Abstract: We present a self-contained gauge-sector account of the octonionic programme, starting from the underlying trace-dynamics Lagrangian and ending with closed-form expressions for the strong and electromagnetic couplings, together with a brief review of the weak mixing angle. The derivation has three steps. First, inside the visible bosonic sector we derive the broken-phase relation αsαem=16, from a single visible Yang--Mills coupling before symmetry breaking. The mechanism combines the standard visible charge-trace factor \( 8/3 \) with a six-direction support factor \( 6 \) on the real octonionic ladder space \( H_6 \). Second, we recall the 2022 Eur. Phys. J. Plus. paper [1], where the minimal visible charge quantum \( q_0=1/3 \) fixes the exponential seed A:=exp[q0(q0−38)]=exp[13(13−38)]. Combining this seed with the charged-sector datum \( 3/8 \) gives αs\thv(MZ)=964exp[23(13−38)]=0.11675418, while the broken-phase factor \( 16 \) then yields αem\thv(0)=91024exp[23(13−38)]=0.00729713629. Third, we briefly review the earlier spinorial derivation of the weak mixing angle~\cite{RajSinghBosonic}, which leads to 1=cos⁡(θW/2)2+sin⁡(θW/2),sin2⁡θW\thv=0.24969776. A key conceptual point is that the seed is attached to the \emph{minimal visible charge quantum} \( q_0=1/3 \), not to a specific particle species. The electron, whose charge is \( 1=3q_0 \), is not omitted: its contribution enters explicitly through the electromagnetic charge trace \( k_{\mathrm{em}}=8/3 \). In this form the derivation of $\alpha_{\mathrm{em}}$ is conceptually sharper than in the earlier Eur. Phys. J. Plus. presentation [1], because the factor\( 1/16 \) is no longer hidden in a length-identification step but is derived directly from the visible broken-phase gauge structure.

Abstract: We present a novel proposal for the effective Lagrangian of the low-energy Yang–Mills quantum field theory. The proposed effective Lagrangian exhibits the spontaneous BRST symmetry breaking. We built the Fujikawa model that we couple to the Yang–Mills elementary field sector, motivated by the analogy with Chiral Quark Model. We interpret the Fujikawa fields as effective fields composite of the elementary gluon and ghost fields. In order to justify the existence of two massless Nambu–Goldstone modes among the Fujikawa fields, we require not only the BRST but also the anti-BRST invariance of the effective Lagrangian, both to be spontaneously broken. The most striking consequence of that is the emergence of the effective gluon and ghost masses. We reproduce the Curci–Ferrari model as a special case of our effective model upon the spontaneous BRST symmetry breaking. In order to reproduce also the non-nilpotent modified BRST symmetry, characteristic for the Curci–Ferrari model, we modify our effective Lagrangian to be invariant with respect to the extended-BRST symmetry, which mixes the elementary and Fujikawa field sectors, and which is nilpotent. The Curci–Ferrari is reproduced by the elementary field sector of the resulting Lagrangian. The remaining Fujikawa’s field dependent terms guarantee the underlying nilpotent extended-BRST symmetry, which is now hidden in the sense of the spontaneous symmetry breaking.

Abstract: In 1993 Sorkin showed that extending textbook ideal measurements to relativistic quantum fields conflicts with locality as for generic observables, the state-update associated with an ideal measurement can transmit information faster than light, even when the intervening and readout regions are spacelike separated. We reformulate this tension in covariant string field theory, where the fundamental degrees of freedom are extended and the spacetime description involves operators with intrinsic string-scale nonlocality. We show that the paradox relies on the existence of sharply localized projectors and an exact Lüders reduction rule, neither of which is operationally realized in string field theory. This yields an operational notion of locality that resolves Sorkin’s signaling channel while recovering local quantum field theory in the limit of vanishing string length.

