Abstract: We present the Primacohedron, a unified framework linking p-adic string resonances, zeta-function spectra, and emergent spacetime geometry. By extending non-Archimedea string amplitudes and constructing a spectral correspondence that maps Riemann–Dedekind zeros to a Hermitian operator, the model reproduces GUE-type fluctuations temporally and closed-string coherence spatially. A curvature–spectral duality yields emergent geometry, holographic behaviour, and dynamically Bekenstein-saturating learning. The framework further incorporates Diophantine geometry: radicals and height functions arise as spectral-energy sums of prime resonances, and the abc-inequality emerges as a curvature-stability condition on an adelic manifold. An adelic operator pair encodes analytic zeros and heights simultaneously, suggesting a geometric route toward Riemann Hypothesis (RH) and abc via curvature regularity. Finally, we extend the structure using perfectoid geometry and p-adic Hodge theory. Perfectoid tilting links mixed- and equal-characteristic layers through a curvature-preserving duality, while Hodge filtrations provide a cohomological interpretation of spectral dimensionality and arithmetic time. Together, these developments position the Primacohedron as a geometric, cohomological, and operator theoretic paradigm for understanding analytic and Diophantine phenomena within a single adelic spacetime.

Abstract: Three Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene establish three complementary fine-structure constants, two of which are negative. Each introduces its own specific set of Planck units. Hence, two sets of basic Planck units are real, and two are imaginary. The elementary charge is the same in all those sets of Planck units, establishing equality between the products of each fine-structure constant and the speed of light it is associated with. All fine-structure constants are related to each other through the constant of \( \pi \), which indicates that they do not vary over time. The negative complementary fine-structure constant inferred from graphene reflectance is dual to the fine-structure constant. The assumption of universality of the black hole entropy formula to the remaining two stellar objects emitting perfect black-body radiation, less dense than a black hole (neutron stars and white dwarfs), renders their energies exceeding their mass-energy equivalence. To accommodate this unphysical result, we introduced an imaginary mass, and we defined three complex energies in terms of real and imaginary Planck units, storing the surplus energy in their imaginary parts. It follows that black holes are fundamentally uncharged and have a vanishing imaginary mass. We have derived the lower bound on the mass of a charged black-body object, the upper bound on the radius of a white dwarf, and the equilibrium density for all three complex energies. The complex force between real masses and imaginary charges leads to the complex black-body object's surface gravity and generalized Hawking radiation complex temperature. Furthermore, based on the Bohr model for the hydrogen atom, we show that complex conjugates of this force represent atoms and antiatoms. The proposed model considers the value(s) of the fine-structure constant(s), which are otherwise neglected in general relativity.

Abstract: This comprehensive work presents detailed mathematical formu- lations and technical refinements addressing critical theoretical chal- lenges in the Expanded Quantum String Theory with Gluonic Plasma (EQST-GP) framework. We provide complete derivations for the neg- ative energy density mechanism, Majorana gluon dark matter proper- ties, and rigorous compatibility analysis with Swampland Conjectures. The enhanced model incorporates moduli stabilization with uplifting potentials, refined gravitational wave predictions, and precise numeri- cal verifications using symbolic computation. All derivations maintain mathematical rigor while ensuring phenomenological consistency with cosmological observations and experimental constraints.

Abstract: This paper investigates the Teleparallel Robertson-Walker (TRW) F(T) gravity solutions for a cosmological electromagnetic source. We use the TRW F(T) gravity field equations (FEs) for each k-parameter value case and the relevant electromagnetic equivalent of equation of state (EoS) to find the new teleparallel F(T) solutions. For flat k=0 cosmological case, we find analytical solutions valid for any scale factor. For curved k=±1 cosmological cases, we find new exact and far future approximated teleparallel F(T) solutions for slow, linear, fast and very fast universe expansion cases summarizing by usual and special functions. All the new solutions will be relevant for future cosmological applications implying any electromagnetic source processes.

