Why 3+1 Dimensions? A Pre-Metric Relational Reconstruction of Space, Time and Causal Structure

Abstract

This technical note develops a pre-metric relational reconstruction of the 3+1 structure of spacetime. It asks whether three-dimensional spatial structure and temporally distinct ordering can arise from relational organisation without being assumed from the outset. Under stated conditions of stability, reference neutrality, internal reorientation, indecomposability and minimality, the smallest admissible compact simple non-Abelian orientational sector is identified with su(2) — a compact non-Abelian Lie algebra associated with rotations — providing three independent internal generators. A non-degenerate relational frame and stable comparability then allow this internal rank to acquire a three-dimensional spatial interpretation. Temporal order is proposed to emerge only after spatial rank saturation, when continued differentiation no longer increases spatial dimensionality and asymmetric reachability introduces transformational precedence. A further argument shows how bounded relational accessibility and stepwise propagation can generate a finite pre-metric causal frontier, later acquiring the meaning of an effective limiting speed ceff. The work is presented as a conditional structural reconstruction rather than a complete dynamical theory. It distinguishes the genealogical origin of spatial dimensionality from that of temporal ordering and identifies the open steps required for metric, causal and gravitational completion.