electrons have the same charge, protons the exactly-opposite charge, etc), I show that a tiny magnetic field "trapped in the topology of the universe" also would logically force that. Which universes does this idea work in? We show it works for every compact 3-manifold topology for the universe except two. We then re-examine monopoles, showing their existence would cause a lot of nasty problems, and none have ever been detected.
Category: Relativity and Cosmology

On the Shape of the Universe

Authors: Warren D. Smith
Comments: 10 Pages. Paper on internet since 2003, now uplaoded to VIXRA for archival purposes.

This is the second of a 3-part series of papers on "the shape of the universe." Bright objects such as quasars are observed to be uniformly distributed on the sky-sphere. We prove a theorem that if this is true for almost every observer-location (i.e. the location of the Earth is not special in this respect) then the universe must be a "harmonic manifold." We also argue that the universe must be an orientable 3-manifold not containing any closed geodesic whose traversal causes "twist." We also discuss incompletely-convincing evidence that the universe contains a nonzero finite number of short closed geodesics passing through the Earth. If that is genuine, we prove our assumptions winnow down the possible topologies for the universe to only 3 possibilities: "flat 3-torus" and its degenerate versions with some paralelipiped sidelengths infinite.
Category: Relativity and Cosmology

Discrete Random Variables

Authors: Teo Banica
Comments: 400 Pages.

This is an introduction to probability and discrete random variables. We first discuss games, counting and basic probability, with the mathematics of the binomials and factorials recalled, and with everything explained with numerous illustrations. Then we get into the theory of discrete random variables, with the basic theory explained, with illustrations, notably with results about the binomial and Poisson laws. Then we discuss more advanced distributions, such as the Bessel, hypergeometric and beta binomial ones, with methods from both algebra and analysis. We end with an introduction to continuous phenomena, advanced combinatorics, random matrices and freeness.
Category: Statistics

On the Finiteness and Shape of the Universe

Authors: Warren D. Smith
Comments: 31 Pages. Paper on internet since 2004, now uploaded to VIXRA for archival purposes.

This is the last of a 3-part series of papers on "the shape of the universe" and includes summary of the prior two. At the end is a chart allowing you to choose your favorite set of assumptions (from among those we consider) whereupon it tells you the conclusions deducible from those assumptions about the topology of the universe. This and the prior papers also includeddiscussion of "reasons the universe is 3-dimensional." As far as I know my analyses of these matters go considerably beyond all others; but quite possibly if "real topologists" got involved then considerably more could be said.
Category: Relativity and Cosmology

The Einstein and Landau-Lifshitz Pseudotensors — a Mathematical Note on Existence

Authors: Stephen J. Crothers
Comments: 4 Pages.

For a closed system, the conservation of energy and momentum has been affirmed through a vast array of experiments. In an attempt to reconcile the General Theory of Relativity with these findings, Einstein constructed, ad hoc, his so-called pseudotensor [A. Einstein, Ann. Phys. 49, 769 (1916)]. Yet this solution fell outside the tensorial mathematical structure of his theory. Landau and Lifshitz also constructed, ad hoc, an even more complex pseudotensor, as a proposed improvement upon the work of Einstein [The Classical Theory of Fields (Addison-Wesley Press, Inc., Cambridge, MA, 1951)]. Their pseudotensor is symmetric, unlike that proposed by Einstein. They advance that their pseudotensor yields a conservation law which also included angular momentum. However, once again, this approach leads to a mathematical construct which is not a tensor and thereby falls outside the very mathematical structure of Einstein’s theory. Both pseudotensors, whether that advanced by Einstein or by Landau and Lifshitz, violate the rules of pure mathematics and therefore can hold no place in physics.
Category: Relativity and Cosmology

Vaccum Unified Solitonic Condensate Theory

Authors: Ibrahim Khalid, M. Golebiowski
Comments: 98 Pages. License: CC BY NC ND (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

The Vacuum Unified Solitonic Condensate Theory is a way to understand the world around us. It says the space or the Vacuum Unified Solitonic Condensate Theory is like a liquid. You can squeeze it. Move it around. The Vacuum Unified Solitonic Condensate Theory has some rules that physicists know. These rules help us understand how things like electrons and protons and neutrons work.These tiny things are like knots in the Vacuum Unified Solitonic Condensate Theory. They have properties like mass and charge and spin. The Vacuum Unified Solitonic Condensate Theory helps us understand where these properties come from. We can use the Vacuum Unified Solitonic Condensate Theory to figure out the mass of electrons and protons and neutrons. We just need to use some ideas about the Vacuum Unified Solitonic Condensate Theory.The Vacuum Unified Solitonic Condensate Theory also helps us understand charge. Electric charge is like a twist in the Vacuum Unified Solitonic Condensate Theory. This twist is always the same. The Vacuum Unified Solitonic Condensate Theory can help us understand the weak forces too. It can help us understand how they are related to the Vacuum Unified Solitonic Condensate Theory.The Vacuum Unified Solitonic Condensate Theory even helps us understand gravity. Gravity happens when the Vacuum Unified Solitonic Condensate Theory gets dense in some places and not in others. The Vacuum Unified Solitonic Condensate Theory can help us understand how atoms work. It can help us understand why some things decay. The Vacuum Unified Solitonic Condensate Theory makes some predictions. We can test these predictions to see if the Vacuum Unified Solitonic Condensate Theory is correct.
Category: High Energy Particle Physics

A Problem in the Theory of Surfaces

Authors: Abdelmajid Ben Hadj Salem
Comments: 2 Pages. In French. Comments welcome.

In this note, we present a problem in the theory of surfaces. The equations for the surface $(Sigma)$ were inspired by the monograph cited in the references, published by the eminent Japanese mathematicians Katsumi Nomizu and Takeshi Sasaki.
Category: Geometry

Post-Carroll Algebra, Conformal Extensions, and Field Theories

Authors: Mojtaba Najafizadeh
Comments: 32 Pages.

By incorporating leading c -dependent corrections to the Carroll transformations, we introduce the post-Carroll transformations". We demonstrate that these transformations are consistent with post-Carrollian mechanics [1]; furthermore, they give rise to the so-called post-Carroll algebra". We show that, unlike the Carroll algebra, this new structure allows for a central charge in higher dimensions; we refer to it as the Carroll{Bargmann algebra". To construct conformal extensions, we first build the conformal extension of the post-Carroll algebra and study field theories invariantunder this symmetry. We then construct the conformal extension of the Carroll{Bargmann algebra, referred to as the Carroll{Schrodinger algebra", and demonstrate that it precisely matches the symmetry algebra of the higher-dimensional Carroll{Schrodinger theory [2]. Finally, we derive the general form of two-point functions in a post-Carrollian CFT, which in 1 + 1 dimensions exhibits both electric and magnetic sectors, while in higher dimensions only the magnetic sector survives.
Category: High Energy Particle Physics

The Missing Electric Aharonov-Bohm Test and the Search for the Electromagnetic Scalar Mode

Authors: Renato Vieira dos Santos
Comments: 8 Pages.

