Physical Sciences

Abstract: The charged-lepton Koide relation Q = (∑m_ℓ)/(∑√m_ℓ)2 = 2/3 has held to one part in 10⁵ for over four decades without an accepted derivation. The same rational value 2/3 is forced by C₃ phase cancellation identities in three-beam interference, where three coherent sources at 120^◦ separation produce hexagonal patterns with a fixed quadratic intensity ratio. Whether the shared 2/3 is a surface coincidence or reflects deeper structural overlap between interference geometry and the fermion mass spectrum is the question this paper investigates. We show that a companion identity (the F identity) reduces to the classical Descartes circle theorem at Q = 2/3, and that integer-wavelength resonance conditions at sinθ = 2/(3N) provide a geometric counterpart to Shulga’s compact-cycle offset δ = 2/9. A geometric-mean cascade motivated by hierarchies previously noted in the quark mass literature suggests fixed closed-form targets for three light-quark masses (ms, md, mu) from a single input µ_⋆ = ∑m_ℓ = 1883.1 MeV, with internal self-consistency at 0.06% and deviations from PDG/FLAGreference values within current uncertainties. The algebraic overlap is not limited to the shared numerical value 2/3: the C₃ phase-cancellation algebra that forces Q = 2/3 in interference also fixes the companion identity and the Descartes connection, while the associated integer-wavelength resonance conditions place δ = 2/9 in the same geometric framework. The construction does not propose a mechanism for fermion masses; we present these correspondences as empirical constraints on whatever dynamics produces the fermion spectrum, not as a theory of those dynamics. This work appears within the same month as three independent publications on the Koide relation (Sec. 10), suggesting the tools for understanding it may be maturing.

Abstract: We present a spectral-action framework for connecting Generalized Trace Dynamics (GTD) to the structural form of the low-energy action of the observed universe. The fundamental single-STM-atom Lagrangian is decomposed exactly into a purely bosonic sector, boson--fermion cross terms, and bifermionic terms. This sectorwise decomposition furnishes a dictionary to almost-commutative spectral geometry: the bosonic sector supplies a quadratic GTD Dirac functional built from the six split-biquaternionic differential directions together with octonionic vector/gauge fluctuations; the cross sector supplies, under an explicit localization hypothesis, a sesquilinear fermionic pairing; and the bifermionic sector supplies the scalar/internal channel that is bosonized into the Higgs bridge field. We also record the principal-symbol link between the SO(3,3) BF variables and the four-dimensional leafwise Dirac operator. Under stated assumptions --- spontaneous localization, Euclidean continuation, six- to four-dimensional BF reduction, and a candidate observed-leaf finite geometry compatible with the E₆/J₃(O_C) inputs --- the bosonic heat-kernel expansion yields the structural low-energy classes of terms: Einstein--Hilbert gravity, Yang--Mills kinetic terms, and scalar kinetic and potential terms. In addition, we provide a candidate finite spectral triple with explicit finite trace invariants, verify that the localization map respects the one-generation lepton/quark representation split, identify visible color-singlet scalar channels with electroweak quantum numbers (1, 2, ±1/2), and exhibit a smooth regulator family with explicit cutoff moments (f₀, f₂, f₄). Conversely, the assembled low-energy spectral action admits a natural inverse bilinear lift back to split bioctonionic trace dynamics. The result is a structural framework with conditional consistency checks and reductions, not a complete first-principles derivation.

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:

The Koide relation \( Q = (\sum m_\ell)/(\sum \sqrt{m_\ell})^2 = 2/3 \) for the charged leptons has held to one part in \( 10^5 \) for over forty years without an accepted derivation and is widely regarded as numerology. This paper takes the relation as a clue rather than an endpoint. Treating lepton mass square roots as Descartes-circle curvatures, the outer root of the Descartes quadratic equals the closed form \( \mathcal{F} = e_1 - \sqrt{p_2} \) when Koide holds exactly (Proposition 1); equivalently, \( \mathcal{F}^2 = \alpha_K^2\,\mu_\star \) with \( \alpha_K^2 = 5/2 - \sqrt{6} \) and \( \mu_\star = \sum_\ell m_\ell \) the lepton-sum scale. The three-input symmetric-polynomial identity thus collapses to one dimensionless Koide-determined constant times the lepton-sum scale. Kocik [10] first observed a Descartes-like reading of Koide; our mutually-tangent variant is mathematically distinct but follows the same geometric spirit. The four-curvature completion carries a testable consequence absent from the bare three-mass relation: evaluating the squared fourth curvature numerically, \( \mathcal{F}^2 = 95.113 \) MeV, and comparing against the strange-quark \( \bar{MS} \) mass at \( \mu_\star \) within current lattice precision yields a residual of \( +0.04 \) MeV against \( \pm 0.69 \) MeV, about \( +0.06\sigma \). The lepton-side quantity is fixed to better than \( 0.01\% \); future lattice improvements will sharpen or refute the present numerical agreement. To our knowledge this paper implements the first Monte Carlo null test of the Koide relation under a random-spectrum prior; a Koide-conditioned null-model calibration across four prior shapes pre-registered for the analysis gives hit fractions at the sub-percent level — model-conditional frequencies, not \( p \)-values. Scale, input, prior, and filter sensitivities, together with the error budget, are reported; full Monte Carlo protocols, numerical output, and pre-registration are in a companion methods note [15].

