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[3] ai.viXra.org:2506.0059 [pdf] replaced on 2025-06-21 02:47:13
Authors: Edrianne Paul B. Casinillo
Comments: 6 Pages.
The Reserve Arithmetic System (RAS) presents a novel framework for addressing division by zeroby encapsulating the numerator within a symbolic zero, termed a reserve. This proof-of-concept paper outlines the foundational mechanics of RAS, including its arithmetic operations and key algebraic properties. The notation 0⟨x⟩ denotes a "zero with reserve x," signifying that while the numerical result is zero, the original numerator is retained symbolically. This approach extends classical arithmetic to include well-defined division by zero, preserving informational content and enabling new algebraic structures.
Category: General Mathematics
[2] ai.viXra.org:2506.0014 [pdf] submitted on 2025-06-04 00:12:26
Authors: Edrianne Paul B. Casinillo
Comments: 8 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
Division by zero is traditionally seen as undefined, causing discontinuities in boththeoretical mathematics and practical computation. The Reserve Arithmetic System(RAS) offers an alternative framework: instead of rendering division by zero undefined,it represents such cases with symbolic values that encapsulate the numerator in aconceptual structure called a reserve. This paper presents a formal framework for RAS,showing that it constitutes a unital commutative semiring-like structure, preserving algebraic properties such as closure, associativity, commutativity, distributivity, and identity. The reserve preserves symbolic information about operations involving zero denominators, enabling traceable, consistent reasoning in symbolic computation and related domains.
Category: General Mathematics
[1] ai.viXra.org:2506.0010 [pdf] submitted on 2025-06-04 00:01:52
Authors: Charles R. Tibedo
Comments: 42 Pages. Solvers Included in PDF for public verification
The Riemann Hypothesis (RH), which posits that all non-trivial zeros of the Riemann zeta function ��(��) lie on the critical line Re(��) = 1/2, stands as the most consequential unsolved problem in pure mathematics. Its resolution would not only deepen our understanding of prime number distribution but also unify disparate domains of mathematics and physics. This work resolves RH through a novel synthesis of septimal-adelic spectral synthesis, hypotrochoidic geometry, and modular stress conservation, anchored by the normalization of the Riemann Siegel framework to cyclic boundary conditions (modulo 1) and validated computationally (��(10−80)). This work provides a proposed formal resolution of the Riemann Hypothesis, validated through axiomatic proofs and computational syntheses. The synthesis of hypotrochoidic geometry, septimal cohomology, and adelic spectral theory establishes a new potential field for exploration in analytic number theory.
Category: General Mathematics