A generalized notion of degenerate wormholes is introduced, defined by the vanishing of the metric determinant g at the throat. It is described by the polynomial, g^2 modified Einstein field equations. Building on this framework, we show that both the Einstein Rosen bridge and the Klinkhamer defect wormhole are exact vacuum solutions of the g^2 modified equations, valid globally including at the degenerate throat, while the Klinkhamer configuration additionally admits traversable geometries with b>2M, where b sets the length scale of the wormhole throat and M is a mass parameter. In contrast, standard Morris Thorne and thin shell wormholes, governed by the conventional (non regularized) Einstein equations, are intrinsically non degenerate and necessarily supported by exotic stress energy. Within a unified regularized system with matter, both thin shell and Klinkhamer wormholes appear as two qualitatively distinct classes of states: non degenerate with exotic matter versus degenerate with vacuum sharing the Einstein Rosen bridge as a common limiting configuration. This unified viewpoint clarifies why classical null energy condition no go theorems apply only to the non degenerate sector and suggests the possibility of stationary degenerate traversable wormholes that do not require NEC violation.

We investigate the evolution of cosmic structures in the power-law modified gravity model $f(R)=R+R^{1+δ}/R_c^δ$, where the dimensionless parameter $δ$ characterizes deviations from General Relativity. The background cosmological dynamics and the evolution of linear matter density perturbations are studied within the framework of metric $f(R)$ gravity. The modified perturbation equation is derived by introducing an effective gravitational coupling associated with the additional scalar degree of freedom, and the evolution of the growth factor, logarithmic growth rate, growth index, and the observable quantity $fσ_8(z)$ are investigated. The results show that the curvature correction enhances the growth of matter perturbations while remaining compatible with the observed late-time accelerated expansion for suitable values of the model parameter. The theoretical viability of the model is established through the ghost-free condition, Dolgov--Kawasaki stability criterion, positive scalaron mass, stable de Sitter solution, and chameleon screening mechanism. Comparison with representative viable $f(R)$ gravity models shows that the present theory achieves a consistent cosmological evolution with a single deviation parameter. The predicted modifications in the growth of structures and the effective gravitational coupling provide observable signatures that can be tested by forthcoming large-scale structure and weak-lensing surveys, providing a means to test curvature-induced modifications of gravity.

We construct the Higgs and Yukawa sectors as a structural completion of an algebraic three-generation model based on the complex Clifford algebra $\mathbb{C}\ell(10)$ with an intrinsic $S_3$ family symmetry. This addresses a common limitation of algebraic frameworks, in which Standard Model fermion multiplets and gauge symmetries may be described naturally, while the Higgs and Yukawa sectors remain less developed or absent. In the present framework, three algebraically distinguished fermion sectors are permuted by $S_3$, while the Standard Model gauge generators remain generation-independent. Higgs components are realised as right-action operators mapping weak-doublet fermion sectors into the corresponding weak-singlet sectors, and Yukawa coefficients are extracted using a Hilbert--Schmidt trace pairing. This yields two first-generation Higgs doublets with electroweak quantum numbers $(1,2,-1)$ and $(1,2,+1)$ under $SU(3)_C \times SU(2)_L \times U(1)_Y$, together with a Type-II-like separation between down-type and up-type Yukawa channels. Acting with the order-three family generator then generates a family-resolved Higgs sector organised into cyclic $S_3$ orbits. In the cyclically averaged Higgs limit, the Type-II-like Yukawa selection rule is preserved, while the generation-space Yukawa matrix is fixed algebraically and is non-diagonal in the algebraic generation basis. Under the usual implementation of electroweak symmetry breaking, the neutral Higgs couplings are aligned with the corresponding mass matrices, so tree-level flavour-changing neutral currents are not expected in this limit. The result is a constrained algebraic starting point for future $S_3$-breaking flavour phenomenology.

In this work, we prove the equivalence between the zero distributions of the Riemann zeta function ζ(s) and a two-dimensional (2D) Ising model with a mixture of ferromagnetic and randomly distributed competing interactions. At first, we review briefly the characteristics of the Riemann hypothesis and its connections to physics, in particular, to statistical physics. Second, we build a 2D Ising model, M_(FI+SGI)^2D, in which interactions between the nearest neighboring spins are ferromagnetic along one crystallographic direction while competing ferromagnetic/antiferromagnetic interactions are randomly distributed along another direction. Third, we prove that all energy eigenvalues of this 2D Ising model M_(FI+SGI)^2D are real and randomly distributed as the Möbius function μ(n), the Dirichlet L(s,\c{hi}_k ) function as well as the Riemann zeta function ζ(s). Fourth, we prove that the eigenvectors of the 2D Ising model M_(FI+SGI)^2D are constructed by the eigenvectors of the 1D Ising model with phases related to the Riemann zeta function ζ(s), via the relation ω(γ_2j) between the angle ω and the energy eigenvalues γ_2j, which form the Hilbert-Pólya space. Fifth, we prove that all the zeros of the partition function of the 2D Ising model M_(FI+SGI)^2D lie on an unit circle in a complex temperature plane (i.e. Fisher zeros), which can be mapped to the zero distribution of the Dirichlet L(s,\c{hi}_k ) function and also the Riemann zeta function ζ(s) in the critical line. In a conclusion, we have proven the closure of the nontrivial zero distribution of the L(s,\c{hi}_k ) function (including the Riemann zeta function ζ(s)).