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| ai.viXra.org > Mathematical Physics > 2606 |
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[8] ai.viXra.org:2606.0053 [pdf] submitted on 2026-06-21 00:39:11
Authors: Allan Plankhart
Comments: 3 Pages. (Note by ai.viXra.org Admin: Part of the diagram is cut off)
We present a timeless mathematical construction of a pre-Big Bang universe. Starting from theempty set, we generate an infinite alternating chain of "something" and "anti-something", buildhigher-dimensional arrays, collapse to dimensions 1—3 due to Hurwitz’s theorem, introduce complex-valued fields with a U(1) symmetry ("oscillations" without time), ascend the division algebra ladderC → H → O, recover the Standard Model gauge group via Cohl Furey’s functor, extract electro-magnetic and spinor fields, and finally let time emerge as a monoid action R≥0 on the field space.The entire pre-temporal phase is a static diagram of functors and natural transformations; no timeparameter appears until the last step.
Category: Mathematical Physics
[7] ai.viXra.org:2606.0047 [pdf] submitted on 2026-06-19 03:26:58
Authors: E. P. J. de Haas
Comments: 144 Pages. https://doi.org/10.5281/zenodo.20747973
The biquaternion (BQ) algebra, realized at the Pauli-spin level as the full matrix algebra (M(2,mathbb{C})), provides a compact framework in which classical field theories can be organized through a common hierarchy of algebraic products and channel decompositions. This paper develops that hierarchy for both electrodynamics and relativistic fluid dynamics and explores the structural relationships that emerge between them.In electrodynamics, the BQ hierarchy reproduces the electromagnetic field, Lorentz force law, Maxwell equations, wave equations, and interaction-conservation relations from a sequence of algebraic products involving the four-current and four-potential. In relativistic fluid dynamics, the corresponding hierarchy generates the continuity equation, Bernoulli relation, vorticity transport, Euler—Lamb momentum balance, acoustic wave equations, helicity relations, and stress-energy conservation through analogous products of the fluid four-velocity and momentum-density field.The central result is that both theories share the same four-level algebraic structure and the same geometric channel decomposition. The force equations (JB=F) and (UH=F) emerge as the unique four-channel level of their respective hierarchies, revealing a common organization that is largely hidden in conventional formulations. The analysis further identifies the acoustic refractive index (n_s=c/c_s) as a hierarchy-wide parameter linking relativistic and practical fluid dynamics, and locates the greater complexity of fluid dynamics in the self-referential coupling between velocity and momentum-density fields.The paper does not introduce new physical postulates. Rather, it proposes that the BQ channel decomposition provides a unified organizational framework that makes structural relationships between electrodynamics and fluid dynamics explicit, offering a common language for force, energy, momentum, vorticity, helicity, and wave phenomena across both theories. In this sense, the work is intended as a conceptual and educational framework as much as a mathematical formalism, and as a foundation for future applications in gravitation, magnetohydrodynamics, and AI-assisted symbolic physics.
