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[3] ai.viXra.org:2507.0137 [pdf] submitted on 2025-07-31 19:34:52
Authors: Lukáš Vik
Comments: 8 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present symbolic and p-adic reformulations of the Lonely Runner Conjecture (LRC),introducing auxiliary constructions to analyze visibility dynamics. By encoding runners assymbolic sequences and shifted p-adic expansions, we define new criteria for loneliness andexplore methods to suppress or structurally control its duration. This formulation allows forentropy-like analysis, symbolic compression, and potential finite verification heuristics
Category: General Mathematics
[2] ai.viXra.org:2507.0111 [pdf] submitted on 2025-07-25 16:54:54
Authors: Abdullah Bin Usman
Comments: 21 Pages. (Note by viXra.org Admin: Please cite and list scientific references)
Dual State Analysis (DSA) is presented as a novel mathematical framework aimed at addressing the interpretability and symbolic limitations of standard quantum computing formalisms. In contrast to the Hilbert-space approach, which relies on complex amplitudes and probabilistic measurement, DSA represents logical and computational systems through the simultaneous quantification of presence (P) and absence (A), extended by a symbolic phase parameter (Θ). We rigorously establish the axiomatic foundations of DSA and prove that structurally distinct expressions (e.g., x − x and y−y for x̸ = y) remain inequivalent, thus resolving a key cognitive and algebraic paradox associated with contextual zero. The Dual State Number (DSN) formalism is fully developed, including definitions for addition, subtraction, scalar multiplication, and novel dual-sum operations. Using this framework, we symbolically emulate quantum-like phenomena such as superposition, entanglement analogues, and phase-based interference entirely through deterministic symbolic arithmetic. We also propose a conceptual photonic implementation, mapping DSA operations to optical components. Comparative analysis highlights DSA’s strengths in interpretability, deterministic evolution, and symbolic scalability, with potential applications in algorithm design, finance, education, and symbolic AI. This work lays a formal foundation for a new class of deterministic, phase-aware, and interpretable symbolic computation inspired by quantum behavior.
Category: General Mathematics
[1] ai.viXra.org:2507.0079 [pdf] submitted on 2025-07-14 11:41:04
Authors: Maxim Govorushkin
Comments: 89 Pages. 10.5281/zenodo.15837292
We present a complete, self—contained demonstration of the Riemann Hypothesis built around a new "local operatoru2009+u2009OS—analyticity" framework. Starting on L²(0,∞) we define a family of compact integral operators Ku209b whose Fredholm determinant exactly equals the ratio Ξ(s)/Ξ(1—s). We then derive an absolutely convergent cluster expansion for lnu2009det(I—Ku209b) in the continuous polymer gas, extend it to the critical strip ℜu2009su2009≥u2009½, and establish rigorous Borel summability via sharp Carleman estimates and contour—shift bounds that exclude renormalon—type singularities.Next, we verify all five Osterwalder—Schrader axioms for the Euclidean correlators generated by lnu2009det(I—Ku209b), and apply GNS reconstruction to obtain a contracting, strongly continuous semigroup whose self—adjoint generator D serves as the Hilbert—Pólya operator. Spectral analysis of D—compact resolvent, absence of continuous spectrum, monotonicity of eigenvalues under s—derivatives and Kreĭn—Rutman positivity—yields a one—to—one correspondence between its discrete spectrum and the nontrivial zeros of ζ(s), proving they lie on ℜu2009s=½ and are all simple.A key technical innovation is the "Nomadic Method" (Homeless), which uses overlapping local charts and transition maps to localize cluster bounds, Borel expansions and OS positivity into a single uniform proof. All analytic constants, operator—norm estimates and s—derivative controls are made explicit, closing gaps that prevented previous global approaches from reaching ℜu2009s=½.deposit includes:— the full LaTeX source with appendix J (lemmas J.1—J.14, J.3u2032, J.9u2032), — Python scripts for discretizing Ku209b and computing first eigenvalues, — Jupyter notebooks reproducing the first twenty nontrivial zeros, — a pytest—based CI workflow ensuring everything compiles and tests pass out of the box. Linking these elements provides both the mathematical rigor and computational reproducibility needed to validate our Hilbert—Polya construction and settle the Riemann Hypothesis.
Category: General Mathematics