Abstract: The Riemann Hypothesis, a core unsolved problem in number theory, asserts that all nontrivial zeros of Riemann's zeta function lie on the critical line of the complex plane. This paper proposes an axiomatic theory of RDSUT (symmetric equivalence, derived generation, singularity completion, and closed-loop reduction) based on the E₈ exceptional Lie group. By constructing an 8-dimensional orthogonal embedding of A₂ subsystems to form an E₈ root system, we prove the discrete-continuous duality theorem and rigorously derive the equivalence between the E₈ fundamental scale and the Riemann critical line, ultimately completing a geometric equivalence proof of the Riemann Hypothesis. All conclusions are supported by rigorous mathematical derivations and numerical experiments (30 A₂ subsystems constructing an 180-dimensional truncated E₈ version with constant discretization error and precise fundamental scale of 1/2), forming a complete logical closed loop of "axioms-constructions-theorems-validation-proofs." The construction logic, fundamental scale, and discrete-continuous duality of the truncated version are entirely consistent with the complete E₈ (40 A₂ subsystems, 240 root vectors), demonstrating that the conclusions can be directly generalized to the complete E₈. The innovations of this paper include: (1) Establishing an orthogonal embedding construction method between A₂ subsystems and the E₈ Lie group; (2) Proposing the RDSUT axiomatic system to standardize the geometric structure of E₈; (3) Revealing the geometric origin of the Riemann critical line in E₈; (4) Providing a proof path that combines theoretical rigor with experimental reproducibility.

Abstract: The Primacohedron provides a unifying adelic framework in which number-theoretic, spectral, and geometric structures arise from prime-indexed resonance modes. Building on its interpretation of the non-trivial zeros of the Riemann zeta function as the spectrum of a Hilbert–Pólya–type operator, this work extends the construction to Diophantine geometry and the abc conjecture. We show that radicals and height functions naturally correspond to spectral-energy sums of prime resonances, while the abc inequality emerges as a curvature-stability condition on an underlying adelic manifold. Within this spectral–Diophantine duality, violations of RH or abc manifest as curvature singularities of a unified spectral–height geometry. We further introduce an adelic operator pair (H_spec, H_ht) encoding L-function zeros and arithmetic heights simultaneously, propose a curvature-anomaly correspondence linking analytic and Diophantine pathologies, and outline a programme suggesting how a completed Primacohedron—extended to motivic L-functions and Vojta theory—could imply both RH and abc through a single geometric regularity principle. This synthesis positions the Primacohedron as a candidate framework for an arithmetic spacetime whose curvature governs both the analytic behaviour of zeta and L-functions and the Diophantine behaviour of rational points, offering a geometric route toward long-standing conjectures in number theory.

Abstract: Chrology establishes a foundational epistemological paradigm that interprets the universe as the structured manifestation of its own original existence and ontological presence. Through five hierarchically nested domains Extramicroscopique, Microscopique, Normal, Macroscopique, and Extramacroscopique Chrology organizes reality into a re-cursive matrix of 25 analytical subfields per domain. This architecture enables high-resolution synthesis of phenomena spanning quantum decoherence, molecular self-assembly, biochemical networks, ecological systems, planetary morphogenesis, and supergalactic topology. This model integrates historical insights from Egyptian Civilizations, Galilei’s heliocentrism to Laplace’s celestial mechanics with contemporary datasets (JWST, Gaia, COBE, Hubble) and observational evidence spanning lunar thermodynamics, planetary magnetosphere behavior, asteroid morphology, Martian geostructures, cryogenic satellites, and galactic clusters. Microscopic data includes viral capsid dynamics, bacterial signaling, atomic orbitals, molecular vibrations, quantum tunneling, electron spin asymmetries, and nanofluidic fields. Terrestrial observation across Europe, Africa, America, and Asia validates the recurrence of these domains via unaided human perception. Mathematical decomposition and dimensional harmonics underpin a unified model demonstrating that at any given portion of the universe in existing and presence regard-less of scale or location is mapped through these five （5） domains. Chrology defines a universal scientific architecture that frames existence as a nested continuum structured by recurrence, symmetry, and observability. It unifies predictive diagnostics and transdisciplinary synthesis across physics, biology, chemistry, geotechnics, and cosmology. Phenomena are interpreted as manifestations of the universe’s intrinsic logic. This logic is formalized into scalable grammar, enabling recursive modeling, symbolic integration, and epistemic advancement throughout the natural sciences.