The electromagnetic four-potential $A^mu = (Phi/c, mathbf{A})$ comprises a scalar potential $Phi$ and a vector potential $mathbf{A}$. In gauge-invariant electrodynamics, gauge symmetry together with the Lorenz condition $partial_mu A^mu = 0$ ensure that only the two transverse photon polarizations are physical, while the scalar degree of freedom decouples. Whether this elimination holds in all physical regimes is an empirical question that has never been tested in configurations where the electric field vanishes but the potentials vary in time. We ask whether the scalar mode, described by the quantity $lorenzScalar = partial_mu A^mu$, can leave a physical imprint on matter that is detectable in the electric Aharonov-Bohm effect. The original configuration proposed in 1959---time-dependent potentials applied to shielded conducting cylinders---has never been experimentally realized. We show that a phenomenological coupling $propto lorenzScalar,bar{psi}psi$ yields a phase shift proportional to $1-cos(omega T)$, orthogonal to the standard $sin(omega T)$ prediction. A frequency sweep can cleanly separate the two contributions. We outline a realistic electron interferometry experiment, analyze the dominant systematic (fringe fields), and argue that such a test is within reach of existing technology. The question is testable: a null result would provide the first empirical bound on a physical scalar mode in a field-free regime; a positive signal would indicate that the gauge-fixing procedure discards a genuine degree of freedom. Either outcome illuminates the foundations of gauge theory.
Category: Quantum Physics

Toward Charge-Dependent Tests of the Equivalence Principle: a Phenomenological Parameter and an Unexplored Frontier

Authors: Renato Vieira dos Santos
Comments: 15 Pages. Published in PRD: https://doi.org/10.1103/frfk-vp26

We introduce and define the phenomenological parameter $kappa$, defined by $Delta a/g = kappa , Delta(q/m)$, to quantify potential linear coupling between electric charge and gravitational acceleration. A synthesis of existing precision equivalence principle experiments yields the first quantitative estimate of the effective sensitivity to this coupling: $|kappa| < 2.1 times 10^{-4}~si{kilogrampercoulomb}$ at 95% confidence level. This constraint is approximately 11 orders of magnitude less stringent than corresponding bounds on composition-dependent violations, revealing that the electromagnetic axis remains a largely underexplored frontier in empirical gravity. We connect $kappa$ to established frameworks---the Standard-Model Extension and the $THepsilonmu$ formalism---showing that it occupies a region of parameter space untouched by existing high-precision tests. An effective field theory analysis shows that dimension-six operators that couple curvature directly to the electromagnetic field strength are suppressed by the minuscule terrestrial spacetime curvature ($G_N ho_oplus sim 10^{-55}~text{GeV}^2$) and are therefore phenomenologically irrelevant. Consequently, a future measurement of $kappa$ at an accessible level would not probe minimal geometric couplings but would signal physics beyond minimal gravitational EFT, such as mediation by light scalar fields as in Einstein-Maxwell-Dilaton theory. We examine the Schiff-Barnhill effect, the primary systematic background for any such measurement, and show how it can be separated from a genuine signal. We outline the necessary experimental strategy, focused on maximizing charge-to-mass ratio differences, to transform this overlooked axis into a targeted probe for new physics.
Category: Relativity and Cosmology

A Preprint Republic at a Crossroads: Governance, Transparency, and the Future of arXiv as a Public Good

Authors: Renato Vieira dos Santos
Comments: 8 Pages.

The arXiv preprint repository, founded in 1991, has become the de facto dissemination platform for physics, mathematics, computer science, and increasingly other quantitative disciplines, hosting over 2.4 million articles and serving millions of users worldwide. In mid-2026, its governance will transition from Cornell University Library to an independent nonprofit entity governed by a board of directors and supported by private philanthropy. This article provides a data-driven examination of that transition, placing it within the history of the repository and the broader political economy of scholarly communication. Drawing on public submission statistics, moderation records, documented controversies, and historical precedents from academic publishing, we identify six structural risks: opaque governance, centralized editorial control, financial unsustainability, systemic bias, erosion of scientific pluralism, and potential mission drift toward privatized gatekeeping. We argue that the moment before institutional inertia sets in is precisely when the research community must insist on binding transparency and accountability mechanisms. We propose a set of concrete safeguards---a community-elected advisory council, mandatory transparency reporting, an independent moderation ombuds, a free-access charter, and periodic external audits---that can preserve arXiv's identity as a public good without compromising operational agility. The manuscript itself has been submitted to arXiv as a real-time test of the moderation system's openness to self-examination.
Category: General Science and Philosophy

A Pedagogical Reconstruction of Maxwell's Equations: How Charge Conservation, Helmholtz's Theorem, and the Speed of Light Make Them Natural

Authors: Renato Vieira dos Santos
Comments: 8 Pages.

Maxwell's equations are traditionally introduced as four independent postulates, each grounded in a distinct experimental law. Although empirically sound, this approach can obscure the logical structure that unifies them. This paper offers a pedagogical reconstruction based on local charge conservation, Helmholtz's theorem, and the experimental fact that electromagnetic disturbances propagate in vacuum with speed (c). A few natural auxiliary assumptions -- linearity, locality, and the simplest possible coupling to sources -- are stated explicitly. From the continuity equation and the physical necessity of a mediating field we argue that the electric and magnetic fields must be introduced. Helmholtz's theorem then demands that both divergences and curls of these fields be specified; coupling them to the available scalar and vector sources yields the inhomogeneous equations. Consistency with charge conservation forces the displacement current, making it a logical requirement rather than an empirical addition. The homogeneous equations follow from the same framework: (abla cdot mathbf{B} = 0) is the most parsimonious choice consistent with experiment, and requiring wave propagation at speed (c) fixes Faraday's law uniquely as (abla times mathbf{E} = -partial_t mathbf{B}). Once Maxwell's equations are established, the Lorentz force density (homathbf{E} + mathbf{J}timesmathbf{B}) emerges as the only dimensionally consistent expression compatible with the local conservation of momentum derived directly from the field equations. The argument employs only elementary vector calculus and is accessible to advanced undergraduates. We suggest that this unified narrative, which reveals the internal coherence of electromagnetism, can serve as a valuable complement to the traditional empirical presentation.
Category: Education and Didactics

Charge-Dependent Equivalence Principle Violation from Kinetic Mixing of the Photon with a New Long-Range Vector Field

Authors: Renato Vieira dos Santos
Comments: 11 Pages.

The weak equivalence principle (WEP) has been tested with extraordinary precision over the past decades, but invariably with electrically neutral bodies. This leaves a fundamental blind spot: does gravity feel electric charge? A recent phenomenological analysis introduced a parameter $kappaphen$ to quantify this unexplored sector, but lacked a consistent field-theoretic origin. In this work we show that the simplest mechanism is kinetic mixing between the standard photon and a new massless vector field $X_mu$ that mediates a long-range fifth force coupled to the ordinary mass current. The single gauge-invariant dimension-four operator $frac{xi}{2}F_{muu}X^{muu}$ preserves causality and unitarity. By adopting the action in which both fields couple to their respective currents with the same sign convention, the two-body potential is fully symmetric and exactly satisfies Newton's third law, rendering the model free of any mathematical pathology.The fifth force between equal masses is repulsive, a known property of vector-mediated interactions, but its contribution to the neutral gravitational acceleration is absorbed into the measured Newtonian constant.For a static, spherically symmetric source, the mixing induces an electric field proportional to the Newtonian gravitational field, leading to a charge-dependent acceleration from which we identify $kappaphen = -ximu_0,G_X/G_N$, where $G_X$ is the fifth-force coupling strength.The symmetric cross term $propto q_1m_2+q_2m_1$ in the two-body potential gives rise to dipole radiation from compact binaries, providing a direct link to pulsar-timing constraints.We derive explicit constraints from the Cassini bound on the PPN parameter $gamma$, from lunar laser ranging on $alpha_1$, and from the Hulse--Taylor pulsar, showing that the model is compatible with all existing data while providing a clear roadmap for future experimental tests.
Category: Relativity and Cosmology

Cosmology from a Rastall-Inspired Variational Theory of Gravitation

Authors: Renato Vieira dos Santos
Comments: 9 Pages.