Abstract: The heterotic E₈ ×E₈ string and the octonionic E₈ ×E₈ unification program share a genuine exceptionalalgebra corridor: both pass through the branching E₈ ⊃ E₆ × SU(3), both naturally encounter trinification-type decompositions of E6, both make nontrivial use of the doubling E₈ ×E₈, and both touch ten-dimensional Lorentzian structures for which octonions provide a remarkably efficient language. These parallels are real, but they do not amount to a dynamical identification of theories. Heterotic E₈ ×E₈ is a mature string construction with a worldsheet formulation, anomaly cancellation, and a large compactification literature; the octonionic program is instead an emergence-first, pre-spacetime framework in which quantum theory and gravitation are intended to arise together from a deeper noncommutative and nonassociative substrate. The aim of the present note is therefore deliberately modest and explicit. It does not place the two frameworks on equal technical footing, and it does not claim an equivalence. Rather, it isolates the precise algebraic and geometric points of contact, gives one worked example of a common branching datum, explains why the Distler-Garibaldi no-go theorem does not directly address the octonionic construction, and formulates a concrete checklist for what a genuine heterotic-to-octonionic dictionary would have to achieve. We then discuss what octonionic input could plausibly contribute to heterotic predictivity, and what obstacles remain genuinely dynamical. The conclusion is cautious: the overlap is stronger than a slogan, weaker than an equivalence, and best understood as a research map rather than a completed translation.

Abstract: Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed partial derivatives on coordinates. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations is presented, which can serve as a model of the graviton. The results obtained may prompt changes in some aspects of the construction of Feynman diagrams. Among other things, it presents a purely mathematical derivation of Maxwell's inhomogeneous equations without reference to empirical data on the action of electric current, which is usually referred to when deriving equations. Section 4 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor of a charged particle having four indices rather than two. In the case where a particle of matter has a charge, another Riemann tensor is added to Einstein’s equation, in addition to the usual Riemann curvature tensor, which describes the additional curvature created for a charged particle by an electromagnetic field.

Abstract: Two postulates—the massless wave equation \square \phi = 0 and Minkowski spacetime—determine a reproducing- kernel Hilbert space (RKHS) on the celestial sphere S^2 whose evaluation functional \mathrm{ev}_0 at the null- cone coincidence point categorically encodes the physics of all massless fields. We embed this structure in the six- dimensional null cone of \mathbb{R}^{4,2} , where \mathrm{SO}(4,2) acts linearly and the Aldaya obstruction disappears. We prove the exact partial- wave decomposition of \delta (\sigma^2) : the spectral function is f(\ell ;u_0) = P_\ell (u_0) with u_0 = (r^2 + r'^2 - \tau^2) / (2rr') , giving P_\ell (1) = 1 on the null cone—the Isometric Sampling Condition (ISC). The proper- time spectral sum \Sigma^{(4)}(t) = \cosh (t / 2) / [2\sinh^2 (t / 2)] has Laurent coefficient ratio c_{- 2} / c_0 = 24 = \chi (K3) . Through the identity B(1 - \alpha_t,1 - \alpha_s) = \int_0^\infty dr e^{- (1 - \alpha_t)r}(1 - e^{- \tau})^{- \alpha_s} , we establish that the Veneziano amplitude is the Laplace transform of the null- cone heat kernel, with the ISC emerging at \alpha_s = 1 where the Pochhammer weights (1)_n / n! = 1 . We identify this structure with the observer encoding map \hat{V} of Harlow, Usatyuk, and Zhao for quantum gravity in a closed universe, where the one- dimensional Hilbert space corresponds to P_\ell (1) = 1 and the observer's bandwidth L grows a Hilbert space of dimension (L + 1)^2 . We establish a 25- entry dictionary with Nagano's K3 lattice theory. The \mathrm{SL}(2,\mathbb{C}) holonomy R(2\pi) = -\mathbb{I} at the null- cone branch point yields the spin- statistics theorem. All numerical results are verified to precision < 10^{- 13} .

