The Bullet Cluster in the Geometric Relay Programme: Intrinsic Decoupling, Wronskian Bifurcation, and the Collisionless Nature of R*

This document develops the Bullet Cluster pillar of the Geometric Relay Programme (GRP), built on the Entangled Relativity action L_m^2/R and its scalar relay dynamics. The Bullet Cluster (1E 0657-558) is widely regarded as one of the strongest observational arguments for particle dark matter. Its weak-lensing mass map is offset from the X-ray gas and follows the collisionless galaxy component, suggesting the presence of a dominant non-baryonic component that passed through the collision without being slowed by ram pressure. The purpose of this note is to show that, within the GRP, this behaviour is naturally reproduced without invoking dark matter particles and without introducing adjustable parameters. The argument rests on five elements. First, the Minazzoli decoupling criterion L_m = T separates the cluster into two dynamical regimes. The galaxies and stellar populations behave as pressureless dust, satisfy L_m = T, and leave the relay frozen. The shocked intracluster gas satisfies L_m != T because of thermal pressure and velocity dispersion, activating only a weak relay perturbation. Second, the relay tension R* = Omega_m - Omega_b is collisionless by construction. It is not a particle substance and therefore has no self-interaction cross-section. In this sense, R* behaves as an effective cold collisionless component at the level required by the Bullet Cluster. Third, the Wronskian W(F,W) = F dW/dt - W dF/dt of the dual relay detects the dynamical bifurcation between the collisionless component F and the fluid gas component W. With r = F/W much larger than the separatrix r* = 0.296, the collisionless branch dominates the gravitational potential. Fourth, the correction induced by relay activation in the shocked gas is estimated to be negligible: delta x ~ 10^-4 kpc compared with the observed ~100 kpc offset. The GRP therefore reproduces the Bullet Cluster offset at current observational precision. Fifth, the mechanism is universal and applies to dissociative clusters beyond the Bullet Cluster, including Baby Bullet-type systems, El Gordo, the Musket Ball Cluster, and potentially complex cases such as Abell 520. Two discriminating predictions are identified: core profiles in post-collision subhalo centres, and a correlation between inner lensing excess and local velocity dispersion rather than a universal dark-sector cross-section. The document classifies the Bullet Cluster pillar as A/B within the GRP: conceptually closed and quantitatively constrained at the level of the offset, but still requiring a full N-body or field simulation of sigma(x) in cluster collisions to reach complete A-level status.