Abstract: Background. Unifying number theory, string amplitudes, and spacetime emergence remains a central challenge in fundamental physics. Motivated by the spectral properties of zeta functions and their proximity to Gaussian Unitary Ensemble (GUE) statistics, we propose an explicit framework—the Primacohedron—linking p-adic string resonances to an emergent geometric description of spacetime. Methods. We extend the non-Archimedeanamplitude formalism for open/closed p-adic strings, develop a spectral correspondence mapping Dedekind/Riemann zeros to eigenvalues of a Hermitian operator H, and introduce a learning framework (Corridor Zero/One) for reconstructing spacetime spectra. Additional sections explore the arithmetic-holographic connection, spectral geometry, and cosmological implications. Results. The expanded model unifies arithmetic quantum chaos, random matrix theory, and holography. Temporal fluctuations arise from open p-adic resonances following GUE statistics, while spatial coherence emerges through closed zeta sectors. A curvature–spectral duality defines emergent geometry, black-hole microstructure yields porous horizons, and algorithmic learning saturates the Bekenstein bound dynamically. Conclusions. ThePrimacohedron thus establishes a spectral route from number theoretic operators to spacetime dynamics, blending p-adic strings, zeta-function operators, random matrices, and holographic complexity into a single coherent synthesis. In addition, Primacohedron also suggests a concrete pathway toward a Hilbert–P´olya-type operator and offers a physically motivated set of sufficient conditions under which Riemann Hypothesis would follow.

Abstract: This paper develops a rigorous mathematical framework unifying physical spacetime, quantum mechanical structures, and conscious experience. Beginning with the hypothesis that perception resides within an infinite-dimensional Hilbert space H∞, and that physical spacetime M3,1 is an emergent cognitive construct, we propose the total manifold as a triple tensor product: M3,1⊗H∞⊗O(H), where O(H) is the space of projection operators associated with conscious observers.Observers are modeled not merely as Dirac delta functions but as evolving strings of projection operators, with entropic gradients driving perceptual forces. These projection operators define a local arrow of time via entropy increase and are cyclically structured in time-crystal fashion over the Yuga cosmology. Inter-observer relations are captured using graph theory and gauge-theoretic techniques, inducing emergent gauge fields and topological solitons. A significant feature of the theory is the introduction of the Micro-Mini-Black-Holein- Brain (MMBHB), enabling quantum gravitational dualities with cortical dynamics. Modular Hamiltonians, entanglement currents, and operator algebras are employed to analyze the non-unitary perceptual evolution in a fundamentally non-commutative geometry. The resulting psychophysical feedback mechanisms are shown to affect null geodesics and possibly contribute to the observed cosmological constant.This work contributes a formal psycho-geometric ontology, bridging consciousness studies, quantum information, and fundamental physics.

Abstract: Since its inception, Quantum Mechanics (QM) has engaged many philosophers the subspecialty of a few. More recently, QM has attracted the attention of a few neuroscientists modelling neuronal and higher brain function. This pedagogical review aims to make QM more accessible to neuroscientists and philosophers less familiar with its basics. Emphasis is on QM measurement and entanglement. We write at an intermediate technical level between elementary textbooks and contemporary journals. Against authoritative advice, we use the “crutch of visuality” to ease comprehension of notoriously difficult ideas in QM, e.g., complex Hilbert space, an apparatus setting the outcomes of an experiment. To keep more advanced readers interested, we sow philosophical comments throughout the text. These touch on under-discussed themes (e.g., complex numbers in QM), seek to clarify pedagogically neglected matters (e.g., particles of definite energy), and strike occasional possibly novel points about QM (e.g., the metaphysical double-humility of QM, the quantum state as an ontologico-epistemological hybrid). We strive ultimately to show that even orthodox QM is more concretely graspable and philosophically better thought-out than many have judged.