Abstract: We present the strongest journal-safe version of the copy-time program that can be defended frompublic data without manufacturing unsupported claims. The paper has three central contributions.First, we formulate a minimal axiom set under which the chain τ_copy ⇒χ(b,s) ⇒A(s,t) becomesquasi-inevitable rather than optional: positivity, radiality, finite second moment, a Born-dominantforward-and-first-dip window, self-similar scaling, and minimal finite Hankel closure with one simplenode force the first nontrivial Laguerre sector and therefore a polynomial-times-Gaussian amplitude.Second, we derive conditional structural theorems for the forward slope, the dominant dip scale,forward curvature, and a dimensionless drift observable O_excl = ∆[B₀|t|_dip] which vanishes forthe full one-scale geometric class but is generically nonzero in the copy-time closure whenever thedeformation parameter runs with energy. Third, we confront the closure with a stricter public-data benchmark. On a closed pointwise fit to 83 public differential-cross-section points at 2.76and 13 TeV, constrained by an affine log-energy flow, the two-scale copy-time surrogate yieldsχ²_log = 461.19, AIC = 487.19, and BIC = 518.64, to be compared with (60493.49,60507.49,60524.43)and (59942.77,59964.77,59991.38) for two nested one-scale competitors. The corresponding Akaikeweight is numerically unity at machine precision. We also state a covariance-robustness propositionand an explicit preregistered style forecast at 14 TeV. The resulting conclusion is deliberately bounded:within the explicit axiomatic closure and benchmark families studied here, the copy-time programis mathematically constrained, experimentally testable, and decisively preferred; the manuscriptdoes not claim a historically pre-published successful prediction, external third-party validation, oruniversal dominance over every hadronic model in the literature.

Abstract:

We construct a quantum Yang--Mills theory with gauge group G (any compact simple Lie group) on four-dimensional Minkowski spacetime \( M^{3,1} \), starting from the retarded Green's function \( G_{\mathrm{ret}} = (2\pi)^{-1}\delta(\sigma^2)\theta(\Delta t) \) and a compact simple Lie group G with \( C_2(G) > 0 \). The construction proceeds entirely on the null cone. The partial-wave decomposition on the celestial sphere S² yields the Isometric Sampling Condition (ISC) \( P_\ell(1) = 1 \) for all \( \ell \). The Whittaker cardinal-function theorem together with the commutation relation \( \lbrack L^2, T^a\rbrack = 0 \) guarantees that the ISC holds for arbitrary coupling g. We replace the path integral by a discrete sum over the reproducing-kernel Hilbert space (RKHS) on S², where the interacting propagator is the unique solution of a Fredholm integral equation of the second kind. Complete monotonicity of the spectral measure is established through a rigorous proof chain: self-adjointness of \( H = H_0 + gW \) via the Kato–Rellich theorem (using the Hilbert–Schmidt property \( \|K_0 V\|_{\mathrm{HS}}^2 \approx 4.73 \)), the spectral theorem for self-adjoint operators, and the Bernstein theorem. The ISC determines the conformal weights \( \Delta_\ell = \ell + 1 \) and, via the Hurwitz-zeta evaluation, yields the exact one-loop \( \beta \)-function coefficient \( b_1 = 11C_2(G)/(12\pi) > 0 \)—asymptotic freedom—for all non-Abelian gauge groups. We prove that for \( g > 0 \) and \( C_2(G) > 0 \), the Yang–Mills self-interaction forces information off the null cone into the timelike interior (verified numerically via the convolution \( \delta(\sigma^2)*\delta(\sigma^2) \)), activating the angular spectral gap \( E_0 = 1/2 \) inherent in the SO(3) representation theory. The Aldaya–Calixto–Cervéro obstruction theorem establishes that the full Poincaré group (including spatial translations \( P_i \)) is unitarily represented on the constrained Hilbert space as "good operators,'' while the special conformal generators \( K_\mu \) undergo dynamical symmetry breaking—providing the physical mechanism for the mass gap. The mass gap is \( \Delta = \Lambda = \mu\exp\lbrack-2\pi/(b_1 g^2)\rbrack > 0 \) for all \( g > 0 \) and all compact simple Lie groups. All Wightman axioms are verified for the interacting theory. Every step in the proof chain uses published theorems of functional analysis; no new mathematical conjectures are required.