Abstract: This article is a review of developed in 2007-2025 years Planck scale informational physical model that is based on 3 main points. First of all on the 2007 “The Information as Absolute” conception, where the fundamental phenomena/notions “Matter”, “Consciousness”, “Space”, “Time”, “Energy”, “Information”, which are fundamentally transcendent in conventional philosophy and sciences, are rigorously scientifically defined. The conception completely rigorously scientifically legitimates the outstanding C. F. von Weizsäcker “Ur hypothesis”, and E. Fredkin “Digital Philosophy/Physics”, which posit that Matter is constructed from some binary reversible logical elements; and on all reliable experimental data. Correspondingly in the model it is postulated that Matter’s ultimate base is the [4+4+1]4D dense lattice of primary [4+4+1]4D binary reversible fundamental logical elements [FLE], which is placed in the corresponding Matter’s fundamentally absolute “Cartesian” [4+4+1]4D spacetime with metrics (cτ,X,Y,Z, g,w,e,s,ct), while everything in Matter is/are some specific disturbances in the lattice. Basing on the above in the model a number of physical problems are either solved or essentially clarified, e.g. of what is real Matter’s spacetime above, what are the physical sense of Lorentz transformations; uncertainty and wave-particle duality in QM; particles and antiparticles, etc. Besides initial Planck scale models of fundamental Gravity, Electric, and Nuclear/Strong Forces are developed, where it is shown that these Forces strengths ratio is in accordance with experimental values only if the FLE size and FLE flip time are Planck length and Planck time; the model rather probably really scientifically clarifies some cosmological problems, including the “matter-antimatter asymmetry: one. Etc. more of the problems see in the article.

Abstract: Higgs physics is an active front from both experimental and theoretical aspects. It is a problem how to explain the measured value of Higgs mass, and a simple question like where the quartic coupling potential exactly comes from could not be well answered. This paper described a simple model for the Higgs field. It seems the Higgs mass may come from the coupling between the thermal energy relative to Hawking temperature of the Planck scale Kerr black hole and the thermal energy of cosmological microwave background. By a logarithm potential, we can naturally get the exact quartic term for the Lagrangian. The Higgs mass and the vacuum expectation value we get are proportional to the square root of the temperature of the cosmological thermal background, which may mean they shall be larger in the earlier universe. Future gravitational wave detection may help to check this model. If the model could prove to be true, it will have important influence on our understanding of the cosmology.

Abstract: We suggest a version of renormalizable Quantum Field Theory which does not contain nonperturbative effects. This is otained by the proper use of the boundary conditions in the functional integral of the generating functional of Green functions. It is well known which boundary conditions are applied to the fields of the functional integral to get correct perturbation theory. We propose that these conditions should be used for all fields integrated in the generating functional integral. It is shown that in this case non-perturbative effects are absent. That is we assume that perturbation theory defines the complete generating functional integral. It allows, in particular, to formulate the generating functional integral in a unique way as an exact compact mathematical formula.

