Mathematical Physics

2507 Submissions

[3] ai.viXra.org:2507.0088 [pdf] submitted on 2025-07-16 05:32:11

Spiral Dance of Cantor’s Cardinals: Textures of the Continuum— IV

Authors: Moninder Singh Modgil, Dnyandeo Dattatray Patil
Comments: 31 Pages.

We develop a resolution-aware quantum field framework grounded in the hierarchy ofCantor cardinals, wherein the continuum is stratified by a sequence of decreasing resolutions{ϵi = 1/ℵi}. This architecture allows us to reformulate classical and quantum field theoriesby replacing infinitesimal constructs with tiered differentiable operators. Beginning with themodification of scalar field dynamics via ϵi-derivatives, we extend the formulation to gaugefields, BRST.We construct ϵi-based renormalization group flows, wherein coupling constantsevolve not only with energy scale but across resolution layers. Statistical field theory is enrichedwith ϵi-dependent entropy, including Gibbs and von Neumann entropy, and extendedto Langevin-type stochastic systems with regularized entropy production. In the geometricdomain, we introduce ϵi-curvature tensors and Einstein equations over resolution-aware manifolds.Finally, we formulate topological and categorical field theories indexed by ϵi, integratingsmooth topos logic, stratified cohomology, and resolution-sensitive path integrals. This frameworkoffers a logically consistent, divergence-free, and epistemic
Category: Mathematical Physics

[2] ai.viXra.org:2507.0086 [pdf] submitted on 2025-07-17 00:04:42

The Arithmetic of Order: A Unified Origin for the Standard Model’s Gauge Symmetries, Hypercomplex Numbers, and Codewords

Authors: Faysal EL Khettabi
Comments: 9 Pages. https://efaysal.github.io/HCNFEK2024FE/HypComNumSetTheGCFE

This report details the Arithmetic of Order (AoO) as a discrete arithmetic framework that provides a unified origin for the Standard Model’s gauge symmetries, hypercomplex numbers, and optimal error-correcting codes. The AoO posits that physical and mathematical structures emerge from the ordered combinatorial progression 1 → n → n+1 and its associated powerset hierarchy, P(Ωn), without assuming a pre-existing continuum. We show how this framework, realized through Farey sequences under the modular group SL(2,Z), lifts naturally to its universal central extension, the braid group B3. This topological extension provides a direct mechanism where the three Reidemeister moves generate the gauge groups U(1), SU(2), and SU(3). The discussion section contextualizes these results, highlighting the framework’s power in unifying the language of physical transformations (hypercomplex numbers) with principles of information stability (codewords), and notes the convergence of our conclusions with parallel topological models of physics.
Category: Mathematical Physics

[1] ai.viXra.org:2507.0068 [pdf] submitted on 2025-07-13 03:45:52

Physics as Quantized Measurement: The Farey Sequence as a Realization of the Arithmetic of Order

Authors: Faysal EL Khettabi
Comments: 9 Pages.

This article formalizes the Farey sequence hierarchy ($Fseq{n}$) as a concrete realization of the enquote{Arithmetic of Order} (AoO) framework. This framework leverages the constructive machinery of ZF set theory, without assuming the Axiom of Infinity, to model physical reality as a procedural, finitistic unfolding. The core physical principle advanced is that the set-theoretic complement between successive powersets, $powerset(Omega_{n+1}) setminus powerset(Omega_n)$, is the mathematical image of a resolution jump in physical measurement. This provides a rigorous engine for Cycle Clock Theory, demonstrating how a coherent, non-overlapping, and generative temporal process emerges from finite principles. By showing that the infinite is an emergent property, the AoO offers a unified, operational foundation for mathematics and physics.
Category: Mathematical Physics