Abstract: We present a non-supersymmetric Grand Unified Theory based on the gauge group U(4) that offers a unified origin for the electroweak scale, the strong interactions, the hierarchy of fermion masses, and the confinement of color. We propose that the symmetry breaking pattern SU(4) → SU(2)_H is radiatively selected, partitioning the vacuum into a standard gluon sector and a new topological "Warden" sector. We identify the excitations of this new sector (the Wardens) not as simple gauge bosons, but as Hopf solitons (Hopfons) formed by the scalar Goldstone degrees of freedom. Crucially, we employ the Cho-Duan-Ge decomposition to separate the gauge potential into a topological ’restricted’ mode and dynamical valence gluons. We demonstrate that the transverse valence modes acquire a large constituent mass from the magnetic background and effectively decouple from the renormalization group flow, leaving the evolution dominated by the scalar topological degrees of freedom. Furthermore, we show that the fermionic statistics of the Warden fields are not a violation of quantum field theory but a rigorous consequence of the Finkelstein-Rubinstein mechanism, where the non-trivial Hopf invariant (Q_H = 1) induces a geometric Berry phase that mandates Fermi-Dirac quantization to preserve S-matrix unitarity. This topological identification resolves the spin-statistics tension and justifies the use of scalar beta-function coefficients, leading to a precise three-loop unification of gauge couplings at MGUT ≈ 3.2 × 10¹⁶ GeV. The theory posits a "Tilted Universe" mechanism wherein the electroweak scale is generated by a geometric misalignment between the Higgs vacuum and the rigid Warden condensate. The misalignment angle is dynamically locked to the flavor sector, predicting a fundamental stiffness scale of FUV ≈ 1.1 TeV. We demonstrate that this high-energy stiffness generates a lower dynamical confinement scale via dimensional transmutation, Λ_IR ≈ 330 MeV, which correctly predicts the scalar glueball mass (M_GB ≈ 1699 MeV) and string tension (√σ ≈ 440 MeV) from first principles. In the matter sector, a single flavor-democratic Yukawa coupling at the GUT scale is shown to deterministically evolve into the observed 17-order-of-magnitude hierarchy of quark and charged lepton masses. The model resolves the flavor puzzle by predicting a hierarchical CKM matrix for quarks and, via a non-universal "Hopf Portal," an anarchic PMNS matrix for leptons. The framework makes two sharp, falsifiable predictions: the absolute stability of the proton, and the existence of a heavy topological resonance at 8.2 ± 0.4 TeV, accessible at future hadron colliders. The proposed model, along with its associated phenomenological effects and searches for new particles within the emergent "Warden" sector, accurately derives from first principles the Top Quark mass (172.68(22) GeV), the Weak Mixing Angle ( sin² θ_W = 0.23125), and other experimentally derived quantities with high accuracy. It can be examined at current and upcoming high energy physics experiments focused on Beyond the Standard Model (BSM) physics, such as at the High Luminosity LHC at CERN and the Future Circular Collider (FCC). To ensure complete theoretical transparency and reproducibility, the full Python computational framework executing these derivations is made publicly available[92].

Abstract: We present an exhaustive derivation of the spectral action functional within the rigorous framework of non-commutative Riemannian manifolds (A,H,D). By employing a non-perturbative heat kernel expansion for Dirac-type operators, we demonstrate that the Einstein-Hilbert-Palatini action is an emergent property of the spectral zeta function at its principal meromorphic poles. We extend this formalism to include the dynamical Barbero-Immirzi parameter γ as a pseudoscalar field coupled to the Nieh-Yan topological invariant. The paper further investigates the categorification of spin-foam vertex amplitudes using SU(2)_q quantum group invariants. We rigorously prove that the transition from Lorentzian to Euclidean geometry is a KMS-state thermalization process within von Neumann algebras of type III1. Finally, we discuss the role of Mukai-Fourier transforms in Calabi-Yau fibers as a mechanism for generating particle masses in the spectral Standard Model.

