Abstract: Presented is a geometric reformulation of the Born rule for finite-dimensional quantum systems. The state space is identified with complex projective space equipped with its canonical Fubini–Study geometry. Three structural axioms — locality in projective distance, invariance under projective unitaries, and additivity/non-contextuality for orthogonal decompositions — are shown to reduce the problem of transition probabilities to the framework of Gleason's theorem, thereby uniquely determining the transition probability P ([ψ] → [ϕ]) = |⟨ϕ|ψ⟩|². The argument provides a transparent geometric reformulation and interpretation of Gleason’s theorem. I then show that any relativistic Dirac spinor theory automatically realizes this geometry locally, so the Born rule is inherited without further assumption. Finally, I formulate a precise conjecture for an octonionic analogue involving the exceptional group G₂ (or F₄/E₈), then illustrate the idea with a simple finite toy model derived from the seven imaginary octonions and the Fano plane.

Abstract: This work presents a new geometric model describing the quantum states of electrons levitating on the surface of liquid helium, based on the introduction of the Kuznetsov tensor. The proposed approach unifies elements of quantum theory, differential geometry, and the thermodynamics of open systems, allowing quantum coherence to be interpreted as the invariance of the metric field under entropic deformations. The introduced Kuznetsov tensor describes singular perturbations of the metric and the interaction between the electron wave function and the surface potential of the medium, enabling a quantitative assessment of system stability and decoherence. The Kuznetsov entropic functional, a generalization of the Ricci flow, is developed to link scalar curvature, wave function density, and entropic potential. It is shown that the stationary solutions of this functional correspond to stable coherent qubit states, while the dynamics of quantum transitions can be represented as local curvature waves in the state space. Based on the proposed model, a stability criterion for qubits is formulated, taking into account the effects of singular deformations and entropic density. The analysis of coherence time as a function of the Kuznetsov tensor parameters reveals optimal stability regions at moderate values of β and γ. The presented results demonstrate the feasibility of a geometric description of quantum information and open the way to the creation of self-organizing qubits, in which stability is maintained by the internal symmetry of the metric field.

Abstract: This paper proposes and develops a novel physical quantity called "Xuan-Liang" (mystery quantity) aimed at providing a new theoretical framework for the unified description of dark matter and dark energy. Starting from the classical concept of "work", we extend it to "accumulation of power along a spatial path" and derive the expression for Xuan-Liang from first principles: X = 13 mv3, with dimension [M][L]3[T]−3. We treat the Xuan-Liang field as a continuum description of this quantity and argue that its equation-of-state parameter w should vary smoothly with energy density ρX. Based on the hyperbolic tangent function, we construct a specific dynamic phase-transition model w(ρX), allowing the Xuan-Liang field to behave as dark matter (w ≈ 0) in the early high-density universe and as dark energy (w ≈ −1) in the late low-density universe. We rigorously solve the Friedmann equations within this model, providing an analytic implicit solution for energy density evolution and performing numerical simulations. Results show that this model naturally reproduces the cosmic evolution from matter dominance to dark energy dominance, compatible with current observational data (e.g., Planck CMB data). The core prediction of this model is a smoothly evolving equation of state w(z) that can be precisely tested by next-generation cosmological surveys (e.g., Euclid, LSST). Furthermore, the path-integral origin of Xuan-Liang suggests new connections to topological properties and quantum gravity theories.

Abstract: Two damped dynamical systems have been studied for a long time in the literature: The damped Kepler system and the damped Harmonic oscillator. The damped factor is usually in the first order in the velocities and it is either time dependent or coordinate dependent whereas the potential is either autonomous or time and space dependent. One example is the Bateman-Caldirola-Kanai(BCK) model [1] which uses a time dependent Lagrangian/Hamiltonian for studying different aspects of dissipative systems at classical and quantum level. In all damped dynamical systems the role of first integrals is crucial because it simplifies the dynamics and allows for the integration or in general the study of the dynamical equations. To determine the first integrals of a dynamical system there are various methods. To mention a few, the Noether theorem, the specification of the functional form of the first integral I and the subsequent requirement dI/dt = 0 which leads to a system of differential equations whose solution provides the first integral, the Prelle-Singer method, and the δ−formalism developed by Katzin and Levin. For most of the damped systems, at least for the cases of the Kepler and the Harmonic Oscillator, there is a Lagrangian which describes the dynamical equations. This means that for these systems one may use the Noether theorem to compute the first integrals. This is the approach of the present work. We consider the time dependent damped dynamical systems described by a Lagrangian L = A(t)L0 where L0 is an autonomous Lagrangian - usually associated with the undamped system- and prove a Proposition which allows the computation of the Noether integrals in a systematic way using the geometric properties of a Riemannian space. The method can be applied to curved spaces where not much has been done. It is important to note that the metric of this Riemannian space is defined by the kinetic energy of the dynamical system and it is not the metric of the space where the motion of the system occurs. In this way the dynamics of the system ”locks” with its own geometry. The purpose of the Proposition is to present an algorithm which applies in definite steps and determines the Noether integrals using standard calculations. The application of the algorithm is demonstrated in the case of Kepler motion. We consider the simple case of Kepler motion with constant damping and a complex one with a time dependent damping of the form A(t) = γ t . For the time dependent case we find new results. For example the homothetic vector produces Noether integrals only for the specific values of γ = −1,−1 3 .