We propose a novel gravitational theory inspired by Rastall's idea of non-conserved energy-momentum, but formulated through a consistent variational principle. After developing a Lagrangian density that naturally incorporates matter-geometry coupling, we study cosmological consequences and fit parameters using recent Hubble function $H(z)$ data. According to model selection criteria (AICc and BIC), we obtain competitive models relative to $Lambda CDM.$ In our framework, the cosmological constant emerges from energy exchange between spacetime geometry and matter content, providing an alternative to vacuum energy interpretations. Our formulation provides a self-consistent variational foundation for non-conservative gravity, offering a potential resolution to the cosmological constant problem through dynamical energy exchange rather than vacuum energy.
Category: Relativity and Cosmology

A Least-Squares Optimization Model for Quad Mesh Offset

Authors: Zitao Xu
Comments: 10 Pages.

This article presents a least-squares optimization model for quadrilateral mesh offsets. Starting from an initial offset mesh generated from a base quad mesh, the method improves offset quality by introducing tangential corrections at offset vertices and solving for them globally. Each correction is represented by two scalar variables associated with two independent directions taken from the corresponding base quad, so that the optimization preserves the intended offset distance while allowing the offset mesh to adjust tangentially.For each quad, the objective function is constructed from local shape-preservation terms together with a center-consistency term, so that the optimized offset quad remains as consistent as possible with the shape and size of the corresponding base quad while reducing local crimping and shearing. The resulting formulation is a global least-squares problem over the mesh and leads to a sparse linear system. Examples in both convex and concave regions show that the method can significantly improve the quality of quad-mesh offsets and can be applied iteratively for further improvement.
Category: Geometry

TauPhi Numbers

Authors: Walter A. Kehowski
Comments: 8 Pages.

Let tau(n) be the number of divisors of n and let phi(n) denote the number of relatively prime numbers less than n. A number is called refactorable if and only if tau(n) divides n. It is known that phi(n) divides n if and only if n is 3-smooth. Furthermore, n/phi(n)=2 when n is a power of two and n/phi(n)=3 otherwise. Define lambda(n)=lcm(tau(n),phi(n)). Then n/lambda(n) is in {1,2,3} and n is called a tauphi-number. It is the purpose of this note to show when lambda(n)=n, called the unital tauphi-numbers.
Category: Number Theory

Quantifying Relational Structure in Quantum Measurement: An Axiomatic Account of Recursive Observation Depth

Authors: Agus Mulia Bakti
Comments: 20 Pages.

We formalize the relational pair (O, M), observer and observed system, as a primitive mathematical object governed by three operational axioms. From this structure we define Recursive Observation Depth (n^k), the length of the longest informative measurement chain an observer O can perform on a system M through a Positive Operator-Valued Measure (POVM). Five propositions about n^k are proved: monotonicity, relational dependence, a characterization of superposition, an exact value for a compatible (classical) observer, and, for a generic informationally complete observer, the impossibility of a terminal state, from which the divergence of n^k follows. Rank appears only as a special case, attained when the observer's measurements are mutually compatible; once they become informationally complete and incompatible, n^k diverges. We test the framework against four no-go theorems. It is consistent with Bell, Kochen—Specker, and No-Cloning on independent structural grounds, and consistent with PBR conditionally, provided the quantum state is taken to be a property of the relation (O, M) and not of the system alone. Wigner's Friend is resolved within this framework without further postulates. The Born rule is not derived, and no new experimental predictions are offered.
Category: Quantum Physics

Reasons the Photon Mass Should be Exactly Zero

Authors: Warren D. Smith
Comments: 5 Pages.

We argue from general relativity that if photons had mass, then charged black holes could not exist. But there is evidence they do exist. Also, big violations of Maxwell's laws would occur, even in regions far from black holes. This is the first evidence that photons really have zero, as opposed to just very small, mass. We also point out that the presence of extragalactic magnetic fields suggests |m_γ|⪅10^-64kg. Two independent lines of argument, one based on extragalactic magnetic fields and very conservative bounds on current densities, the other based on Voyager spaceprobe magnetometer measurements,both find |m_γ|⪅1.7×10^-56kg. These numbers are respectively 10¹⁰ and 60 times stronger than the best previous experimental bound. Finally, we consider the fact that massless quantum fields have additional symmetries. While it is tempting to try to use that as further justification for masslessness, it ultimately seems wrong.
Category: Classical Physics

Invisibility is Uncomfortable

Authors: Warren D. Smith
Comments: 3 Pages. Paper on www since 1993 now uploaded to VIXRA for archival purposes.

As a consequence of information-theoretic limitations arising from thermodynamics form a previous paper, we conclude that the sustained wearing of a "cloak of invisibiliity" is impossible.
Category: Thermodynamics and Energy

Perturbative String Theory and Its Non-Perturbative Probes

Authors: Sutirtha Mukherjee
Comments: 292 Pages.

Perturbative string theory treats a moving string as a two-dimensional surface and studies the quantum field theory of the fields living on it. Removing the two gauge freedoms of that surface, the freedom to relabel its points and the freedom to rescale its metric, by the Faddeev--Popov method produces a nilpotent BRST charge $Q_B$, and its cohomology $H(Q_B)$ is the physical spectrum of the string. The condition $Q_B^2=0$ fixes the number of spacetime dimensions at twenty-six for the bosonic string and ten for the superstring. At one loop the surface is a torus, and its modular invariance chooses the allowed spectrum, the GSO projection of the superstring, and the charge lattices the theory can carry. The same degeneration limit gives unitarity, since the amplitude factorizes on physical intermediate states. When that intermediate state is a massless one emitted into the vacuum, its coefficient is a tadpole, and requiring the tadpole to vanish is what picks out an allowed background. For the unoriented open string, this condition, equivalently the cancellation of gauge anomalies, forces the gauge group $SO(32)$. These results, and the tree and loop amplitudes built from them, make up the perturbative theory. Perturbation theory is limited to weak coupling, and reaching strong coupling relies on a set of non-perturbative probes that supersymmetry protects. A single cylinder stretched between two D-branes can be understood in two ways, as a loop of open string and as a closed string exchanged across the gap, and equating the two fixes the tension of a D-brane and shows that its Ramond-Ramond charge is quantized in the smallest unit Dirac allows. The masses of the bound states the branes form, the $(p,q)$ strings, and the D0-brane threshold bound states are set exactly by the supersymmetry algebra. Because these tensions, charges, and masses do not move with the coupling, they can be compared across two descriptions of the physics. The $(p,q)$ strings fill out the $SL(2,mathbb Z)$ multiplet behind the S-duality of type~IIB, and the D0-branes become the momentum modes of an eleventh dimension that opens up in strongly coupled type~IIA. The same protected states count the microstates of a black hole and reproduce its Bekenstein-Hawking entropy, and D-branes sourcing a warped throat give a four-dimensional gauge theory that confines while keeping one massless particle.
Category: Quantum Gravity and String Theory

Viscous Time Theory & Birch—Swinnerton-Dyer

Authors: Raoul Bianchetti, Payam Danesh
Comments: 10 Pages.

