Holographic Bit Threads on Aperiodic Joints: A Geometric Origin for the Area Law

This paper extends Julian Barbour’s relational formulation of General Relativity—wherein gravity arises from evolving 3-dimensional conformal geometries—by identifying the "hitherto unrecognized fundamental symmetry principles" of the York degrees of freedom with the aperiodic order of the Einstein Monotile (Ξ). We propose that the growth of Shape Complexity from the Janus Point is not merely a gravitational phenomenon but a fundamental aperiodic topological tiling transition. By mapping the configuration space of N-body systems onto Combinatorial Complexes, we demonstrate that the "Rigid Gauge" provided by aperiodic fixity ensures the global nilpotency of the BRST operator (Q²=0), thereby resolving the Gribov ambiguities inherent in periodic manifolds. We further show that the "Creative Core" of gravity acts as a topological low-pass filter, "lifting" the central charge of the vacuum from a dissipative early state (c≈-0.1) to a stable state (c=1) via the constructive gain of arithmetic murmurations. This provides an algebraic origin for inertial mass as the work required to shift monotile boundaries against the vacuum’s topological tension, ultimately deriving the "Arrow of Time" from the intrinsic optimization of arithmetic coherence.By mapping holographic "bit threads" onto this discrete structure, we demonstrate that the Markov gap identified by Hayden (2021) is minimized when the causal set joint terms —specifically the cothθ contributions Dowker (2025)—align with the topological zero-modes of the bulk TQFT. In this framework, the Standard Model Lagrangian emerges as the effective action of gapless excitations localized at the hinges and corners of the aperiodic vacuum, providing a purely geometric origin for mass and gauge symmetry.