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[5] ai.viXra.org:2603.0098 [pdf] submitted on 2026-03-27 21:01:55
Authors: Chang Yu
Comments: 5 Pages.
The Toeplitz Conjecture has remained a focal point in geometry for over a century, yet traditional analytical approaches often overlook the fundamental logical constraints of geometric entities. This study addresses the "Inscribed Square Problem" by shifting the focus from empirical measurement to a first-principles logical audit. By revising the logical definition of "inscription" and treating a square as a rigid four-node structure with inherent 90-degree angular equilibrium constraints, we demonstrate that the existence of such a structure is not a probabilistic occurrence but a topological necessity of curve closure. Our results introduce the concept of "270-degree Space Compensation", showing that any simple closed curve that returns exactly to its starting point must generate sufficient spatial tension to accommodate the vertices of an inscribed square; curves failing to meet this threshold are logically invalid closed entities. We conclude that within a valid geometric domain, a closed curve and its inscribed square are logically inseparable. This finding reframes the conjecture from an existence proof to a deterministic outcome of topological cost. Logically, this necessitates any geometrically valid closed curve to sweep through topological coordinates sufficient to accommodate the four vertices of a square; conversely, a curve that cannot sweep through these coordinates fails to meet the 270-degree threshold.
Category: Topology
[4] ai.viXra.org:2603.0063 [pdf] submitted on 2026-03-14 11:34:51
Authors: Kobie Janse van Rensburg
Comments: 9 Pages.
We study a discrete dynamical system on the three-strand braid group B3 in which generators act on an integer valued framing vector through a local slot-based transfer rule.Terminal states are pairs (P, f) consisting of a permutation P ∈ S3 and a framing vector f ∈ Z3. Under trace closure, strands belonging to the same permutation cycle form a single loop, and physical states are defined modulo cyclic relabeling of the starting point of each loop.We prove a complete classification theorem: two terminal states are physically equivalent if and only if a finite sector-dependent invariant I agrees. The invariant decomposes according to the cycle type of P into three sectors (identity, transposition, three-cycle), and every equivalence class admits a unique canonical representative computable by a constant-time algorithm. A key structural finding is that the three-cycle sector carries a discrete cyclic-ordering obstruction not reducible to symmetric polynomial invariants of the framing differences. The analytic proofs are independently verified by exhaustive BFS enumeration of 631 terminal states within word length 10.This work provides the classied-state foundation for a broader programme connecting combinatorial braid kinematics to RP3 topology and Skyrmion physics within the Topological Inversion Model (TIM).
Category: Topology
[3] ai.viXra.org:2509.0061 [pdf] submitted on 2025-09-22 17:02:24
Authors: Justin Sirotin
Comments: 25 Pages. (Note by ai.viXra.org Admin: Please use standard math/equation/formula typesetting such as LaTeX)
This report provides a comprehensive synthesis of contemporary approaches to incorporating fermionic matter into non-perturbative quantum gravity. We begin by establishing the foundational role of the Nieh-Yan topological density in the first-order gravitational action, demonstrating its connection to the Barbero-Immirzi parameter and its implications for parity symmetry. We then perform a detailed canonical analysis of the Einstein-Cartan-Holst-Dirac system within the Loop Quantum Gravity (LQG) framework, elucidating how fermions source spacetime torsion, deform the symplectic structure, and introduce profound coupling ambiguities. The covariant perspective is explored through the spinfoam formalism, where we describe a minimal coupling for fermions as excitations on the boundaries of quantum spacetime, ensuring consistency with the canonical picture. Finally, we investigate the Asymptotic Safety paradigm as a potential mechanism for ultraviolet completion, focusing on the functional renormalization group flow of Einstein-Cartan theory. We argue that torsion acts as a unifying physical concept across these formalisms and that the principle of Asymptotic Safety may provide a physical selection criterion to resolve the ambiguities inherent in the canonical theory. The analysis reveals that a fundamental parity violation may be a generic prediction of quantum gravity coupled to the Standard Model.
Category: Topology
[2] ai.viXra.org:2507.0093 [pdf] submitted on 2025-07-19 02:19:11
Authors: Maxim Govorushkin
Comments: 44 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references) 10.5281/zenodo.16080619
We present a fully constructive proof of the Hodge conjecture for rational cohomology classes of type ((p,p)) on smooth projective complex varieties. Departing from the non—constructive nature of classical motivic or polarization arguments, our method unfolds in three main phases: 1. "Good" cover and Čech—de Rham translation. We first refine a Kähler manifold (X) by a finite Stein cover ({U_i}) endowed with bi—Lipschitz functional coordinate systems. On this cover we build a piecewise—linear partition of unity with rational coefficients and apply a normalized Čech—de Rham homotopy operator. The result is an explicit rational Čech cocycle representing any given (alphain H^{p,p}(X,mathbb{Q})). 2. Local analytic CI—cycles and homeLESS filtration. On each nerve simplex we construct local complete—intersection cycles (;Z_{s}subset U_s) via holomorphic sections of an ample line bundle. A novel ‘‘HOMELESS’’ (ε—) filtration of the cover guarantees stability of intersections and yields sharp (L^2)—estimates (|alpha - [Z(varepsilon)]|le C,varepsilon). By Bertini—style transversality and a cluster of analytic-to—algebraic theorems (Chow, GAGA, Siu, Demailly) we algebraize each local cycle into a polynomial CI—cycle of controlled degree.3. Global rational assembly and integer representative. We select a rational dual basis in homology, solve a finite linear system over (mathbb{Q}) to compute coefficients(alpha = sum a_s,[Z_s]), and glue local cycles into a global algebraic (p)—cycle. Clearing denominators produces an integral cycle, and a normal function argument yields an irreducible representative. Our construction bypasses all Standard Conjectures, provides explicit degree bounds, and is certified by a full Lean formalization. Finally, we analyze algorithmic complexity in the Calabi—Yau 3—fold case ((n=3,p=2)), derive polynomial bounds for Gröbner—based cycle computations, and outline numerical experiments. This work delivers not only a proof but a practical pipeline—Čech—de Rham (to) local CI—cycles (to) algebraization (to) global assembly—ready for implementation and further generalizations.
Category: Topology
[1] ai.viXra.org:2504.0091 [pdf] submitted on 2025-04-24 19:38:15
Authors: Yuval Fradkin
Comments: 10 Pages. (Note by ai.viXra.org Admin: Please cite and list sceintific references)
This article introduces a topological approach to proving the Poincaré Conjecture in dimension 3. Unlike previous solutions that rely on geometric flows and curvature (notably Ricci flow), the approach here uses only tools from classical algebraic and geometric topology. The proof is based on the principle that every loop in a simply connected 3-manifold can be contracted within a locally embedded disc. This sets the foundation for a global classification that culminates in identifying the manifold with the 3-sphere.
Category: Topology
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