Abstract: We present a geometric field theory in which the action and field equation are constructed from a vector field and its covariant derivative and have full general covariance in a higher-dimensional spacetime. The field equation is the simplest possible generalisation of the Poisson equation for gravity consistent with general covariance and the equivalence principle. It contains the Ricci tensor and metric acting as operators on the vector field. If the symmetrised covariant derivative is diagonalisable across a neighbourhood under real changes of coordinate basis, spacetime coincides with a product manifold. The dimensionalities of the factor spaces are determined by its eigenvalues and hence by its algebraic invariants. Tensors decompose into multiplets which have both Lorentz and internal symmetry indices. The vector field decomposes into conformal Killing vector fields for each of the factor spaces.The field equation has a `classical vacuum' solution which is a Cartesian product of factor spaces. The factor spaces are all Einstein manifolds or two-dimensional Riemannian manifolds. All have a Ricci curvature of roughly the same order of magnitude, or are Ricci-flat. A worked example is provided in six dimensions.Away from this classical vacuum, connection components in appropriate coordinates include $SO(N)$ gauge fields. The Riemann tensor includes their field strength. Unitary gauge symmetries act indirectly on tensor fields and some or all of the unitary gauge fields are found amongst the $SO(N)$ gauge fields. Symmetry restoration occurs at the zero-curvature `decompactification limit', in which all dimensions appear on the same footing.

Abstract: We explore the hypothesis that all physical constants may be derived from a single dimensionless parameter: the normalized time period ˜ T = T tP of a cyclic universe. This work reviews the theoretical background, develops models for key constants including α, G, Λ, h, e, mp/me, and kB, and discusses the implications of deriving physical law from cosmic periodicity. Building on earlier models of cyclic time and restorative potentials, we show that ˜ T governs both microscopic recurrence structures and macroscopic physical constants, enabling the derivation of {G, ℏ, α,Λ,mp,me, kB} from a post-collapse cosmological boundary condition. The restorative potential Φ(t), previously modeled via a divergence at the end of the cycle t → T−, is shown to encode a universal quantization spectrum through a Taylor expansion, modular embeddings, and spectral collapse at ˜ T. Conscious observers are represented geometrically as Dirac delta functions embedded in a symplectic recurrence manifold, where their roles within the cosmic drama are projected as time-evolved quantum histories. We further demonstrate that the structure of time near the collapse limit maps onto tree-like causal graphs of souls, culminating in a modular procession toward an entropy-free boundary state. Connections to string dualities, holography, Fourier-dual entropy flows, neural recurrence, and non-commutative time operators are examined. In this formulation, ˜ T replaces arbitrary physical input with a single parameter encoding global cyclic memory, thereby offering a minimal yet comprehensive rewriting of fundamental physics.

Abstract: We propose a novel, constructive framework toward resolving the 3D Kakeya conjecture by introducing a differentiable sweep embedding based on octonionic conjugation, fractal geometry, and spectral analysis. Unlike prior analytic approaches focused primarily on dimension bounds, our method constructs explicit sweep sets that rotate a unit needle through all directions in three-dimensional space while maintaining a strictly positive volume. The embedding leverages the non-associative structure of the octonions and the triality symmetry of the exceptional Lie group , producing smooth rotational maps with a fractal, self-similar structure. Using Jacobian determinant analysis, we rigorously derive a lower volume bound expressed through the Riemann zeta function , revealing a spectral correspondence between geometric measure theory and quantum statistical mechanics. This embedding framework not only satisfies directional completeness but also introduces a spectral interpretation of volume via angular mode summation. Our construction offers a differentiable, algebraically rich, and generalizable approach to Kakeya-type problems, with implications for harmonic analysis, number theory, and high-dimensional mathematical physics.