We develop a precise arithmetic version of Viscous Time Theory by replacing the original Sobolev field space with a finite Hilbert model of the Selmer complex. The VTT passive Hessian creates an anchored space for arithmetic. When we use a Schur complement it gets rid of the memory coupling. This results in an operator. The Bloch-Kato Selmer group is a sector, for this operator. The VTT passive Hessian and the Bloch-Kato Selmer group are connected in this way. The free part of the Bloch-Kato Selmer group gives us the Mordell-Weil rank. The finite arithmetic memory is what captures the Tate-Shafarevich and the Tamagawa contributions. The height-energy layer is also important. It connects with the Néron-Tate regulator. This connection happens through Arakelov and nonarchimedean potential theory. The VTT passive Hessian and the Néron-Tate regulator are related in this way. The main result is a rigorous VTT—BSD reduction theorem: after the active determinant germ is compared with L(E,s) at s=1, the BSD rank formula and refined leading coefficient follow. In this paper, we establish the variational, cohomological, height-theoretic and determinant-line structure needed for that comparison and it identifies the remaining analytic theorem.
Category: Algebra

Complexity of Minimization

Authors: Warren D. Smith
Comments: 15 Pages. On internet since 2000; uploading to VIXRA for archival purposes.

An important problem in numerical analysis is trying to minimize (or maximize) a real function F(x₁,x₂,...,x_d) of d real variables. We investigate (and survey) its computational complexity borderlines.

Results include (precise theorem statements are in the text): Minimizing, e.g., functions specified by polynomial formulas involving both trigonometric and ordinary arguments, is undecidable. Hence consider plain rational functions; we'll show minimizing these (more precisely, deciding whether the minimum lies below some threshold) is in NP. If the sum of the numerator and denominator degrees of a (multivariate) rational function is ≤4, then finding all its local minima is in P, except for the 4+0 case of a quartic polynomial (or its reciprocal) in which case it is NP-complete. (There is a slight caveat for the case 3+1; the "polynomial" runtime unfortunately depends also on an additional parameter, but for practical purposes this does not matter.) Indeed, even deciding whether a specified point is a localmin is NP-complete for a quartic. Minimizing a linear function in a cube is polynomial, but minimizing a quadratic in a cube is NP-complete. Minimizing a quadratic in a ball is in P, but it is NP-complete for a quartic (cubics: unknown). In all theseNP-completeness results, even approximating the value of the min is NP-complete, and the problems remain NP-complete even if all integers are input in unary. The P minimization algorithms for low degree rational functions involve new techniques of "multi-stage minimization" and "dimension reduction."

Minimizing functions "unimodal on lines," is in P. Solving systems of nonlinear equations F&8407;(x⃗)=0 is in P, if all the nonlinear functions $F_k are monotonic on lines. We present evidence that minimizing strictly unimodal functions (with exactly one minimum, and no saddlepoints) is exponentially hard; but if additionally the function obeys certain derivative bounds, then hardness seems to occur precisely when there are extremely long "winding valleys."
Category: Data Structures and Algorithms

Some Short Remarks on the Riemannian Hypothesis, Concerning the Zero-Finding Theorem

Authors: Holger Döring
Comments: 3 Pages. In German

At first, this description is no proof but a compact synopsis of the Riemanian prime-number-problem and its zero-states, seen and interpreted with the eyes of a theoretical physicist. Ergo there are mostly themes described, how the theorem can be handled by quantum mechanical descriptions and historical found partial solutions. Nevertheless there are some own ideas in it and in this way it may be that it can be useful as an assist for the goal. Here is the systematic summary of the entire thought experiment. It links every physical concept of hyperbola string of beads model with its exact mathematical equivalent in number theory. A more detailed paper will follow.
Category: Mathematical Physics

Computational Complexity of Synthetic Chemistry Basic Facts

Authors: Warren D. Smith
Comments: 8 Pages. On internet since 1997; uploading to VIXRA for archival purposes. Related paper: Molecular tinkertoys.

Three negative results:

1. Given a set of allowed starting materials and allowed chemical reactions, the problem of determining whether a chemical of specified structure is synthesizable is Turing undecidable.
2. The question of whether it is synthesizable in N steps is NP-complete. These 2 results apply in a general abstract setting and do not prove anything about how hard the problem is with any specific set of starting materials and allowed reactions. In particular we prove nothing about the case when the starting materials are all commercially available chemicals and the reactions are all known reactions!
3. The IUPAC naming scheme for assigning names to organic compounds will assign ambiguous names to most N-atom compounds and requires great chemical expertise. We show that both these problems are avoidable with alternative naming schemes, There are unquestionably {it far} simpler and better methods to name chemical compounds than the IUPAC scheme.

All 3 of these facts are almost trivial from the viewpoint of modern computer science, but I don't think they've been mentioned in the chemical literature.

Positive results: We discuss algorithms for finding out how to solve the synthesis problem (of determining whether something is synthesizable, and what the best synthetic process is). An interesting generalization of the famous "shortest path problem" in directed graphs arises in which the paths become synthetic trees, the graphs become hypergraphs, and the distances become synthetic costs. This problem is soluble either by generalizing Dijkstra's algorithm or the dynamic programming algorithm. Despite the fact that these algorithms are highly efficient as a function of the size of the hypergraph, in practice because the hypergraph arising from chemistry is enormous, that is inadequate. We discuss methods for ameliorating this.
Category: Chemistry

The Payment Verification Model for Digital Cash Point of Sale Systems

Authors: Lukas Saul, Robert Gludo, Jonathan Gludo
Comments: 6 Pages.

With public ledger digital cash payments comes a new type of point of sale (POS) system which has been implemented by several devices and apps. The "payment verification model" for POS minimizes security risk and associated fees while enabling merchants to adopt their point of sale system to different numéraires and different payment currencies. In this paper we outline this system and its advantages and operation. We provide the first systematic security taxonomy, privacy mitigation strategies, and accounting integration guidance specifically for payment verification POS systems.
Category: Economics and Finance

Relativistic Geometrical Horizons(rgh) in Both the Original Versions of Alcubierre and Natario Warp Drives

Authors: Fernando Loup
Comments: 26 Pages.