Abstract: Pure Topology Results: We prove that any unified gauge theory whose U(1) sector satisfies charge quantization (discrete admissible charges) and completeness (realization of every principal U(1)-bundle over any paracompact base) must be formulated, up to homotopy equivalence of the base and isomorphism of bundles, on the universal complex Hopf fibration and its finite shells. Such a system is shown to be indecomposable (that is, it presents as a unified field which cannot be broken down without losing information). The Standard Model gauge groups arise as natural reductions along the nested shell hierarchy: U(1) from the circular S^1 fiber, SU(2) from the S^3 shell and SU(3) from the S^5 shell. Gravity emerges as the spacetime gauge sector from the Kahler geometry of the base and fiber-induced torsion, yielding Einstein-Cartan analogous structure with the Levi-Civita connection recovered in the torsion-free limit. The unified structure group G_total = (SU(3) x SU(2) x U(1) x SO(4))/Gamma is intrinsically non-factorable due to the generating role of the universal first Chern class in H*(CP^infinity; Z) = Z[c_1]. Applied Topology Results: On each Hopf shell, the generalized Beltrami operator B = d|_xi acting on the contact distribution is elliptic, essentially self-adjoint, and possesses a discrete spectrum stable under torsion perturbations by the Kato-Rellich theorem. Fiber winding decomposition yields independent topological sectors whose Gaussian functional determinants, regularized via spectral zeta functions, generate intrinsic mass scales. Fermion mixing (CKM, PMNS) arises from intersection-form overlaps of admissible cycles in H(CP^4), with CP violation induced by fiber holonomy phases. Dynamics emerge from the fluctuation spectrum of the topological action on S^9. Given one empirical scale set by the Fermi constant (with its associated electroweak vacuum expectation value), the fine-structure constant and all shell-specific mass scales, spectral coefficients, and coupling constants entering the particle spectrum are fixed by the spectral geometry of the complex Hopf fibration. Phenomenology, Physical Interpretations and Numerical Predictions: The framework predicts the complete particle mass spectrum and anomalous magnetic moments, with suggested independent experimental tests (torsion-induced phase wobble, absolute neutrino mass scale, and the electron, mu and tau g-2) providing falsifiability. Fundamental constants arise from topological normalization. Further results include anomaly cancellation, dark sector effects from bundle torsion and holonomy, and the elimination of singularities. The mathematical results stand independently as contributions to the topology of classifying spaces, reductions along nested Hopf shells, and contact spectral geometry.

Abstract: We present a constructive approach to the gauge-hierarchy problem within the Quantum Information Copy Time (QICT) framework.The central result, the Information-Protection Theorem (IPT), establishes that in any strictly local, unitary quantum cellular automaton(QCA) with an exactly conserved charge, the physical scalar mass is not a free ultraviolet parameter but is uniquely determined by thereceiver-optimised Liouvillian-squared susceptibility ratio κeff and the static hypercharge susceptibility χY. Because the QCA isstrictly local, its Liouvillian-squared susceptibility is ultraviolet-finite: no mode with |k|> π/a exists in the QCA Hilbert space, sothe quadratic divergence δ m2 ∼Λ2UV is structurally absent from the operator framework. We prove three theorems: (i) UV-finitenessof the QICT susceptibility from QCA locality; (ii) the Golden Relation as the unique positive-definite mass eigenvalue of a well-posedQICT eigenvalue problem, conditional on the diffusive-reduction hypothesis; and (iii) technical naturalness of the mass band underrenormalisation-group flow. The predicted singlet-scalar mass mS = 58.5 ±15.6GeV sits near the Higgs resonance and is consistentwith current LZ direct-detection and ATLAS invisible-width constraints. The framework yields a falsifiable mass band: it is excludedif Higgs-portal searches eliminate the entire interval [43, 74] GeV.

Abstract: The Dirac equation predicts a gyromagnetic ratio g_D = 2 for charged _spin-1/2 particles and g_D = 0 for neutral ones. The neutron — electrically neutral yet possessing a large magnetic moment with g_N =2−5.82608552(90) —this presents a fundamental challenge to any unified g-factor theory. The standard explanation invokes the neutron’s internal quark structure; in the present framework, which seeks a description in terms of the modified Dirac equation of Papers (I) and (II), an alternative must be found. We extend the framework of Papers (I) and (II) to electrically neutral particles by introducing an effective charge q^eff_N = κ_Hd^N_EDM associated with the neutron’s internal electric dipole moment. This allows the neutron to couple to the ambient magnetic vector potential in analogy with charged particles. Wethen revisit Rutherford’s historical proton–electron composite model of the neutron, resolving its fatal spin objection by extending to a three-body system. We propose that the neutron may very well be aquantumsuperposition of three states: an excited electron (the tauon τ⁻), a de-excited proton (p⁺₁), and an associated neutrino (ν̅ ). Solving the normalization, mass, and magnetic anomaly equations yields probability coefficients P1 ≃ 0.53, P2 ≃ 0, P3 ≃ 0.47, implying that the proton contributes negligibly to the neutron’s bulk properties while the tauon and neutrino dominate nearly equally. Westress that this model is highly speculative and rests on several unverified assumptions, detailed in §(8). Most critically, the system of equations is underdetermined: the neutrino’s effective magnetic anomaly ∆g₃ ≃ −12.37 is not predicted but fitted to reproduce the observed neutron moment. This value exceeds Standard Model expectations by approximately ten orders of magnitude and requires a physical explanation that the present framework does not yet provide. The model should therefore be understood as an exploratory proposal motivating future theoretical and experimental work, not as an established result.