Abstract: Machine learning capabilities are expanding into scientific domains at an accelerating pace. When applied to high energy physics pattern discovery, they will generate candidates faster than traditional evaluation can absorb. ML finds patterns in past data. It is inherently post hoc. Whether those patterns reflect structure or coincidence is unknowable at discovery time. This limitation applies equally to human and computational pattern finding. What differs is scale. ML candidate generation is effectively unbounded, while human evaluation capacity remains fixed. When generation rate exceeds evaluation bandwidth, binary accept or reject degenerates to random sampling. Information theoretically, the only response that preserves ranking under a finite evaluation budget is stratification. By focusing on stratification rather than binary filtering, rule adjustments can be made retroactively, thresholds tuned as results accumulate, and evaluation bandwidth focused on top ranked candidates. This paper attempts to codify those criteria, proposing seven computationally evaluable standards for stratifying ML generated patterns. The goal is not to deliver verdicts but to prioritize which candidates merit preregistration and longitudinal tracking. The framework preserves the essential paradigm: pattern plus theory equals potentially real physics. Patterns alone, however striking, remain candidates until theoretical understanding arrives. Making these criteria explicit enables prefiltering at scale while creating a collaborative resource rather than a competitive one. ML capabilities extend what physicists can search while preserving how physicists evaluate. We offer this provisional framework for community calibration, with the goal of developing validation infrastructure before the capability fully arrives.

Abstract: We summarize the key aspects of the ongoing octonionic E₈ ×ωE₈ unification program. This summary was fed to Open AI’s GPT-5.4 Pro with a prompt to give a critical appraisal of this program, and to compare it with string theory. We include the prompt and the detailed response from AI in an Appendix.

Abstract: We present a strengthened collider-facing extension of the uploaded Quantum Information Copy Time (QICT) program to compressed Higgsino searches at \( \sqrt{s}=13 \) TeV. The underlying QICT manuscript identifies its theorem-level core as a copy-time definition \( \tau_\text{copy} \), a Liouvillian-squared susceptibility \( \chi^{(2)}_Q \), and a conserved-charge speed-limit bound; the infrared and phenomenological sections are explicitly conditional closures. We therefore do not claim a theorem-level Higgsino prediction. Instead, we construct a fully explicit QICT-to-collider closure map, validate a detector-level surrogate against public CMS and ATLAS compressed-Higgsino reach anchors, perform a quantitative public-contour recast, propagate dominant surrogate uncertainties, and isolate a new branch-transition observable \( B_{\ell t} = N_{\ell t} / (N_{\ell t} + N_{\text{track}}) \). Within this strengthened framework, the residual post-public-limit search prior remains two-branched: an ultra-compressed branch near \( m_{\tilde{\chi}_1^\pm} \approx 200-240 \) GeV with \( \Delta m^\pm \sim 0.35-0.9 \) GeV, and a few-GeV branch near \( m_{\tilde{\chi}_1^\pm} \simeq 150-220 \) GeV with \( \Delta m \simeq 2-6 \) GeV. The manuscript is deliberately modest about logical status, but it is now quantitative, reproducible, and falsifiable.

Abstract: We study the effect of trapped neutrinos on the properties of the deconfinement phase transition from hot $\beta$-equilibrated, electrically neutral hadronic matter to quark matter. To describe the thermodynamic properties of hot hadronic matter, an extended relativistic mean field (RMF) theory is used, which also incorporates the isovector–Lorentz-scalar $\delta$-meson effective field. The three-flavor quark phase is described within the framework of the local Nambu--Jona-Lasinio (NJL) model. It was assumed that the surface tension at the quark-hadron interface is so strong that the phase transition occurs according to Maxwell's construction. The thermodynamic properties of the quark and hadronic phases were calculated for both neutrino-trapped and neutrino-transparent regimes at various temperatures ranging from 0 to 100 MeV and baryon number densities from 0 to 1.8 fm$^{-3}$. The impact of trapped neutrinos on the thermodynamic properties of the coexistence state has been investigated. It has been demonstrated that the baryon chemical potential in the coexistence state decreases as temperature increases. The critical endpoint parameters in the $T-n_B$ plane of the phase diagram were obtained for the case of trapped neutrinos (74 MeV; 0.269 fm$^{-3}$) and for the case of the absence of neutrinos (75.6 MeV; 0.255 fm$^{-3}$).

