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[2] viXra:2606.0029 [pdf] submitted on 2026-06-08 00:56:25
Authors: Julinho Jorge Luís
Comments: 20 Pages.
This work proposes a complementary extension of the factorial function to the real line, based on the separation of the bilateral integral of e^(-Ax) into two disjoint domains. For A>0, the classical Gamma function Γ(z) emerges in the right half-plane; for A<0, we introduce the Symmetric Gamma Function Γ ˉ(z)=e^(-iπ(z+1)) Γ(-z), defined in the left half-plane, with poles at the positive integers. The duality between the integral kernels underlies a regularization mechanism for logarithmic divergences: Γ_R (z)=1/Γ ˉ(z-1) and Γ ˉ_R (z)=1/Γ(z+1), replacing each pole by an exact finite value. An alternative real representation B(x)=∫_0^∞e^(-u) u^(-x) du (convergent for x<1) is constructed, together with a trigonometric factor C(x)=2sinu2061(πx)+cosu2061(πx), defining F(x)=C(x)B(x) for x≤0, yielding F(-n)=(-1)^n n!, thereby unifying the complex branch, real regularization, and Laurent expansion formalisms. The relation F(-n)⋅Resu2061(Γ,-n)=1 establishes the fundamental duality, which emerges directly from the construction of the integral kernels. The method is validated in arithmetic progressions, QED, and QCD. The extension to k-loops is systematic via exponentiation: Γ_R^((k) ) (-n)=[(-1)^n/n!]^k. Distinct physical prediction: In d=5 dimensions, the vacuum energy density changes sign — a repulsive force where the MS-bar predicts attraction. This inversion has direct implications for the stability of extra dimensions in string theory and brane models.
Category: Quantum Gravity and String Theory
[1] viXra:2606.0012 [pdf] replaced on 2026-06-04 20:19:31
Authors: Nigel B. Cook
Comments: Corrected contents page links to sections and also a bibliography added.
We construct a complete quantum field theory in which all observed interactions - electromagnetic, weak, strong, gravitational, and cosmological - arise from a single mechanistic principle: energy conservation in vacuum polarization. The theory replaces the ad hoc structure of the Standard Model and the geometric assumptions of general relativity with a unified framework based on the group chain for Euclidean signature SO(3, 3) ~ SU(4) ⊃ SU(3) × U(1)G ⊃ U(2) ⊃ SU(2) × U(1)Y . Matter representations, gauge charges, running couplings, particle masses, gravity, and dark energy all emerge from this structure without arbitrary parameters. This paper is a new presentation of the ideas in vixra paper 1111.0111 dated 2011, with updates, a few corrections, and further development. That earlier paper should still be consulted for further references, graphics and details documenting the origins of the various nascent ideas.
Category: Quantum Gravity and String Theory
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