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[4] ai.viXra.org:2605.0043 [pdf] submitted on 2026-05-20 13:22:51
Authors: Chad Meyer
Comments: 28 pages, 47 references. Seven Standard Model observables (α, α_s, sin²θ_W, λ_H, m_μ/m_e, |V_us|, G_N) reproduced from one transcendental fixed point with zero free parameters. Companion paper on the polar-square geometric identity in preparation. Develop
We derive the fine-structure constant α from six properties that anyquantum coupling constant must satisfy: scale invariance, self-reference,cumulativity, probability normalization, self-consistency, and contractivity.These chain together with zero tunable parameters to produce a uniquefixed-point equation erfi(b)² = e^(b²), whose solution b* generates afive-letter algebraic alphabet. A convergent Riccati series makes thederivation analytic, extending α to unlimited precision.Three closure theorems force every component of the master equationα = R/(D_M − R α L^d): (I) a universal escape identity constrains theprojection ratio R; (II) Hurwitz's division algebra theorem forces d = 4;(III) Euler's ζ(2) = π²/6 organizes the master denominator D_M.Theorems II and III unify as facets of the graded real spectral tripleon S¹_{2π}, whose chirality grading forces the spectral convention uniquely(the competing convention is excluded at 134σ). The Padé form of theself-energy Σ_2 — previously the last remaining assumption — is nowderived: the fixed-point equation forces constant coupling (β = 0),which forces the Dyson self-energy series to be geometric, closing toa rational function with a single simple pole. A spectral dimension jump(Gabriel's Horn) connects the logarithmic branch point of the Riccatiseries (radius r = 2π = det'(D), exponent consistent with p = −1;extracted from 50 coefficients via Wynn acceleration) to the convergentspectral zeta value ζ(2) that organizes D_M.The result, 1/α = 137.035 999 075, agrees with CODATA 2018 at 0.4σ andwith Parker et al. (2018) at 1.1σ. The 4.9σ tension with CODATA 2022constitutes a falsifiable prediction. The same alphabet reproducesα_s(M_Z) (0.11σ), sin²θ_W (0.25σ), λ_H (0.03σ), m_μ/m_e (0.2σ),|V_us| (0.9σ), and Newton's constant (0.15σ) — seven observables froma single transcendental number with zero free parameters. No theorempostdating 1954 enters the derivation chain; the spectral-geometricinterpretation of the denominator draws on Connes (1994).
Category: Mathematical Physics
[3] ai.viXra.org:2605.0025 [pdf] submitted on 2026-05-12 19:50:05
Authors: Carl Andrew Brannen
Comments: 11 Pages. To be submitted to Journal of Mathematical Physics
Continuous gauge symmetries are usually introduced through Lie groups acting on quantum fields. In this paper we show that the algebraic structure underlying non-abelian gauge transformations already arises naturally inside the complex group algebra of a finite non-abelian group. The dihedral group $D_4$, the symmetry group of the square, is used as an explicit example.The complex group algebra $C[D_4]$ decomposes into irreducible matrix blocks under the Artin—Wedderburn theorem. While the character table describes only the subspace of class functions, the full group algebra contains additional intra-class directions invisible to the character table. For $D_4$ these directions form a three-dimensional subspace which, after elementary normalization, satisfies the Pauli algebra and generates continuous SU(2) transformations inside the two-dimensional irreducible block.The construction is carried out explicitly using only the multiplication table of $D_4$. Continuous rotations arise from exponentials of finite group algebra elements, without assuming any continuous symmetry at the fundamental level. The mechanism generalizes to any finite group possessing higher-dimensional irreducible representations, where the associated matrix blocks naturally support the corresponding su(N) Lie-algebra structures.