Abstract: Wave–particle duality in interferometric systems is commonly formulated through complementarity relations linking fringe visibility and path distinguishability. In realistic experiments, interference suppression arises not only from unitary which-path marking but also from environment-induced decoherence. We derive an angle-dependent pure-dephasing model from a microscopic system–bath Hamiltonian, obtaining a Lindblad master equation with geometric coupling dependence. Moving beyond the Markovian limit, we utilize a second- order time-convolutionless (TCL2) expansion with a structured spectral density to show that geometric scaling persists in non-Markovian regimes, potentially leading to geometry-dependent coherence revivals. Furthermore, we explicitly derive the entropy production rate, demonstrating that the transition toward classicality is quantitatively governed by directional entropy flow. The framework remains fully within standard quantum mechanics, introducing no modifications to the Schr¨odinger equation. Experimental falsifiability criteria, including early-time scaling and coherence revivals, are presented.

Abstract: Quantum Mechanics is sufficiently capable of proving quantum gravity by itself without considering actual Einsteinian General Relativistic formalism. Due to the non-applicability of Einsteinian relativity in quantum gravity, in this article, we have described gravity as a correspondence between General (Quantum) Relativity and Quantum Field Theory (QFT) by introducing a (quantum) quadratic form and a (quantum) metric tensor along with dynamic time t. Here, we have developed a Kline-Gordon-like equation and a Dirac-like equation in QFT, which are themselves actually nothing but the quantum gravitational field equations (analogous to Einstein's field equation in General Relativity) for bosons and fermions, respectively. Furthermore, we have developed a Generalized Quantum Gravitational Field Theory, where QFT is conjugated with gravity and Dark Energy (for inconstant cosmological constant), so that it can unify Standard Model with gravity and Dark Energy in 'General Unified Theory' as SU(5)=SU(3)×(SU(2)⊕iSU(2)) through a Gravito-weak symmetry group. In addition, we have shown that unbounded operators, such as, i) the (quantum) relativistic mass and time, ii) the quantum scalar curvature and the proper time, iii) the (quantum) relativistic mass and its inversely stretched/shrank (3+1)D curvilinear quantum spacetime, all in pairs are satisfied their individual Uncertainty Principles, i.e., they cannot have definite and constant values at the same time. We have also proved that the present theory of Quantum Gravity is 'multiplicatively renormalizable'.

Abstract: The anomalous gyromagnetic ratio (g-factor) of elementary particles---\textit{most famously the electron's deviation from the Dirac value}: g_D = 2---has long been one of the most precisely measured and theoretically challenging quantities in fundamental theoretical physics. In Quantum Electrodynamics (QED), this anomaly is explained through radiative corrections represented by an infinite series of Feynman diagrams, culminating in the Dyson expansion. In the present work, building on the foundations laid in Paper (I), we develop an alternative, non-perturbative framework that traces the origin of the g-factor anomaly to the electromagnetic self-energy of an extended charge distribution coupled to a hypothetical cosmic vector field. By identifying the zeta-parameter [proposed in Paper (I)], which parameter governs the deviation from the Dirac value: g_D = 2---with the spin-coupled electromagnetic self-energy, we derive a power series for the particle anomalous magnetic moments that reproduces the full Dyson expansion series of QED without invoking Feynman diagrams or perturbative methods. The coefficients of this series are shown to arise from the multipole moments of the charge distribution, as described by the Azimuthally Symmetric Theory of Electrostatics (ASTE-model) introduced herein. This framework naturally implies that particles with non-zero anomalous g-factors---including the electron---must possess a finite spatial extent. Using existing g-factor data for the proton, neutron, and lambda hyperon, we obtain a preliminary theoretical estimate of the electron radius: r_e = (1.20(5) e-22 m, in remarkable agreement with the stringent experimental upper limit: r_e < 1.50e22 m. We strongly believe that these results offer a fresh new perspective on the nature of fundamental particles, the origin of radiative corrections, and the possible existence of a cosmic field mediating spin-electromagnetic interactions across the fabric of spacetime.

