TSO: The Emergent Euclidean Metric at a Black-Hole Horizon: A Time–Space Oscillatory Representation

TSO: The Emergent Euclidean Metric at a Black-Hole Horizon: A Time–Space Oscillatory Representation What happens to geometry when the geometry itself breaks down? The event horizon of a black hole raises a fundamental question in theoretical physics: what happens to the Schwarzschild metric at the event horizon, and what geometric structure, if any, remains when the familiar spacetime description ceases to apply in its usual form? This paper seeks to shed light on that question by investigating whether the familiar geometric structures of spacetime are fundamental, or whether they can emerge from a deeper mathematical organisation. In conventional descriptions, the geometry of spacetime provides the framework within which gravitational phenomena are expressed. But what if the Euclidean metric is not the starting point, but a structure that exists only under particular conditions? This paper explores that question through Time–Space Oscillations (TSO), a mathematical framework that represents time and space through coupled oscillatory relations. The central proposal is that a local Euclidean metric emerges conditionally, through the compatibility and mutual normalisation of distinct oscillatory sectors. When those conditions no longer hold, the Euclidean metric may cease to apply without requiring the underlying cyclic structure to disappear. The paper examines how this distinction can be expressed mathematically in the setting of a black-hole horizon, including the role of phase relations and their connection to Dirac and Pauli structures. Rather than assuming Euclidean geometry as an immutable background, the construction investigates how such geometry may arise from more elementary relations—and what mathematical structure remains when it does not. The aim is not merely to describe a horizon using geometry, but to investigate the conditions under which that geometry becomes possible. This work presents a theoretical construction within the TSO framework. Its interpretation and implications remain open to mathematical scrutiny and independent assessment. Readers interested in black-hole physics, spacetime geometry, mathematical physics, and alternative approaches to the foundations of quantum theory are invited to examine the derivations and assess the proposal for themselves. For Readers not familiar with the TSO (Time Space Oscillations) Framework and its relation to General Relativity, Electro Mechanics and Quandum mechanics please read: Time–Space Oscillations: A Geometric and Deterministic Approach to Relativistic and Quantum PhenomenaZenodo DOI: 10.5281/zenodo.17534734The foundational article introducing the TSO framework and its applications to both relativistic and quantum phenomena. Time Dilation and the Nature of Gravitational and Inertial ForcesZenodo DOI: 10.5281/zenodo.17543059Explores the TSO perspective on time dilation and the interplay of inertial and gravitational forces. Time–Space Oscillations and ElectromagneticsZenodo DOI: 10.5281/zenodo.17591325Applies the TSO model to electromagnetic phenomena, revealing novel interpretations of wave propagation and field interactions. Time–Space Oscillations and Quantum Mechanics Zenodo DOI: 10.5281/zenodo.17670668Extends TSO to encompass quantum mechanical principles, including Heisenberg uncertainty, Schrödinger dynamics, path integrals, entanglement, zero-point energy, and the Quantum Zeno effect.