Category: Mathematical Physics
[6] ai.viXra.org:2606.0040 [pdf] submitted on 2026-06-16 20:34:28
Authors: J. W. McGreevy
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a first-principles derivation of 3+1-dimensional Lorentzian spacetime andEinstein—Cartan gravity from the arithmetic structure of the Bost—Connes C∗-dynamicalsystem. Starting from the primitive object (ABC,σt) and its KMS states, we construct ascale-invariant quadratic form on the idèle class space via the GNS inner product. Irregularprimes appear as topological defects through class-group shifts, inducing a shear in thequadratic form that maps canonically to a point τp ∈ H. For the resonant prime 691 this shear aligns with the hexagonal fixed point τ = ρ, where the Z/6Z stabilizer providestopological protection against adelic screening.The local 3-defect patch (primes 37, 59, 691) carries a non-abelian su(2) connection.Global adelic closure projects the three independent directions onto a single screened ray (time) while preserving the protected 3-dimensional spatial structure generated by the Eisenstein lattice at ρ. Explicit differentiation of the quadratic-form embedding with respect to the shear parameter yields the six generators of the Lorentz algebra so(1,3), with compact rotations locked at resonance and non-compact boosts arising from unbounded shear.The second-order Magnus expansion of parallel transport produces a residual torsion vector whose components are fixed by phase mismatches between the three incommensurate modular frequencies. Promoting the local Quantum Geometric Tensor metric on the 3-defect patch together with the screened ray yields a derived 4D Lorentzian geometry. On the resulting resonant spectral triple we evaluate the Chamseddine—Connes Spectral Action. Functional variation with respect to the emergent metric produces the Einstein—Cartan field equations, with the cosmological term fixed by the converged quadratic sum over the 47 irregular primes below 691 and the torsion source fixed by the linear sum. The next heat-kernel coefficienta4 generates quadratic gravity corrections that are parametrically suppressed by the samearithmetic factors, establishing ultraviolet safety. All physical structures—spacetime signature, Lorentz symmetry, torsion, and gravitational dynamics—emerge as necessary consequences of adelic consistency acting on the arithmetic defects of the Bost—Connes system. No external gravitational or field-theoretic ansatzis introduced.
Category: Mathematical Physics
[5] ai.viXra.org:2606.0029 [pdf] submitted on 2026-06-10 21:28:40
Authors: J. W. McGreevy
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a fully rigorous, self-contained analytic proof that the non-trivial zeros of the Riemann zeta function are precisely the globally stable skin modes of an inner-fluctuated Dirac—Zeta operator on the compactified modular curve X(1)∗. The proof proceeds entirely within the framework of unbounded operators on weighted Sobolev spaces, regularized Fredholm determinants on Hilbert—Schmidt ideals, the Riemann—Roch theorem, and a vanishing theorem for fractional states. The simple pole at s = 1 is interpreted as a topological point gap that triggers the NonHermitian Skin Effect. Spectral stability under discretization is established via uniform pseudospectrum control. The point-gap winding number is realized as the Fredholm index of the regularized resolvent. Riemann—Roch divisor constraints on meromorphic sections force the Generalized Brillouin Zone onto the critical line. A vanishing theorem shows that isolated fractional (non-color-singlet) states are excluded from the domain, thereby locking every non-trivial zero to R(s) = 1/2. The construction is compatible with Alain Connes’ spectral triples and yields, as a byproduct, an explicit derivation of the Clausius relation from the spectral action along the adiabatic paths of the modular loop. All steps are topological and independent of any particular representative of the operator.
Category: Mathematical Physics
[4] ai.viXra.org:2606.0022 [pdf] submitted on 2026-06-08 01:19:11
Authors: Mohammed Belabbaci
Comments: 5 Pages.
The IEEE 754 floating-point standard, foundational to modern computing since 1985, treats division by zero as an undefined operation, returning error states or special infinity values. This convention, while practically expedient, creates mathematical discontinuities and computational barriers across multiple domains. We propose IEEE DZ20i, a continuous extension of IEEE 754 based on the axiom a/0 = a, which maintains computational continuity through zero-valued divisors.This paper presents the mathematical foundation of the DZ20 axiom, demonstrates its consistency within a Meta-Framework for Adapted Mathematics (MCMA), specifies the IEEE DZ20i standard including symmetric human-AI co-validation mechanisms, and explores applications across algorithmic computing, energy systems, hardware design, robotics, and space exploration.Implementation results show that IEEE DZ20i eliminates a broad class of runtime exceptions while preserving backward compatibility with IEEE 754 operations outside the zero-divisordomain.
Category: Mathematical Physics
[3] ai.viXra.org:2606.0017 [pdf] submitted on 2026-06-07 17:08:30
Authors: Xiangqian Zhang，Mingming Zhao，Linchao Ge
Comments: 5 Pages.