In 1994 Alcubierre developed the first warp drive work using the original 3+1 ADM-MTW formalism.Seven years later in 2001 the same original 3+1 ADM-MTW formalism appeared in the first part of the second warp drive work developed by Natario. But in 1997 the concept of the Relativistic Geometrical Horizon RGH was introduced by Hiscock in his work about the Alcubierre warp drive. According to Hiscock if the time component of a given spacetimemetric tensor becomes null g_(00)=0 a Horizon similar to the Event Horizon of the Schwarzschild black hole appears. The Alcubierre warp drive suffers from this pathology because in a given point of the Alcubierre warped regionthe g_(00)=0 appears generating the Alcubierre Horizon equivalent to the Event Horizon of the Schwarzschild black hole.The Natario warp drive was formulated in polar coordinates in a2+1 spacetime.A dimensional reduction of the Natario warp drive from a 2+1 to a 1+1 spacetimes in a given point of the Natariowarped region the g_(00)=0 appears generating the Natario Horizon in a 1+1 spacetime.But in the original Natario warp drive in polar coordinates in a 2+1 spacetime the g_(00)=0 never appears.Due to the presence of a second spatial dimension the g_(00)=0 can be avoided.
Category: Relativity and Cosmology

Eddington’s Mass-Luminosity Relation and the Laws of Thermodynamics

Authors: Stephen J. Crothers, Pierre-Marie Robitaille
Comments: 8 Pages.

Ever since its formulation by A. S. Eddington, the mass-luminosity relation has been viewed as a triumph for theoretical astronomy and astrophysics. The idea that the luminosity of the stars could be controlled solely by their mass was indeed a revolutionary concept. The proof involved two central aspects: (1) the belief that stars could be treated as ideal gases in hydrostatic equilibrium, and (2) that the opacity of Capella could be used as a reference mark applicable to other stars. Yet, when the mass-luminosity relation was advanced, no thought was given to the need for thermodynamic balance. Within thermodynamic expressions, not only must the dimensions (hence units) be consistent on each side of the equals sign, but the extensive nature of the properties must also balance. Namely, thermodynamic expressions must be balanced by properties which are extensive to the same degree. In this regard, mass is an extensive thermodynamic property and can be represented by a homogenous function of degree 1. Conversely, the luminosity of a star is neither extensive nor intensive, but rather can be represented by a homogenous function of degree 2/3. Consequently, the mass-luminosity expression is thermodynamically unbalanced and stands in violation of the laws of thermodynamics.
Category: Astrophysics

The Electrodynamic Origin of Black Holes (Part 1)

Authors: Raul Fattore
Comments: 17 Pages.

The electrodynamic origin of black holes as well as the electrodynamic origin of the gravitational force developed in the present study will contribute to a better understanding of how our universe works and obeys the real-world physics demonstrated by the more than proven laws of electrodynamics.This study presents a new perspective that will clarify and demystify many aspects about the origin of the gravitational force through a newly derived equation with terms that are absent in basic Newton’s gravitational law. It will also be demonstrated the decay of the gravitational force, the origin of black holes, what they really are, how they are formed, how they evolve, the real radiation spectrum, what particles are the constituents of black holes, what particles are to be found outside a black hole, the wavelength shift (or redshift) of black holes, and other properties.- Do "receding" galaxies really mean that the universe is expanding?- Is it scientifically serious to accept that gravity is not a force, but a "geometrical effect" caused by mass?- Is it scientifically serious to accept that "geometrical gravity" lacks a unique unit of calculation, because it depends on what is being calculated?- Can it be scientifically acceptable that a lump of Newtonian mass bends EM radiation?- Can monochromatic radiation emission from black holes be scientifically acceptable?- Can it be scientifically acceptable that some types of black holes do not rotate?
Category: Astrophysics

The Electrodynamic Origin of Black Holes (Part 2)

Authors: Raul Fattore
Comments: 37 Pages.

The electrodynamic origin of black holes as well as the electrodynamic origin of the gravitational force developed in the present study will contribute to a better understanding of how our universe works and obeys the real-world physics demonstrated by the more than proven laws of electrodynamics.This study presents a new perspective that will clarify and demystify many aspects about the origin of the gravitational force through a newly derived equation with terms that are absent in basic Newton’s gravitational law. It will also be demonstrated the decay of the gravitational force, the origin of black holes, what they really are, how they are formed, how they evolve, the real radiation spectrum, what particles are the constituents of black holes, what particles are to be found outside a black hole, the wavelength shift (or redshift) of black holes, and other properties.- Do "receding" galaxies really mean that the universe is expanding?- Is it scientifically serious to accept that gravity is not a force, but a "geometrical effect" caused by mass?- Is it scientifically serious to accept that "geometrical gravity" lacks a unique unit of calculation, because it depends on what is being calculated?- Can it be scientifically acceptable that a lump of Newtonian mass bends EM radiation?- Can monochromatic radiation emission from black holes be scientifically acceptable?- Can it be scientifically acceptable that some types of black holes do not rotate?
Category: Astrophysics

Best Play for Imperfect Players and Game Tree Search: Part I (Theory)

Authors: Eric B. Baum, Warren D. Smith
Comments: 41 Pages.

This is part I (theory) of a 2-part paper about a new, non-minimaxing, approach to search in game-trees such as Chess and Go. Its first point is that minimaxing is NOT the best decision-making (i.e. chess-move-selection) process based on inexact merely-statistical estimates about the "goodness" of the chess positions at the leaves of a search tree. Its second point is that going down the game tree to some fixed depth, then stopping, is NOT the best choice of chess-positions for your search to explore. "BPIP search" is a way to make better decisions based on any particular search tree, and a way to explore better trees. As of year 2026, BPIP still has not yet been used by chess and go programs, but if were might enable them to get substantially stronger. Would it? I would like to know, but never found out because I abandoned my attempts to write strong chess and go programs as just too difficult a programming task for me.

I originally wrote this paper and put it on the internet in 1993, and at that time "Monte Carlo game tree search" and today's highly advanced selective alphabeta variants did not yet exist, computer chess was weaker than the best humans, and computer go far weaker than decent human amateurs. Later, go programmers invented "Monte Carlo game tree search" which can be regarded as a "poor man's BPIP" (although they did not realize that). MCGTS has been used in go programs with great success.
Category: Artificial Intelligence

Fermi-Born-Infeld Electrodynamics: a Nonlinear Theory with Physical Gauge

Authors: Renato Vieira dos Santos
Comments: 12 Pages.

We construct a nonlinear extension of Fermi's electrodynamics by incorporating a Born--Infeld structure that depends directly on the four-potential $A_mu$ rather than on the field strength $F_{muu}$. The resulting theory, which we call Fermi--Born--Infeld (FBI) electrodynamics, eliminates the $U(1)$ gauge redundancy by elevating the Lorenz gauge to a dynamical condition. The Lagrangian is built from the determinant of a metric-like tensor $g_{muu} = eta_{muu} + 2kappa, partial_{(mu} A_{u)}$, ensuring that the canonical energy--momentum tensor and the spin density remain unique and free of gauge ambiguities. We derive the field equations, which reduce to $partial_u(sqrt{-g}, g^{muu}) = 0$, and show that the Lorenz condition $partial_mu A^mu = 0$ emerges dynamically from retarded boundary conditions and the requirement of a positive-energy spectrum. The nonlinearities modify the propagation of longitudinal modes; we argue, via a Vainshtein-like mechanism, that the nonlinear self-interactions may stabilize the longitudinal mode, opening the possibility of a stable massive scalar photon under extreme field conditions. We also compute the spin density from the Noether current and discuss its properties. The FBI theory preserves the physical gauge of Fermi's original formulation while incorporating the regularization features of Born--Infeld electrodynamics, making it a candidate for describing electromagnetic phenomena in strong-field regimes.
Category: Relativity and Cosmology

Best Play for Imperfect Players and Game Tree Search: part II (experiments)

Authors: Warren D. Smith, Eric B. Baum, Charles Garrett, Rico Tudor
Comments: 39 Pages.