Abstract: In this book, we present a mathematically consistent paradigm for describing nature. Modern physics is supported by an immense body of experimental and observational data, alongside a theoretical framework that, at times, aligns with this data—and at other times, diverges from it. The absence of a clear theoretical explanation for the cause of quantum phenomena, combined with the growing mismatch between cosmological observations and theoretical predictions, suggests that a fundamental principle of nature is missing from current physical theories. The Self-Variation Theory introduces such a principle into the theoretical foundations of physics.In this work, we present the core principles and primary consequences of SVT. The theory is built upon three foundational elements:the Principle of Self-Variation,the Principle of Conservation of Energy-Momentum, and a definition of the rest mass for fundamental particles.From these principles, Self-Variation Theory leads to a number of key conclusions:it predicts a specific internal structure of particles that extends across all distance scales,it provides a unified explanation for particle interactions,it accounts for both cosmological data and quantum phenomena, offering a coherent framework that connects them. The theory's predictions regarding the origin, evolution, and current state of the universe are in agreement with available observational evidence. From subatomic scales to astronomical distances spanning billions of light-years, Self-Variation Theory demonstrates a remarkable consistency with experimental and observational data. The structure of the book has been carefully designed to ensure the necessary clarity and precision in presenting the theory. A sequence of interconnected derivations begins with the fundamental principles, proceeds through their synthesis, and culminates in the field equations of the theory—applicable across all distance scales.

Abstract: We investigate the planar Dirac equation with the most general time-independent contact (singular) potential supported on a circumference. Taking advantage of the radial symmetry, the problem is effectively reduced to a one-dimensional one (the radial), and the contact potential is addressed in a mathematically rigorous way using a distributional approach that was originally developed to treat point interactions in one dimension, providing a physical interpretation for the interaction parameters. The most general contact interaction for this system is obtained in terms of four physical parameters: the strengths of a scalar and the three components of a singular Lorentz vector potential supported on the circumference. We then investigate the bound and scattering solutions for several choices of the physical parameters, and analyze the confinement properties of the corresponding potentials.

Abstract: We present an algebraic reformulation of Calabi–Yau geometry grounded in the complexified exceptional Jordan algebra and its automorphism group . In this framework, core features of Calabi–Yau threefolds—including complex structure, Kähler form, SU(3) symmetry, and Ricci-flatness—arise not from differential geometry, but as intrinsic consequences of algebraic symmetries and trace identities. The complex structure is realized via scalar multiplication, the Kähler form through the Jordan trace and triality, and Ricci-flatness as a trace-free condition on commutator curvature in the derivation algebra. A non-vanishing holomorphic volume form is naturally provided by the cubic norm of , ensuring vanishing first Chern class.Rather than treating Calabi–Yau geometry as a set of imposed geometric conditions, this approach suggests it can be emergent from exceptional algebra. The abundance of SU(3)-compatible configurations within the algebra provides a constructive moduli space, offering insight into the multiplicity of Calabi–Yau structures. We further outline how this algebraic framework may connect to quantum gravity and unification, where both spacetime and curvature arise from derivation symmetries. Our aim is not to replace classical results, but to propose a new lens through which their deeper algebraic origin may be understood.

Abstract: Taking the A₂root system as the sole core symmetric carrier, this paper constructs a cross-disciplinary self-consistent loop of "algebraic coding-geometric realization-quantum stabilization-topological verification". Through structure-preserving mappings and Qiskit quantum simulations, it achieves in-depth integration and empirical verification of 3-dimensional Calabi-Yau (CY) manifolds, quantum stable systems, and the Poincaré conjecture. The core results include: 1) A constructive proof of the exclusive correspondence between the A₂ root system and 3-dimensional CY manifolds—the rank 2 of the root system strictly determines the Hodge number , and the Cartan subalgebra forms an inner product-preserving mapping with the Kähler moduli space; 2) Establishment of the A₂ root system-driven quantum system theory: algebraic preparation of 4 Bell states via the structure-preserving mapping , realization of noise-resistant stabilization with fidelity ≈1.0 in noisy environments based on the dual reflection mechanism of polarized root systems, and further construction of a 2-qubit time crystal (TIME crystal), verifying period doubling (response period 4.0 = 2× driving period 2.0) and spontaneous time-translation symmetry breaking; 3) Proof of the equivalence between supersymmetric score locking () and the simply connectedness of CY manifolds (). Combined with the convergence of Ricci flow curvature (error < 10⁻⁴), it provides a specific instance verification of the Poincaré conjecture under "CY manifold + quantum constraints".

