A Crossed-Memory Non-Local Extension of Verlinde's Emergent Gravity as a Toy Model for the Bullet Cluster without Particle Dark Matter

A Crossed-Memory Non-Local Extension of Verlinde's Emergent Gravity as a Toy Model for the Bullet Cluster without Particle Dark Matter Independent Researcher, Magdalena, Buenos Aires, Argentina Priority Note - 27 October 2026 - Magdalena, Buenos Aires, Argentina. Abstract:We present a toy model extending Verlinde's Emergent Gravity (Emergent Gravity and the Dark Universe, arXiv:1611.02269, 2016) with a long-range elastic memory kernel. In standard emergent gravity, apparent dark matter mass M_D is locally tied to baryonic mass M_B via M_D^2(r) = (a0/6G) d/dr [r M_B(r)]. For the Bullet Cluster (1E 0657-56), M_B is dominated by X-ray gas at x=0, while galaxies are at ±200 kpc. The standard formula predicts M_D peaks at x=0, contradicting weak lensing observations (Clowe et al. 2006) which show lensing peaks at ±200 kpc and a void at 0. We propose the "crossed ghost" idea: If emergent space has elastic memory with characteristic length λ ~ 200 kpc, the deformation created by Cluster A (coming from +300 kpc) remains at -200 kpc after crossing, and vice versa. The lensing map shows not the present baryons, but the memory of the opposite baryons. Toy equation:rho_D,mem(x) = 0.3 * ∫ rho_B(x') exp(-|x-x'|/λ) dx' / λ + 0.8 * rho_B(-x, past)with λ = 200 kpc, kernel K(r) = exp(-r/λ)/(2λ) Results:- Figure 1 (Verlinde 1.0): rho_D peaks at gas position (0 kpc). Fails to reproduce Bullet Cluster.- Figure 2 (Verlinde 2.0 with λ=200 kpc + crossed term): rho_D peaks at ±200 kpc, coincident with galaxies, with suppressed central peak (~60%). Qualitatively reproduces Bullet Cluster lensing morphology without particle dark matter. This idea was formulated on 27 Oct 2026 in Magdalena, Argentina, by an independent secondary-school teacher in discussion with Meta AI. This Zenodo record establishes priority of the crossed-memory concept. Open problems: (1) Derive λ from first principles (de Sitter entropy), not tuned. (2) Write full non-local field equation conserving energy and preserving Tully-Fisher M_b ∝ V^4 in isolated galaxies. (3) Test against CMB and other mergers (MACS J0025, Abell 520). References:- Verlinde, E., 2016, SciPost Phys. 2, 016, arXiv:1611.02269- Clowe, D. et al., 2006, ApJ 648, L109 (Bullet Cluster lensing)- Bradač et al., 2006, ApJ 652, 937- Brouwer et al., 2017, MNRAS 466, 2547 (test of Verlinde)- Ettori et al., 2017, A&A 599, A45 (Verlinde fails in clusters) Keywords: emergent gravity, Bullet Cluster, non-local gravity, dark matter alternative, memory effect License: CC BY 4.0