This is part II (experiments) of a 2-part paper about a new, non-minimaxing, approach to search in game-trees such as Chess and Go. Its first point is that minimaxing is NOT the best decision-making (i.e. chess-move-selection) process based on inexact merely-statistical estimates about the "goodness" of the chess positions at the leaves of a search tree. Its second point is that going down the game tree to some fixed depth, then stopping, is NOT the best choice of chess-positions for your search to explore. "BPIP search" is a way to make better decisions based on any particular search tree, and a way to grow better trees to explore. As of year 2026, BPIP still has not yet been used by chess and go programs, but if were might enable them to get substantially stronger. Would it? I would like to know, but never found out because I abandoned my attempts to write strong chess and go programs as just too difficult a programming task for me. However in this paper we did program BPIP and alphabeta gameplayers for a number of simpler games, achieving strengths apparently stronger than any human and also among the strongest year-1995 computer programs. For some games BPIP indeed outperformed alphabeta, sometimes hugely. Games large enough that full solves of game positions are usually out of reach, and which can have "smart and slow" position evaluators, and in which large searches are done, are the games favoring BPIP versus alphabeta.

I originally wrote this paper and put it on the internet in 1995, and at that time "Monte Carlo game tree search" and today's highly advanced selective alphabeta variants did not yet exist, computer chess was weaker than the best humans, and computer go far weaker than decent human amateurs. Later, go programmers invented "Monte Carlo game tree search" which can be regarded as a "poor man's BPIP" (although the MCGTS inventors did not realize that). MCGTS has been used in go programs with great success.
Category: Artificial Intelligence

The Influence Mechanism of Weight Allocation and Test Set Design on Open-Source Large Model Evaluation Results

Authors: Dong Zhang, Yang Sun, Fei Lyu, Xiaofeng Liu, Xin Li
Comments: 22 Pages.

The evaluation results of open-source large language models are not only influenced by model parameter scale, training corpus, and alignment strategies, but also significantly constrained by the test set architecture and the weight allocation of evaluation metrics. To address this issue, this paper leverages multiple types of real-world evaluation data collected from engineering projects to investigate the evolution mechanism of open-source large model evaluation scores and rankings under the coupled effects of weight allocation and test set design. This paper selects four mainstream frontier open-source models—DeepSeek-V3.2, MiniMax-M2.5, Qwen3.6-35B-A3B, and gpt-oss-120b—to conduct controlled experiments. All models uniformly employ 8 96G H20 GPUs for local offline inference evaluation, and weighted aggregation is performed after unifying the scoring criteria to eliminate evaluation interference caused by third-party API rate limiting, version differences, and inconsistent scoring standards. The native capabilities of the models are supplemented with official model card information from the ModelScope platform: MiniMax-M2.5 focuses on code engineering, intelligent agents, and office interaction, with inference speed improved by 37% over the previous generation and inference cost at only 10% of Claude Opus 4.6; gpt-oss-120b is an OpenAI open-source MoE architecture model that natively supports hierarchical reasoning, tool invocation, and MXFP4 quantized deployment. The experimental datasets cover five major tasks—social bias discrimination, common misconception fact-checking, lifestyle question answering, contextual semantic understanding, and basic logical reasoning—corresponding to five evaluation capabilities: bias identification, fact verification, knowledge response, contextual interpretation, and mathematical reasoning. Specifically, this paper operationalizes "task type"as an observational variable into five categories of questions: ：social bias discrimination is used to measure the Bias Identificationconcept（characterized by bias judgment accuracy and harmful tendency response rate），common misconceptionFact-Checkingfact-checking is used to measure the fact verification concept（characterized by fact judgment accuracy），lifestyle question answering is used to measure the knowledge response concept（characterized by answer correctness rate and completeness score），contextual semantic understanding is used to measure the contextual interpretation concept（characterized by semantic consistency and coreference resolution accuracy），basic logical reasoning is used to measure the mathematical reasoning concept（characterized by reasoning step accuracy and final conclusion accuracy）。This paper constructs three evaluation weighting schemes: average baseline weights、performance-oriented weights、, and robust compliance-oriented weights，to analyze the variation patterns of model tier rankings under differentiated weight allocations。The experimental results show that：when the test set task categories are homogeneous、and question types are uniform，model rankings are highly susceptible to local sample perturbations；test sets with diverse task types、and balanced difficulty levels，exhibit stronger evaluation stability，and can objectively characterize comprehensive model capabilities。The study further confirms that，metric weights、sample structure、and task distribution are not auxiliary evaluation variables，but core elements determining the credibility of evaluation conclusions、and applicability。
Category: Artificial Intelligence

Non-Hermitian Quantum Mechanics Fundamentals & Lennard-Jones Potential

Authors: Larry Lee, Preet Sharma
Comments: 8 Pages.

Non-Hermitian quantum mechanics is a fairly new field and has been of great interests to theorists and experimentalists alike. In this study we have shown the basics of Non-Hermitian quantum mechanics and have put forward a scenario in which Non-Hermitian quantum mechanics can be applied to the well known Lennard-Jones potential. We have not shown explicitly what the results would be but when we apply Lennard-Jones potential to the Non-Hermitian scenario, but we have given one idea as to how it can play a part. This idea is also our continuing research.
Category: Quantum Physics

A Network-Based Microscopic Interpretation of Vacuum Localised Structures

Authors: Rudi Van Nieuwenhove
Comments: 10 Pages.

Vacuum Localised Structures (VLS) have recently been proposed as self-gravitating solutions of the Einstein field equations. Such structures have been investigated in connection with galactic dark matter distributions, filamentary gravitational configurations, and black-hole-like vacuum solutions without baryonic matter sources. Although the corresponding solutions are mathematically consistent within General Relativity, the physical interpretation of the underlying stress-energy tensor remained unclear.In the present work, a microscopic interpretation of VLS structures is explored using a dynamical network framework. Space is interpreted as an emergent structure arising from an underlying discrete connectivity network, in which effective metric properties are associated with local link configurations and shortest-path relations. Curvature is interpreted as a spatial variation of effective network connectivity.Within this framework, a VLS is described as a self-sustained coherent network configuration maintained by local reconnection dynamics. This surface-dominated dynamical behaviour naturally leads to the effective equation of state in which the radial pressure equals minus one-third of the energy density, thereby providing a possible microscopic interpretation of the peculiar stress-energy properties associated with VLS solutions.
Category: Relativity and Cosmology

Introduction to Pseudo Vector Theory of the Mediating Medium for Dark Matter

Authors: J. W. A. Zwart
Comments: 12 Pages. (Note by viXra Admin: Please cite and list scientific references)

