Can a Relational Carrier Generate a Six-Dimensional Lorentzian Effective Regime? An Explicit Ordered-Frame Model with Internal Time and Controlled Causal Response

Abstract

Imagine removing everything from the Universe: the stars, the galaxies, matter, radiation, atoms and particles. Then remove space itself. Remove distance, directions, and the familiar distinction between “here” and “there.” Finally, remove even the clock on the wall of the Universe. What remains? This work begins there. It explores a simple but radical possibility: perhaps spacetime is not the place where physics begins. Perhaps spacetime is something physics has to create. The starting point is therefore not a universe and not a geometric space. It is a world of relations: fundamental elements that can interact, transform, reverse transformations, and organize themselves, but that are not initially assigned positions in space. Nothing begins by saying that one element is a metre away from another. There is no left or right, no up or down, and no collection of extra dimensions already waiting somewhere in the background. At the beginning, there are only relations. The first question is whether those relations can begin to behave like space. In the model developed here, particular relational configurations are energetically preferred. When the system enters an ordered regime, its relations organize themselves into five independent directions. Those directions are not supplied as coordinates at the beginning of the construction. Coordinates are reconstructed afterwards, from the organization of the relations themselves. In this sense, space is not the starting point of the model. It is one of its results. But five directions of space are still not spacetime. Something essential is missing: time. And time cannot simply be added as a sixth coordinate after the fact. The model therefore takes a different route. It introduces an internal physical clock, a system whose ordered correlations allow one stage of physical evolution to be compared with another. Relative to that clock, matter can evolve. This is where the two halves of the story finally meet. The relational carrier provides five spatial directions; the internal clock provides one direction of evolution. At long wavelengths, when the fine discrete structure of the carrier can no longer be resolved, the resulting dynamics approaches that of a field propagating through a world with five spatial dimensions and one time dimension. A six-dimensional Lorentzian spacetime appears as an effective physical description. It does not appear because such a spacetime was drawn into the model at the beginning, because six dimensions were simply declared, or because time was hidden inside the mathematics under another name. Space and time enter through different physical mechanisms and become parts of the same effective geometry only when the response of the system is reconstructed. The microscopic world does not disappear completely. At very short scales, the system still remembers that it is made of discrete relations. Propagation is not perfectly relativistic and small deviations remain. As the appropriate continuum regime is approached, however, those deviations can be quantitatively controlled and the familiar idea of causal propagation emerges. The effective world develops a causal structure in which spatial separation and the ordering supplied by the internal clock together determine which events can influence which others. The paper follows this transition both mathematically and numerically. It reconstructs coordinates even after the microscopic elements have been randomly relabelled. It shows that simply having a certain number of neighbours does not create dimensionality: what matters is how those relations are organized. It follows the discrete dynamics toward its continuum limit, examines the role and limitations of the internal clock, reconstructs an effective Lorentzian metric from physical response, and calculates propagation on an explicitly five-dimensional relational carrier containing more than one million sites. The result is not presented as a final theory of spacetime or gravity. Einstein’s equations are not derived, and some of the deepest ingredients of the construction—including the relational ordering law, the canonical matter structure and the internal clock architecture—remain physical inputs. What the model provides instead is a controlled example showing how a system without an initially assigned spacetime metric can develop a regime whose observable response is naturally described by a six-dimensional Lorentzian geometry. Instead of hiding the birth of spacetime inside a single mysterious arrow, pregeometry → spacetime, this work opens that arrow and examines what might lie inside it: relations become organized, organization creates independent directions, directions acquire the meaning of space, an internal clock gives physical meaning to change, change becomes evolution, and spatial response and temporal evolution finally meet in an effective Lorentzian geometry. The question behind the work is therefore a very old one expressed in a different language: what was there before there was a “where” and a “when”? This paper does not claim to provide the final answer. It asks whether “where” and “when” could themselves be consequences of something deeper—and develops one explicit mathematical path by which that might happen.