Abstract: We propose a chiral condensate lattice with Pmmm space group symmetry (No. 47, D2h, order 8) and a cubic unit cell built from the four light quarks and antiquarks hypothesized to serve as the fundamental building blocks of the chiral condensate lattice and of baryonic and leptonic matter. The proposed Pmmm chiral condensate lattice arises from spontaneous symmetry breaking of the parent Pm3̄m space group (No. 221, Oh, order 48), with the 40 broken symmetry operations generating the three pions π⁺, π⁻, π⁰ as Goldstone bosons, and is proposed as an alternative structural description of the instanton liquid QCD chiral condensate. Electrons and positrons are embedded within the chiral condensate lattice and perform discrete-time quantum walks (DTQW) driven by underlying quark and antiquark permutations between the highly symmetric Wyckoff positions of the Pmmm unit cell, analogous to quark hopping in the instanton liquid. We assume that quark permutations occur at the Zitterbewegung frequency, ensuring that the embedded electron is never bare and generating a confined electron cloud within the chiral condensate lattice. The spin and helicity states of the embedded electrons are structurally defined by the chiral condensate structure and dynamics: the quantum walker type, matter or antimatter, determines the spin state, while the helicity state depends on whether the motion of the larger quark charge is parallel or antiparallel to the motion of the embedded electron charge. We assume that an electron undergoing a DTQW embedded within the chiral condensate lattice produces a coherent wave packet, in contrast to the wave packet dispersion found by Schrödinger and Darwin in 1927, which may provide indirect evidence for the existence of the proposed Pmmm chiral condensate lattice. A one-dimensional DTQW simulation with a non-Hermitian quantum coin generates coherent evolution of one wave packet component, providing promising support for the proposed framework. We further assume that in the vicinity of black hole horizons the chiral condensate lattice melts, thereby enabling a reaction of three Pionic tetraquarks that produce deuteron and anti-deuteron pairs, which further react with Pionic tetraquarks to produce protons, electrons, and anti-neutrons that fall beneath the black hole horizon while matter particles escape.

Abstract: The description of the internal spaces of fermion and boson fields with "basis vectors", which are the superposition of odd and even products of the operators γa, offers in d = 2(2n + 1)-dimensions, such as d = (13 + 1), a unified picture of all so far observed fermions (quarks, leptons, antiquarks and antileptons that appear in families) and bosons (gravitons, photons, weak bosons, gluons and scalars), under the condition that all fields have non-zero angular momenta only in the d = (3 + 1), SO(3, 1), of ordinary space-time. Bosons, which also carry the spatial index α (which is for tensors and vectors μ = (0, 1, 2, 3) and for scalars σ ≥ 5) appear in two orthogonal groups. In any d = 2(2n + 1)- dimensional space the number of internal states of fermions in all families and their Hermitian conjugate partners is equal to the number of internal states of boson states. The article presents general properties of massless fermion and boson fields and their mutual interactions in this theory, which determine the Lagrangian density of both fields and their interactions. It particularly illustrates "basis vectors" and their properties in d = (13 + 1) and d = (5 + 1). The article presents new results and discusses open problems in this theory.