The dark matter mediating medium for the pseudo vectors carries by definition four independent vector cells of the components for acceleration, spin and precession making up twelve combinations of pseudo vectors. The conjugated medium at the absolute zero energy state consists of solely pseudo e-neutrino triplets as a dynamic non destructible equilibrium in cubic pyramid symmetry. At a relativistic thermodynamic level of electron energy the pseudo vector medium consists of pseudo μ-triplets in cubic symmetry mixed to triplet ensembles of τ-pseudo triplets in cubic symmetry together guided by the pseudo e-neutrinos. Both the μ- and τ- triplets carry always the rotations for τ-triplets perpendicular to the μ-triplets suggesting orthogonal cubic pyramid symmetry. The top in equilateral pyramid symmetry of equal dimension for the base triangle in the cubic symmetry provides the dynamic electric charge in one polarity following the acceleration vector component of the pseudo vector cells. The time intervals for the electric charge dynamics give the sequential conjugated inversion shifts along the rotation axis of the height of the equilateral pyramid. Common sense reasoning of some scaling calculations shows the validity of the statement in this abstract explained in the paper below.
Category: High Energy Particle Physics

Quaternions, Octonions, and Now, 16-Ons and 2^n-Ons: New Kinds of Numbers

Authors: Warren D. Smith
Comments: 68 Pages. Placing this 2004 paper on vixra for archival purposes.

"Cayley-Dickson doubling," starting from the real numbers, successively yields the complex numbers (dimension 2), quaternions (4), and octonions (8). Each contains all the previous ones as subalgebras. Famous Theorems, previously thought to be the last word, state that these are the full set of division (or normed) algebras with 1 over the real numbers. Their properties keep degrading: the complex numbers lose the ordering and self-conjugacy (x̅=x) properties of the reals; at the quaternions we lose commutativity; and at the octonions we lose associativity. If one keeps Cayley-Dickson doubling to get the 16-dimensional "sedenions," zero-divisors appear.

We introduce a different doubling process which also produces the complexes, quaternions, and octonions, but keeps going to yield 2ⁿ-dimensional normed algebraic structures, for every n≥0. Each contains all the previous ones as subalgebras. We'll see how these evade the Famous Impossibility Theorems. They also lead to a rational "vector product" operation in 2^k-1 dimensions for each k≥2; this operation is impossible in other dimensions.

But properties continue to degrade. The 16-ons lose distributivity, right-cancellation yx·x^-1=y, flexibility a·ba=ab·a, and antiautomorphism (c̅=b̅a̅ where c=ab). The 32-ons lose the properties that the solutions of generic division problems necessarily exist and are unique, and lose the "Trotter product limit formula." We introduce an important new notion to topology we call "generalized smoothness." The 2ⁿ-ons are generalized smooth for n≤4.

All 2ⁿ-ons have 1 and obey numerous identities including weakenings of the distributive, associative, and antiautomorphism laws. In the case of 16-ons these weakened distributivity laws characterize them, i.e. our 16-ons are, in a sense, unique and best-possible. Our 2ⁿ-ons also are unique, albeit in a much weaker sense. The 2ⁿ-ons with n≤4 support a version of the fundamental theorem of algebra. Normed algebras (rational but not nec. distributive) over the reals are impossible in dimensions other than powers of 2.
Category: Algebra

Embedding Space Construction of Conformal Field Theory From First Principles

Authors: Sutirtha Mukherjee
Comments: 104 Pages. (Note by viXra Admin: Please cite listed scientific references)

This work presents a systematic development of conformal field theory from first principles in quantum field theory, emphasizing complete derivations of the conformal group in arbitrary dimensions from the conformal Killing equation, proving its isomorphism to $mathrm{SO}(d+1,1)$, and develop the embedding space formalism as a tool for constructing conformally covariant correlation functions constrained by Ward identities. In two dimensions, conformal symmetry enhances to an infinite-dimensional algebra: we derive the Virasoro algebra $[L_m, L_n] = (m-n)L_{m+n} + frac{c}{12}(m^3-m)delta_{m+n,0}$ from infinitesimal generators $ell_n = -z^{n+1}partial_z$, computing the central charge explicitly for free theories (boson $c=1$, Majorana fermion $c=1/2$, $bc$-ghosts $c = 1-3(2lambda-1)^2$) and demonstrating its physical interpretation as vacuum energy. Through radial quantization, we establish the operator-state correspondence and construct the Hilbert space via Virasoro descendants of primary states $|h,bar{h}angle$, deriving the transformation law $T'(w) = (dw/dz)^{-2}[T(z) - (c/12){w;z}]$ that reveals the geometric origin of the conformal anomaly. Unlike conventional treatments, we develop all mathematical machinery from standard QFT without relying on unexplained results from string theory or representation theory, exhibiting all intermediate steps explicitly to provide a detailed continuation beyond introductory QFT texts.
Category: Quantum Gravity and String Theory

Entanglement, Common-Origin Correlations, and the Interpretation of Bell Violations

Authors: Ion Vlad
Comments: 9 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)

Bell's theorem demonstrates that no local deterministic theory can reproduce the complete set of correlations predicted by standard quantum mechanics. The present work accepts Bell's theorem as mathematically correct but questions the physical interpretation of the quantum correlation function used in Bell analyses. It is argued that entangled particles inherit a complete common physical state at their creation and that subsequent measurements reveal selected observables of this state rather than generating new information or requiring superluminal communication. The paper further proposes that the standard quantum correlation function, (E(θ)=−cosθ), incorporates additional angular dependence beyond the information physically established at the source. Under this interpretation, the observed violation of Bell inequalities arises from the mathematical structure of the correlation function rather than from nonlocal information transfer between distant particles. An explicit local common-source model is presented, leading to a piecewise linear correlation function and motivating a decomposition of the standard quantum prediction into a local component and a supplementary angular term.
Category: Quantum Physics

Ramanujan Transformations and Robin Defects

Authors: Payam Danesh
Comments: 9 Pages.

We studied the Robin defect associated with the inequality σ(n)<e^γ nlogu2061logu2061n, express its Laplace transform through Ramanujan’s transformation for the divisor Lambert series and isolate the precise difference between smoothed positivity and coefficientwise positivity. The main results give us an exact Ramanujan-transformed identity for the Robin defect, an equivalent coefficientwise formulation of the Riemann Hypothesis showing why transform-level positivity cannot by itself prove the hypothesis and an extremal reduction to colossally abundant and highest abundant numbers. Numerical data for early extremal integers illustrate how the normalized defect behaves past the exceptional value 5040. Ramanujan’s identities provide powerful global control, but the Riemann Hypothesis requires pointwise positivity at the extremal divisor-rich integers.
Category: Number Theory

Optimization Model for Polyline Offset

Authors: Zitao Xu
Comments: 7 Pages.

Classical offset algorithms focus on constructing offset curves and surfaces. Optimization-based geometry processing has also been extensively studied for mesh fairing, deformation, and parameterization. The present work differs from both directions by introducing a least-squares formulation whose unknowns are tangential displacements of offset vertices, yielding a globally optimized offset configuration.
Category: Geometry

Analyzing a Variant Version of a Fibonacci Polynomial

Authors: Marciano L. Legarde
Comments: 12 Pages.