Abstract: The Standard Model is a renormalizable chiral Yang–Mills–Higgs quantum field theory defined on a principal fiber bundle over four-dimensional Minkowski spacetime with structure group G_SM = SU(3)_C × SU(2)_L × U(1)_Y. Its Lagrangian is uniquely constrained by local gauge invari-ance, Lorentz symmetry, perturbative renormalizability, and the requirement of gauge and mixed anomaly cancellation. The resulting theory couples non-Abelian gauge connections to chiral fermions transforming in complex representations of su(3) ⊕ su(2) ⊕ u(1), together with a scalar Higgs doublet whose vacuum expectation value induces spontaneous symmetry breaking and mass generation through the Higgs mechanism. In this review, we present a systematic and geometrically motivated construction of the Standard Model action from symmetry principles. The Yang–Mills sector is derived from the curvature two-form associated with the gauge connection on the principal G_SM-bundle, while the fermionic kinetic terms arise from covariant derivatives in chiral representations. We analyze the Yukawa interactions and scalar potential in representation-theoretic terms and interpret sponta-neous symmetry breaking as a reduction of the gauge symmetry accompanied by a reorganization of physical degrees of freedom. At the quantum level, we discuss BRST quantization, gauge fixing, and the derivation of Slavnov–Taylor identities ensuring perturbative unitarity and renormalizability. The one- and two-loop beta functions for gauge, Yukawa, and scalar couplings are computed, and the renormalization group flow is examined across many orders of magnitude in energy. Special emphasis is placed on the cohomological structure of gauge anomalies and their exact cancellation within each fermion generation. We further consider ultraviolet extensions and effective field-theoretic embeddings, including Grand Unified Theories, supersymmetric completions, right-handed neutrinos and seesaw mechanisms, and string-motivated constructions. Throughout, we emphasize the inter-play between geometric structure, renormalization group dynamics, and experimentally accessible observables. This document aims to provide a technically rigorous and conceptually unified reference for researchers in high-energy theory and mathematical physics.

Abstract: Quantum electrodynamics (QED) provides extraordinarily accurate predictions for charged lepton properties, although its formalism offers limited intuitive insight into the geometrical and energetic scales associated with vacuum effects. In this work, a phenomenological representation is introduced to describe the leading-order contribution to the anomalous magnetic moment of charged leptons. By combining characteristic length and energy scales associated with the Compton radius and rest energy with geometric arguments, the Schwinger correction to the electron magnetic moment is recovered. Within this framework, the fine-structure constant acquires the meaning of a characteristic angular scale associated with the effective vacuum dressing of the particle. The construction naturally extends to the muon, indicating the universality of the angular structure underlying anomalous magnetic moments. The model does not replace quantum electrodynamics but tries to provide an effective geometric representation of its lowest-order result, offering an intuitive picture of vacuum dressing and interaction scales.

Abstract: Like the electron, the proton is considered to be a stable particle—the meaning of which is that, it does not decay into smaller constituents—as such—much like the electron, it is naturally expected to be accurately described by the Dirac equation. However, its measured g-factor: g_p = 2 + 3.5856946893(16), significantly deviates from the expected Dirac prediction of: g_D = 2. In this article—which is the first in a ten-part series titled: Toward a Unified and Universal Dirac Equation—we propose that a massive electrically neutral zero-rank scalar coupled Dirac equation could, in principle, explain this deviation (the g-excess) and this requires that the proton have a non-zero radius. The same theory (equation) can be applied to the electron’s non-Dirac g-excess and this requires us to endow the electron with a non-zero radius. That is to say, it is observed that the electron, which is typically regarded as a point particle (i.e., as having a zero radius), exhibits a g-excess: g_e − 2 = 0.002319304362(2). According to the ideas presented herein, this non-Dirac g-excess, implies that the electron may actually be a spatially extended particle system.