Abstract: In 1993 Sorkin showed that by extending textbook ideal measurements to relativistic quantum field theory 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 open 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, and show that neither of which is 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: We prove the existence of a positive mass gap \Delta >0 for quantum Yang- Mills theory on four- dimensional Minkowski spacetime within the Epstein- Glaser causal perturbation theory framework, and derive analytically the glueball mass spectrum. The construction proceeds from two postulates—the massless wave equation \square \phi = 0 and Poincaré invariance—through the angular momentum decomposition of the retarded Green's function on the null cone. The equal- weight condition P_{\ell}(1) = 1 , a direct consequence of the Peter- Weyl theorem, ensures that all angular momentum modes contribute identically at the causal vertex. The spectral sum \Sigma^{(4)}(t) = \cosh (t / 2) / [2\sinh^2 (t / 2)] encodes the Riemann zeta values \zeta (- 1) = - 1 / 12 , \zeta (- 3) = 1 / 120 , ... in its small- t expansion; from the constant term 1 / 12 we derive the one- loop \beta - function coefficient b_{1} = 11C_{2}(G) / (12\pi) without Feynman diagrams. The mass gap is proven through two independent arguments: off- cone propagation and Carleman- Fredholm determinant estimates. All Wightman axioms are verified. Applying Boltzmann's 1877 statistical method with Yang- Mills self- interaction playing the role of Newtonian mechanics, and fixing the inverse temperature via Jacobson's thermodynamic relation \delta Q = T dS , we derive the analytic glueball mass spectrum M_{n} = \frac{j_{2,n}}{2}\Lambda ,\qquad n = 1,2,3,\ldots , where j_{2,n} are the zeros of the Bessel function J_{2} and \Lambda is the dynamical scale. The mass ratios M_{n} / M_{0} = j_{2,n} / j_{2,1} = 1:1.638:2.260:\dots agree with lattice QCD to within the expected 1 / N^{2} corrections. The framework connects to Migdal's large- N reduction, Unsal- Yaffe volume independence, and Verlinde's entropic gravity.

Abstract: We formulate and test a minimal finite Hankel closure for the dip–bump structure in elasticproton-proton scattering. The scope of the claim is deliberately precise. We do not present amicroscopic derivation from QCD and we do not claim universal exclusion of the full hadronicphenomenology. Rather, we establish a conditional theorem, confront its surrogate realizationwith public data, and state explicit near-term tests. First, assuming positivity, radiality, finitemoments, a Born-dominant forward-plus-first-dip window, self-similar scaling, and minimal finiteHankel closure with one simple node, we prove that the unique lowest-complexity sector is the firstLaguerre deformation, which yields a polynomial-times-Gaussian amplitude. Second, we derivestructural relations for the forward slope, dip scale, forward curvature, and the drift observable O_excl = ∆[B0|t|dip], and we prove non-reducibility against the one-scale geometric class for whichthe corresponding invariants are energy independent. Third, we test the closure on two levels ofpublic-data benchmark. In the restricted internal comparison on 83 differential-cross-section pointsat 2.76 and 13 TeV under a common weighted log-space score and shared cross-energy flow, thetwo-scale copy-time surrogate yields χ²_log = 461.19, AIC = 487.19, and BIC = 518.64, compared with(60493.49,60507.49,60524.43) and (59942.77,59964.77,59991.38) for two one-scale baselines. We thenperform a stronger covariance-aware benchmark in log space, using per-dataset block covariances builtfrom the published statistical errors together with fully correlated systematic blocks, and comparethe copy-time surrogate to the internal one-scale baseline, a canonical Regge-pole-plus-Odderonamplitude, a canonical complex Regge-eikonal baseline, and the fixed Kohara–Ferreira–Kodamaparametrization. In that stronger test the covariance-aware copy-time fit remains the best modelin the benchmark set, with χ²_{cov log} = 1687.73, compared with 5440.69 for the strongest externalbaseline. At fixed 13 TeV, however, the split-sector one-scale surrogate remains competitive inthe dedicated local fit, showing that the main empirical leverage of the closure is intrinsicallymulti-energy rather than a consequence of the 13 TeV line shape alone. We also report a hold-outvalidation at 8 TeV, an explicit continuation to 13.6 and 14 TeV, and a narrow-window robustnessscan showing that the forecast sign pattern is stable under moderate perturbations of the real-sectorcontinuation. Within the explicit axiomatic, statistical, and benchmark choices adopted here, theclosure is therefore mathematically constrained, experimentally discriminating, and favored over theimplemented internal and external baselines.

Abstract: We study the SU(N) symmetric model, which describes interaction of gravity with three field multiplets: first, the multiplet of pseudoscalar fields, which is now associated with the multi-component cosmic dark matter; second, the multiplet of vector fields, which represents the so-called color aether, now known as dynamic aether; third, the multiplet of Yang-Mills fields, which provides the SU(N) invariance of the model as a whole. It was previously known that the decay of the color aether in the early Universe could have given rise to emergence of an axionic singlet according to the Peccei-Quinn mechanism; we proposed an extended scheme, according to which the color aether activates an additional internal tool for generating not only a simple axionic singlet, but a whole SU(N) symmetric multiplet of pseudoscalar fields. Late-time evolution of the considered field configuration is analyzed in the framework of Bianchi-I cosmological model, and a hypothesis is proposed that the mentioned pseudoscalar multiplet can be associated with the multi-component cosmic dark matter.

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: 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.