Category: Mathematical Physics
[2] ai.viXra.org:2605.0024 [pdf] submitted on 2026-05-12 19:46:56
Authors: Moninder Singh Modgil, Dnyandeo Dattatray Patil
Comments: 92 Pages.
he theory of roots of unity occupies a central position in algebra, geometry, Fourier analysis, quantum mechanics, and modern mathematical physics. In the ordinary complex plane, roots of unity generate discrete cyclic structures on theunit circle S1 and are governed by commutative rotational symmetry. However, the extension from complex numbers to higher Cayley—Dickson algebras introduces profoundly richer mathematical structures involving noncommutativity, nonassociativity,exceptional topology, higher-dimensional phase manifolds, and zero-divisor geometry. In the present work, we systematically investigate generalized roots of unity in quaternionic, octonionic, sedenionic, and higher hypercomplex systems. Explicit generalized exponential maps are derived for quaternionic and octonionicphase structures, and the resulting root manifolds are analyzed geometrically andtopologically. We show that quaternionic roots naturally generate continuous S2 families associated with the Lie group SU(2) ≃ S3, while octonionic roots generate S6 root geometries connected with exceptional G2 symmetry and nonassociative algebraic structure. Beyond the octonionic level, the emergence of zero divisors radically alters polynomial behavior, leading to extended algebraic varieties rather than isolated algebraic roots. The paper develops generalized Fourier structures, hypercomplexgauge geometry, exceptional fiber bundles, generalized Hopf fibrations, cohomological structures, characteristic classes, K-theory constructions, and homotopy classifications associated with generalized root manifolds.Connections are established between hypercomplex root geometry and Yang—Mills gauge theory, Clifford structures, exceptional Lie groups, spinorial geometry, string theory, noncommutative geometry, generalized quantum phase spaces, and exotic spacetime formulations. The work further explores possible applications to generalized cognitive geometry, hypercomplex neural dynamics, exceptional lattice structures, and multidimensional consciousness manifolds involving octonionic and E8-inspired configurations. The resulting framework suggests that generalized roots in higher Cayley—Dickson systems may provide a unifying mathematical language connecting topology, algebra, gauge.
Category: Mathematical Physics
[1] ai.viXra.org:2605.0006 [pdf] submitted on 2026-05-04 23:43:54
Authors: J. W. McGreevy
Comments: 24 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
The Relativistic Field Theory of Primes (RFTP) is a self-contained arithmetic fieldtheory in which the primes, modular forms, and monstrous moonshine are promoted to thedynamical degrees of freedom of a clutched adelic bundle over the compactified modularcurve X(1). The theory begins with the Eisenstein series E4(τ ), E6(τ ), and E12(τ ) and therigorously derived Ramanujan—Serre congruence τ (n) ≡ σ11(n) (mod 691), which is elevatedto the primary topological defect injecting the delta source ρsource =65520691 δj(τ)(P) into theArakelov Poisson equation. This single defect sources a shared curvature potential Veff(r, ϕ)felt identically by timelike soliton geodesics and null photon rays, generating a conservedRunge—Lenz vector and an SO(4) symmetry that unifies gravitational and electromagneticscales.A least-arithmetic-action functional S[Φ] on the Monster vertex operator algebra V♮yields a self-adjoint radial Dirac operator D on the clutched bundle whose spectrum interpolates the Balmer series and the critical-line zeta zeros. The theory realizes a Connesspectral triple (AVOA, Hsoliton, D), Einstein—Cartan gravity with χ11-sourced torsion, andphoton propagation via explicit multi-field WKB ray-tracing. Modular invariance underSL(2,) is the arithmetic skeleton: it preserves the characteristic bending speed p|d2r/ds2|,the Runge—Lenz vector, the gravitational constant GA = 65520/(8π · 691), the speed ofcausality cA, all spectra, scattering cross-sections, Einstein A/B coefficients, vacuum energy,dark-energy equation of state, and cosmological constant. The wave/ray duality explainsthe integer 3 (triality compactness of the ray limit) and the transcendental tail of π (infinitedimensional wave phase space). The Gutzwiller trace formula over modular periodic orbitslinks the primes to the zeta zeros, while compactness, discreteness, and finiteness of theclutched bundle enforce ultraviolet finiteness at the Planck scale lP .All constants and observables emerge purely arithmetically from the single master functional and the 691 defect, with no external parameters. RFTP therefore provides a parameterfree unification in which modular symmetries dictate the observable universe.
Category: Mathematical Physics