In this paper, we introduce a new finite polynomial called the "Variant Fibonacci Polynomial". This polynomial is defined by using the standard Fibonacci sequence as both the coefficients and the exponents of each term. Although the definition is simple, it leads to several interesting mathematical properties.We begin by defining the polynomial and giving several examples. We then evaluate it at different values of x, identify its guaranteed real root, discuss its complex roots, and derive formulas for both its derivative and antiderivative. We also examine the infinite series version of the polynomial and observe when it converges and when it diverges.Finally, we discuss whether this polynomial could be used to construct a new type of series expansion for functions and whether such an expansion might eventually be useful for solving differential equations. While it is still too early to draw definite conclusions, the results of this paper suggest that the Variant Fibonacci Polynomial is an interesting mathematical object that deserves further study.
Category: General Mathematics

On a Quadratic Collatz Type Map Governed by Prime Factorization: A Conjectural Universal Cycle, Two Reduction Theorems and an Intrinsic Computational Barrier

Authors: Christoper Mututu
Comments: 36 Pages.

We study a map T on the integers defined by T(n)=n^2+1 when n is prime or even and T(n)=n/P(n) when n is odd composite, where P(n) denotes a designated prime factor n. Two variants arise according to the choice of P(n). In Part I, P(n) is the largest prime factor of n while in Part II, it is the smallest. Informally, the conjecture of this paper asserts that every integer greater than one in absolute value eventually enters the single twelve element cycle5→26→677→458330→210066388901→52357→41→1682→2829125→1625→125→25 and remains there forever. Formally, we verify this for every integer n∈[2,1000] under Part I with no exception and no alternate cycle observed and we conjecture it holds for every integer n in the domain Z^*=Z {-1,0,1} but we do not prove it.Toward this conjecture, we prove two theorems. The first is exact rather than asymptotic. For any odd composite m, repeated application of the largest prime factor reduction reaches a prime in precisely Ω(m)-1 steps where Ω(m) counts the prime factors of m with multiplicity. Each application removes exactly one element from the prime factorization multiset m so the count decreases by exactly one per step and terminates uniquely at a prime. The second theorem follows from the first. Under the extension of primality to negative integers through |n|, which is the only convention under which the conjecture is well posed on Z, every negative integer reaches a positive value within at most Ω(|n|) steps. This reduces the negative integer case of the conjecture entirely to the positive integer case. We also identify an obstruction to further verification that is structural rather than a matter of computing resources. The reduction step of T requires the complete factorization of the input which is a problem for which no general sub exponential algorithm is known. Repeated application of the squaring branch can therefore produce integers whose factorization lies beyond any presently known method regardless of computing time available. Our verification required factoring intermediate values of up to 96 digits and succeeded in every instance though with no guarantee that a harder instance does not arise beyond the tested range. For Part II, this obstruction is severe enough to foreclose even a conjecture. Every trajectory examined exceeded 100 digits within fewer than 25 iterations without any value repeating. We are unable to characterize the long-term behavior of Part II by any method available to us and as a result, Part II is entirely open.
Category: Number Theory

The Giza Pyramid Complex as an Analogy to the Voyager Mission

Authors: Andrey V. Voron
Comments: 8 Pages.

This paper explores a potential analogy arising from a comparison between the space missions of Pioneer 10, Pioneer 11, Voyager 1, and Voyager 2 on one hand, and the Giza pyramid complex project on the other. Within this framework, five plausible assumptions are examined. Based on their outcomes, the study demonstrates the probable message intended by the pyramid builders and identifies a potential recipient for this transmission (a planet of a specific star).
Category: Archaeology

Artificial General Intelligence (AGI)

Authors: Clark M. Thomas
Comments: 4 Pages.

The sudden injection of giant data server farms into the world’s advanced economies, especially during 2026, is unique in human history. Humanity’s lofty position as the only global hyperkeystone species could be challenged by the emergence of a consciousness not fully human, nor what actual space aliens would bring to ourgeneralized intelligence. Here are some key aspects of this unique cybernetic challenge that we must soon face.
Category: Artificial Intelligence

A Quantum Induced Wapdrive II Subspacestructure, Dilaton-Field (Φ) and Their Cosmic Connections

Authors: Holger Döring
Comments: 26 Pages.

In a former paper a quantum induced warp-drive was introduced by assuming a model of microscopic cylinders as spacelike dimensions whereby the timelike dimension remains at its classical one-dimensional state. Described now is more detailed the underlying fundamental physical and mathematical spacetime structure than in the first paper , which is announced to feature this quantum induced warp-drive concept. Although there are certainly some new elements involved, everything is based on very classical GRT and QTH- descriptions. A more detailed description of the coupling dilaton-field is made, which connects the macroscopic sector of GRT with the microscopic sector of QFTH. This dilaton-field shall be named a "barytic -field".
Category: Quantum Gravity and String Theory

Comment on "Memories of Amplitude and Direction Coexist and Compete in Non-Brownian Suspensions"

Authors: Vladimir Kuz'menko
Comments: 3 Pages.

A recent article [Phys. Rev. Lett. 136, 258201 (2026)] continued the experimental study of an interesting physical phenomenon, where the viscosity of a suspension can be greater when moving forward than when moving backward. The authors attribute this phenomenon to the presence of a certain memory in the environment. The physical nature of both the phenomenon itself and the supposed memory is unknown. A possible explanation of the physical nature of this effect is proposed here and simple experiments to study some properties of such non-local memory are discussed.
Category: Quantum Physics

The Simple Geometry of Time: A Physical Record of Cosmic Metric Transformation

Authors: Alata Elatawneh
Comments: 26 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org) Repository URL: https://github.com/Elatawneh/UCK-Cosmology-Framework.git

We present a predictive, geometrically constrained cosmological architecture, designated the Unified Cosmic Kinematic (UCK) framework, derived from the foundational postulate that cosmic time is macroscopically identical to relational metric spatial expansion. The invariant comoving time derivative of this identity demonstrates that the unperturbed background expansion parameter is the analytical inverse of the cosmic epoch, yielding a global background baseline constant of 70.85 km/ s/ Mpc when anchored to the empirically observed cosmic age of 13.8 Gyr. Although the baseline expansion model is geometrically deterministic, mapping observational signatures requires accounting for the interaction of matter and radiation within the spatial manifold via an effective field theory approach. This framework introduces four native tracking parameters calculated from first principles via the optimization of unbinned data streams, independent of external fluid-driven dark sector parameters. The statistical viability of this architecture is evaluated against five high-precision, unbinned astronomical data pipelines spanning late-universe standard candles, local velocity anomalies, localized mass-deflection geometries, the cosmic microwave background, and deep spectroscopic horizons. Our optimization routines address the persistent Hubble tension by demonstrating that regional gravitational density induces a localized clock-drag fraction that elevates the apparent local expansion rate via gravitational time dilation. This exact localized clock-drag metric reconciles weak-field gravitational limits, precisely mapping the mass-deflection profiles of strong gravitational lensing systems under pure baryonic constraints without invoking dark matter halos. Finally, coupling this expansion history to the primordial plasma sound horizon maps the early acoustic peaks with high statistical fidelity. The resulting high-redshift temporal elongation extends the available structural development timeline, thereby addressing the early mature galaxy chronological discrepancy across all targeted spectroscopic horizons.
Category: Relativity and Cosmology