What question did this study set out to answer?

This paper aims to establish a theoretical framework for understanding rigid body motion in biconic structures.

July 12, 20260 citationsOpen Access

PFUSRC-071 Topological Phase Transfer: Rejection, Spiral, and Rigid Boundary Operations in Biconic Structures - A Unified Mechanism from Inner-Outer Coin Motion to Structural Response

Key Points

This paper aims to establish a theoretical framework for understanding rigid body motion in biconic structures.
Constructed an inner-outer coin-to-biconic isomorphism theorem.
Developed a first-order linear relaxation dynamical equation to describe topological phase transfer.
Defined a normalized perturbation intensity measure and derived critical tolerance limits.
Identified two response modes: spiral state (order maintained) and rejection state (disorder exceeds tolerance).
Characterized the critical transition criterion with a maximum perturbation intensity of approximately 0.0364.
Mapped structural responses to hierarchical topological architecture, linking microscopic motion to macroscopic observation.

Abstract

This paper uses the classical coin-rolling reference frame paradox as its conceptual entry point and establishes the Inner-Outer Coin-to-Biconic Isomorphism Theorem, rigorously embedding two-dimensional rigid-body rolling motion into the PFUSRC 11-dimensional triple coaxial 45◦biconic topological framework. This paper proves that rigid topological structures admit exactly two fundamental self-consistent response modes under perturbation: the spiral state (perturbation absorbed by the system, transformed into ordered topological flow, maintaining intrinsic structural relations) and the rejection state (perturbation exceeding topological tolerance, structural endogenous coupling severed, outward projection of observational appearance). To render the mechanism quantifiable, verifiable, and computationally tractable, this paper constructs a unified normalized perturbation intensity measure A, and derives the system’s inherent critical tolerance A₌₀ₗ ≈0. 0364, constrained by the layered topological density of the 55 steady-state datum points. A first-order linear relaxation dynamical equation is established to precisely characterize the reversible topological phase transfer between rejection and spiral, including its dynamical evolution and intrinsic time scale. The two response modes are strictly mapped onto the three-layer steady-state topological architecture-core anchor points, intermediate flow-variable datum points, and outer boundary datum points-achieving a closed hierarchical loop from microscopic motion, through mesoscopic response, to macroscopic observation. This paper independently characterizes the spontaneous structural response of rigid biconic topology in the absence of external active manipulation, forming a strict dual with PFUSRC-070 (Topological Allocation Theory). All conclusions are entirely inherited from the axiomatic and constant systems of PFUSRC-00, PFUSRC-001, and PFUSRC002, with no additional foundational assumptions. A critical transition criterion and a complete experimental verification chapter are appended, providing testable, falsifiable physical validation pathways, and offering a unified geometric-paradigm explanation for reference-frame ambiguity, the separation of observational appearance from ontological structure, and topological boundary response mechanisms.

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