Abstract: This work establishes a functional-analytic obstruction within the standard cosmological framework. Concretely, we model the set of all possible FLRW expansion histories H(z; p) as a finite-dimensional manifold immersed in the space of smooth functions. The process of extracting cosmological observables - the co- moving distance DA and the Alcock-Paczynski parameter FAP - defines a smooth map Φ from this parameter manifold into an infinite-dimensional Banach space of constraints C = R × C([a, b]) × RN . Our central theorem shows that the image of Φ is a finite-dimensional submanifold of C, and thus has zero measure in the infinite-dimensional setting. Physically, this means that a generic set of combined early and late-universe observations - represented by a point in C - has zero probability of lying exactly on the FLRW manifold. The resulting inevitable incoherence is not due to noise or systematics, but to the inherent geometric rigidity of the FLRW metric ansatz. This result formalises why joint early and late-time cosmological constraints generically produce irreconcilable tensions within the standard framework.

Abstract: This paper presents a new mathematical framework for describing stellar pathways in the Galaxy based on the Kuznetsov tensor, a geometric–physical construct for modeling systems with singularities and complex curvature evolution. Traditional models rely on Newtonian gravity, relativistic metrics, or N-body simulations, but they inadequately capture discontinuous curvature zones, anisotropic gravitational fluctuations, and topological transitions. The proposed approach introduces a tensor field Kij that characterizes local and global singularities influencing stellar trajectories. The model defines a modified metric evolution equation analogous to a generalized Ricci flow, augmented with a singularity-driving term governed by the Kuznetsov tensor. This enables refined description of star–galaxy interactions, detection of critical curvature corridors, and prediction of pathway branching, stability domains, and large-scale reconfiguration. The framework can be applied to spiral-arm evolution, exoplanet migration, interstellar transfer routes, and engineered stellar navigation systems. Overall, the Kuznetsov tensor provides a mathematically consistent and physically insightful tool for modeling star-route dynamics in singular, evolving gravitational geometries.

Abstract: Similar to electroweak interaction, strong force and electromagnetism can have similar Higgs mechanism mediated interaction. Thus, gluons can acquire mass. And, neural colored gluons have larger mass than colored gluons. Total, we can have eight gluons without red-anti-red gluon. The puzzle of proton or neutron mass can be solved. We can also derive a new SU(5) model to include all the above eight gluons, three W/Z bosons, photon, Higgs boson, three generations of leptons and quarks to make a new 5x5 SU(5) model. Wightman axioms can be fulfilled in this new SU(5) without causing proton decay crisis. We can also add the 4x4 four dimensional spacetime tensor integrating mass-energy density, light pressure, electric fields, and magnetic fields as well as four gradients to make a new contravariant SO(10) model. Weyl tensor and Ricci tensor related to the new SO(10) model are also given. Thus, grand unified theory or theory of everything can be obtained, that is compatible with four dimensional spacetime without extra-dimension needed in string theories.

