Topology

2603 Submissions

[2] ai.viXra.org:2603.0098 [pdf] submitted on 2026-03-27 21:01:55

On the Logical Falsification and Geometric Reconstruction of the Toeplitz Conjecture

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

[1] ai.viXra.org:2603.0063 [pdf] submitted on 2026-03-14 11:34:51

Canonical Classication of Framed Closures in a Three-Strand Braid Transfer Model

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