The paper develops a structural criterion for identifying physically meaningful macrovariables, together with an information-theoretic interpretation. We introduce macrotype maps, defined as quotient maps that identify microconfigurations indistinguishable under admissible transformations and invariant under admissible interactions. This yields a notion of macroscopic identity based on two invariance principles: (i) internal invariance, understood as temporal or symmetry-based stability of defining properties, and (ii) external relational invariance, understood as a stable response profile across admissible coupling contexts. Macrotypes retain precisely the invariant interaction-relevant information for macroscopic behaviour while discarding irrelevant microscopic detail, providing a principled selection criterion for physically meaningful coarse-graining. The point of the construction is not the mere observation that macrostates may be represented as equivalence classes. Rather, the paper proposes a criterion for selecting those equivalence classes whose fibres are physically meaningful because they preserve invariant structure and reproducible interaction behaviour under explicitly specified admissible contexts. The framework is illustrated with examples from both physical and functional domains, and it naturally supports an information-compression interpretation and the possibility of hierarchical and compositional descriptions of macroscopic organization. The approach offers a concise structural contribution to the micro-macro problem by characterising macroscopic identity in terms of transformation- and interaction-stable invariant structures.
We investigate the stabilization of a skyrmion crystal in a centrosymmetric square-lattice magnet under zero magnetic field. By constructing an effective spin model in momentum space, we examine the roles of bond-dependent magnetic anisotropy, higher-harmonic wave-vector interactions, and two distinct types of four-spin interactions. Using simulated annealing, we obtain a low-temperature phase diagram that includes a skyrmion crystal with skyrmion number two, three topologically trivial double-Q states, and a single-Q state. The zero-field skyrmion crystal is characterized by a noncoplanar multiple-Q texture composed of sinusoidal modulations at the primary and higher-harmonic ordering wave vectors. We show that the cooperative effect of the conventional biquadratic interaction and the sinusoidal-wave-selective four-spin interaction is essential for activating the higher-harmonic components and stabilizing the topological phase. We also demonstrate that an out-of-plane magnetic field drives the skyrmion crystal into a topologically trivial double-Q state by suppressing the scalar spin chirality and higher-harmonic wave vector components, while a trivial double-Q state remains topologically trivial under magnetic fields. Our results provide a microscopic route to zero-field skyrmion crystals with high skyrmion number in centrosymmetric square-lattice magnets.
We present a proposed E 8 & times; E 8 kinematic scaffolding for the standard model with pre-gravitation. The notation E 8 & times; omega E 8 used below records a split-complex exchange grading of two factors; it is not a tensor product of groups. Each factor branches through SU(3) & times; E 6 and thence through trinification, E 6 -> SU(3)3. The first factor supplies representation labels associated with the standard-model lineage, and the second supplies a mirror, pre-gravitational lineage. The identification of a single vector-like colour group across the two factors, and the proposed gravitational reading of the right-sector SU(2), remain dynamical hypotheses. Physical chiral fermions are not assigned to the E 8 adjoints: they are modelled by minimal ideals of the complex Clifford algebra C l 6 ( C ) $C{l}_{6}\left(\mathbb{C} ight)$ , while the 496 adjoint dimensions are used only as a representation-label ledger. The numerical split 208 + 288 is therefore a declared roster-matching convention, not an invariant decomposition and not a particle count. Spacetime enters through a selected six-dimensional split-biquaternionic vector space with a separately specified quadratic form of signature (3, 3). Its full frame group is SO(3, 3); deriving a soldering form and the proposed BF/Pleba & nacute;ski dynamics remains open. The conventional Clifford-Dirac operator is constructed independently and exactly: J 2 ( H s ) ${J}_{2}\left({\mathbb{H}}_{s} ight)$ realises the vector space R 3,3 ${\mathbb{R}}<^>{3,3}$ through its determinant, and after choosing H s congruent to M 2 ( R ) ${\mathbb{H}}_{s}\cong {M}_{2}\left(\mathbb{R} ight)$ its matrix gradient and adjugate act between real four-dimensional Weyl modules and factorise the wave operator in both orderings. The rank-two algebra supplies the quadratic spacetime layer, whereas J 3 ( O C ) ${J}_{3}\left({\mathbb{O}}_{\mathbb{C}} ight)$ supplies the proposed internal, generation and cubic spectral layer through the magic-star decomposition of e 8 C ${\mathfrak{e}}_{8}<^>{\mathbb{C}}$ . We distinguish established algebraic identities from model identifications and list the missing real-form, anomaly, localisation and invariant-coupling constructions explicitly.
In this study, the optical soliton solutions of the perturbed fourth-order Schr & ouml;dinger-Hirota equation with a parabolic law in the presence of nonlinear chromatic dispersion are investigated. To obtain analytic solutions, the modified new Kudryashov method, which provides effective solutions for fiber-optic models, and the modified extended tanh function method, which yields a wide variety of solutions through its multiple auxiliary equations, are employed. The combined use of these approaches enables the systematic derivation of a rich family of exact solutions despite the complex structure of the governing model. The analysis reveals the existence of various types of optical solitons, including bright, dark, singular, and kink-shaped structures. Furthermore, modulation instability analysis is performed to explore the stability characteristics of continuous-wave solutions, and the corresponding gain spectrum is examined in detail. The influence of the model parameters on both the soliton profiles and the gain spectrum is thoroughly illustrated through an extensive set of 2D and 3D graphical representations. With no restrictions on the scope of investigation, the findings provide a comprehensive understanding of the dynamical behavior associated with the studied equation, highlighting its potential relevance to nonlinear wave propagation and optical fiber communication systems.
We investigate non-equilibrium entropy in space plasmas by integrating the BBGKY hierarchy with Kappa distribution functions. Binary distribution functions are derived to account for particle correlations, and the Kappa distribution captures suprathermal tails observed in the solar wind. A quantum correction to the classical entropy is obtained from a semiclassical expansion of the Wigner function and evaluated analytically for the isotropic Kappa distribution. The correction, though negligible under typical solar wind conditions, may become significant in high-density, low-temperature regions. Three-dimensional visualizations reveal the evolution of the entropy landscape from turbulent to near-Maxwellian states. The framework is applied to solar wind plasma, showing consistency with observed suprathermal features and entropy production.