This paper systematically analyzes the geometric origins of errors within Topological ResidualTheory (TRT). Taking the Fibonacci number Fu2081u2089 = 4181 as the critical node, we demonstrate thatthe fine-structure constant α is precisely the radian-measure expression of the angle 0.4181°. Theconstant α inherently carries three topological factors: rotation (right-handed cylindrical helicity), symmetry (Markov global stability), and scaling (renormalization and fractality). These factors necessarily generate irreducible projection residuals in real topological realizations. By quantifying the discrepancies between TRT predictions and both 1/137 and experimental values for α, G = μu2080 α², and the Planck constant, we show that all observed errors originate from the intrinsic rotation and topology of α itself. Special emphasis is placed on the multiple algebraic decompositions of 4181 and on relating error magnitudes to the renormalization factor (1/2)α² × 10u207f or α² × 10u207f. Connections to the Lamb shift and Rydberg constant are established to strengthen the theory’s self-consistency and predictive power.
Category: Mathematical Physics
[2] ai.viXra.org:2606.0016 [pdf] submitted on 2026-06-06 13:24:06
Authors: Ginanjar Utama
Comments: 8 Pages.
This paper proves, by explicit Clifford-algebra computation, that the γ0-diagonal complete orthogonal primitive decomposition of real Cl(1,3) has two sectors, while the extension to Cl(1,4) supplies a second independent commuting basis-blade involution, γ1234. Because I5 = γ0γ1234, the generated involution group is Z2 × Z2, not (Z2)3. The resulting decomposition contains four primitive idempotents, each generating a real 8-dimensional minimal left ideal. The count four is itself fixed by Wedderburn theory, since each simple summand has rank two; the explicit content of the result is the basis-blade realization of the family and the demonstration of why a naive three-involution count would predict eight. The central idempotents (1 ± I5)/2 realize the real semisimple splitting Cl(1,4) ≅ M2(H) ⊕ M2(H), where H denotes the quaternions. They do not, by themselves, supply a complex imaginary unit; a complex spinor interpretation still requires choosing an internal quaternionic complex structure within each ideal. Physical identifications with chirality, charge, helicity, or U(1) phase are therefore treated as conjectural.Supporting results include: (i) a Pancharatnam-phase computation from projector triple products in Cl(3,0), confirming the half-solid-angle rule 2|φ| = Ω for geodesic spherical triangles; (ii) a bivector-square classification of geometric-i candidates, with the observation that the chiral split in real Cl(1,3) is a complex-structure operation, not a real idempotent split; and (iii) a short positioning of the result relative to real Cl(0,6) and complex Clu2086 Standard Model programs. The paper is intended as a lower-dimensional real-idempotent audit, not as a derivation of a Standard Model generation.
Category: Mathematical Physics
[1] ai.viXra.org:2606.0014 [pdf] submitted on 2026-06-06 03:46:27
Authors: Xiangqian Zhang, Mingming Zhao, Linchao Ge
Comments: 13 Pages.
Within the framework of Topological Residual Theory (TRT), this study explores the unification ofgravitational and electromagnetic interactions at the level of spacetime geometry. Utilizing the cylindrical helical motion of spatial points (geometrization of Zitterbewegung) as the dynamical foundation, we derive the wave equationvia the variational principle of the displacement field . Using the geometric structure of thelocal cylindrical orthonormal frame induced by local helical symmetry, the electric, magnetic, and gravitational fields are defined as projections of the time derivatives of onto this orthonormal frame. Consequently, the mutual orthogonality of the three fields emerges as a geometric necessity of the frame decomposition rather than an independent postulate. The gravitational field is derived via two independent pathways: (i) the framekinematics and time evolution pathway ( ), and (ii) the second-order derivative pathway of the wave equation ( ). The mathematical consistency betweenthese two pathways indicates that gravity can be interpreted as a second-order residual effectof electromagnetic oscillations propagating at the speed of light. Furthermore, charge is defined as the topological charge (winding number / Hopf invariant) of the helical displacement field on a closed sphere (Euler characteristic ), which manifests as the effective rate of mass change only during the transient state of topological structure formation. Under the compatibility condition of theflux change rate (local topological flux continuity), we naturally derive the vacuum relations and . Based on this unified topological mechanism, the model provides a self-consistent geometric explanation for both charge quantization and particle spin.
Category: Mathematical Physics