Abstract:

We establish through rigorous mathematical proof that no physical constant can be ‘absolute’ in the sense of being simultaneously determinable with infinite precision, independent of measurement scale, and independent of cosmological epoch. Our framework rests on three pillars: (i) information-theoretic bounds (Bekenstein-Holographic principle), (ii) renormalization group analysis, and (iii) functional analysis of oscillatory operators on Sobolev spaces. We introduce the Dynamic Zero Operator (DZO)—a rigorously defined linear operator on H²(ℝ) with oscillatory kernel—and prove that: (a) Borwein π-algorithms converge to DZO fixed points, (b) Riemann zeta zeroes are DZO eigenvalues for specific kernel choice, (c) the geometric constant π is not absolute but emerges as scale-dependent projection π_eff(Λ, R). This establishes a profound trinity: Borwein algorithms ↔ DZO spectral theory ↔ ζ(s) zeroes, unified by modular symmetry and phase cancellation. We provide: (1) complete proof that ΛCDM parameters (H₀, Λ) cannot be fundamental constants, (2) numerical example demonstrating π_eff(Λ) dependence, (3) testable predictions linking Borwein convergence to GUE statistics. This falsifies ΛCDM as currently formulated and provides foundation for scale-dependent effective cosmology.

Abstract: This paper constructs a novel metaphysical framework for the archetype of the “Spiritual Sniper”—a figure whose silent awareness operates with precision at the level of karmic, temporal, and ontological structures. By synthesizing spiritual traditions, quantum mechanics, and differential geometry, the sniper is modeled as a boundary-conscious agent who collapses karmic fluctuations via directed attention at the point-like Now. The sniper’s inner landscape is represented as a dynamic geometry influenced by Ricci flow and entropy gradients, while his outer targeting system is governed by entangled karmic networks and discrete automata on a finite temporal cycle. Key equations adapt tools such as the Trotter formula, von Neumann entropy, Landauer’s erasure principle, the Schr¨odinger equation, and spherical harmonics to spiritual phenomenology. In particular, silence is formalized as the zero-mode of karmic spectral decomposition, and spiritual attention is quantized into discrete units analogous to Planck-scale awareness. Through references to the Bhagavad Gita, near-death experiences, Eckhart Tolle, and Arjuna’s archetype, the sniper’s dharmic path is formulated as a low-entropy, minimal-energy evolution from ego to Supreme Observer. This multi-disciplinary synthesis offers a new perspective on inner transformation and collective karmic disentanglement.

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 introduce a new combinatorial framework for classical mechanics - the Ramsey -Hamiltonian approach - which interprets Poisson-bracket relations through the lens of finite and infinite Ramsey theory. Classical Hamiltonian mechanics is built upon the algebraic structure of Poisson brackets, which encode dynamical couplings, symmetries, and conservation laws. We reinterpret this structure as a bi-colored complete graph, whose vertices represent phase-space observables and whose edges are colored gold or silver according to whether the corresponding Poisson bracket vanishes or not. Because Poisson brackets are invariant under canonical transformations (including their centrally extended Galilean form), the induced graph coloring is itself a canonical invariant. Applying Ramsey theory to this graph yields a universal structural result: any six observables necessarily form at least one monochromatic triangle, independent of the Hamiltonian’s specific form. Gold triangles correspond to mutually commuting (Liouville-compatible) observables that generate Abelian symmetry subalgebras, whereas silver triangles correspond to fully interacting triplets of dynamical quantities. When the Hamiltonian is included as a vertex, the resulting Hamilton–Poisson graphs provide a direct graphical interpretation of Noether symmetries, cyclic coordinates, and conserved quantities through star-like subgraphs centered on the Hamiltonian. We further extend the framework to Hamiltonian systems with countably infinite degrees of freedom - such as vibrating strings or field-theoretic systems - where the infinite Ramsey theorem guarantees the existence of infinite monochromatic cliques of observables. Finally, we introduce Shannon-type entropy measures to quantify structural order in Hamilton–Poisson graphs through the distribution of monochromatic polygons. The Ramsey–Hamiltonian approach offers a novel, symmetry-preserving, and fully combinatorial reinterpretation of classical mechanics, revealing universal dynamical patterns that must occur in every Hamiltonian system regardless of its detailed structure.

Abstract: The structural vibration of industrial droplet dispensers can be modeled by telegraph-like equations to a good approximation. We reinterpret the telegraph equation from the standpoint of an electric-circuit system consisting of an inductor and a resistor, which is in interaction with an environment, say, a substrate. This interaction takes place through a capacitor and a shunt resistor. Such interactions serve as leakage. We have performed analytical investigation of the frequency dispersion of telegraph equations over unbounded one-dimensional domain. By varying newly identified key parameters, we have not only recovered the well-known characteristics but also discovered crossover phenomena regarding phase and group velocities. We have examined frequency responses of the electric circuit underlying telegraph equations, thereby confirming the role as low-pass filters. By identifying a set of physically meaningful reduced cases, we have laid foundations on which we could further explore wave propagations over finite domain with appropriate side conditions.

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.