Abstract: We propose a unifying framework that generalizes classical Euclidean subspace hierarchies into a pseudo-Riemannian landscape by introducing a signature-based stratification of manifolds. For a total dimension D = m+n, we systematically construct the space of submanifolds M_m,n with m spacelike and n timelike dimensions. These give rise to a structured family of pseudo-Grassmannians Gr_(m,n)(M,N), where signature plays a central geometric and physical role. We extend key constructions— such as Pl¨ucker embeddings, local charts, homogeneous space representations, volume forms, and cohomological invariants—to these indefinite-signature settings. Furthermore, we explore implications for symmetry breaking, field theory on signature-changing manifolds, and brane-world cosmologies involving M(1,3), M(2,2), and M(3,1) universes. Applications to twistor theory, supersymmetry, and quantum gravity foams with signature fluctuations are included. The resulting geometric machinery provides new tools for modelling transitions between classical, subtle, and metaphysical layers of spacetime.

Abstract: The Weinberg angle (weak mixing angle) θW is a fundamental parameter of the Standard Model that describes the mixing between the electromagnetic and weak forces after electroweak symmetry breaking. In the conventional framework, sin2 θW is a free parameter requiring experimental determination. We present a derivation of the Weinberg angle from first principles within the Kosmoplex Theory framework, which derives 4D physical constants as projections of geometric structures in 8-dimensional octonionic space, obtaining sin2 θW (mZ ) = sin2(1)/√3π ≈ 0.23064 with zero adjustable parameters at tree level. This value arises from the geometric structure of three fundamental “glyphs”, discrete octonionic operators in the 8D substrate that project to observable numerical values in 4D spacetime through the Octonionic Binomial-Modular Transform (OBMT). We demonstrate agreement with experimental data across energy scales from 1 GeV to the Z-pole (mZ ≈ 91 GeV), with minimal post-OBMT running characterized by a single parameter. The theory predicts sin2 θW (10 GeV) ≈ 0.2343, suggesting a testable discrepancy with current experimental extractions that warrants further investigation.

Abstract: The classical Maxwell equations, while foundational to electromagnetism, exhibit an inherent asymmetry in their treatment of electric and magnetic sources—electric charges and currents are explicit, yet magnetic monopoles remain absent. Prior works, such as those by Milton (2006) and Griffiths (2013), have formally extended Maxwell’s equations to incorporate magnetic monopoles, but they stop short of exploring the equations’ geometric structure and calculus properties under the exterior differential form framework, especially the critical distinction between the classical form dF = 0 (no magnetic sources) and the generalized form dF = μ₀J_m (with magnetic sources). Additionally, these works lack a rigorous construction of the Lagrangian density for the generalized system and a derivation of the equations via Noether symmetry, which are essential for linking the theory to fundamental principles of symmetry and conservation.In this work, we revisit the generalized Maxwell equations with magnetic monopoles from a perspective rooted in Dirac’s emphasis on mathematical consistency, symmetry, and physical intuition. We first contextualize our work within existing literature, explicitly acknowledging the contributions of Milton and Griffiths in formulating the vectorial extension of Maxwell’s equations with magnetic sources. We then advance the field by: (1) systematically analyzing the geometric structure of the generalized equations in exterior differential form—including cohomological properties of the field strength 2-form F and the role of the Hodge dual in preserving duality symmetry; (2) constructing a gauge-invariant Lagrangian density that couples both electric and magnetic sources to the electromagnetic field, and deriving the generalized equations via the principle of least action; (3) applying Noether's theorem to the Lagrangian, showing that duality symmetry implies the conservation of both electric and magnetic charges, and that the equations themselves emerge as a consequence of this symmetry.Our formulation maintains manifest Lorentz covariance and duality symmetry, resolving ambiguities in vectorial descriptions and providing a unified geometric framework for electromagnetism with magnetic monopoles. We verify consistency by decomposing the 4-dimensional differential form equations into 3-dimensional vector form, confirming correspondence with charge conservation and dimensional analysis. Finally, we connect our results to Dirac’s original work on monopole-induced charge quantization, showing that our Lagrangian and symmetry arguments reinforce the necessity of the Dirac quantization condition.