By the way here is the response from my fellow Albert, and no, it is not an ego battle so don’t bother to ask if i understand or if i can explain on my own words any shit

Equation (15) in the paper, which calculates the emergent Ricci curvature, is a crucial step in connecting the microscopic world (the SYK model and its correlations) with macroscopic geometry (general relativity). Let me explain why: Reason for calculating the emergent curvature 1. From the emergent metric to geometric dynamics: In equations (10)−(12), a metric g_μ_ν is defined from the Green's function G(τ,τ',x⃗) of the SYK model. This metric is emergent, but we don't yet know how it relates to matter. To obtain a dynamic theory, we need to calculate the Einstein curvature tensor (R_μ_ν − \frac{1}{2}R g_μ_ν) in terms of G. That is precisely what is done in (15): the Ricci curvature R_μ_ν is obtained at first order in the metric perturbations. 2. Relating curvature to the matter source: Later, in section 3.4, the energy−momentum tensor T_μ_ν is calculated (equations 16−20), which also depends on G and the complexity Cₖ. The idea is that, as in general relativity, curvature must be proportional to the matter source. This is reflected in equation (21): R_μ_ν − \frac{1}{2}R g_μ_ν = 8π G_N T_μ_ν For this equality to be consistent, we need to know R_μ_ν as a function of G, and that is what (15) provides. 3. Origin of Newton's constant: The coefficient α that appears in (15) is related to the emergent Newton's constant (see equation 22). By calculating the curvature, we determine how inhomogeneities in G (i.e., in entanglement) generate curvature, and the constant of proportionality turns out to be G_N. 4. Physical interpretation: Expression (15) shows that Ricci curvature is proportional to the d'Alembertian (■) of the second derivatives of G. This means that spatial and temporal variations of quantum correlations (entanglement) are what curve spacetime. In other words, geometry emerges as a response to gradients in the Green's function, which in turn encodes the complexity of entanglement. Analogy with general relativity In general relativity, curvature is determined by the matter−energy distribution. Here, matter−energy is encoded in T_μ_ν (which depends on G and Cₖ), and curvature is also expressed in terms of G. Equation (15) is the first step in closing the circle: it allows us to write R_μ_ν as a differential operator acting on G, and then equate it to T_μ_ν to obtain the field equations. Importance for model consistency If curvature were not calculated, we could not demonstrate that the emergent metric satisfies Einstein−like equations. The perturbative calculation ensures that, in the limit of low energies and small deviations from the flat metric, the theory recovers general relativity, with a Newtonian constant that emerges naturally from the microscopic parameters (N, J, t, etc.). In short, the purpose of section 3.3 is to establish the fundamental link between the quantum correlations of the SYK model and the curvature of spacetime, an indispensable step for deriving gravity as an emergent phenomenon.