Can a position-dependent lattice spacing in a discrete spacetime model reproduce the gravitational field equations? [closed]

in a discrete spacetime model where the fundamental lattice spacing L(r) is not constant but varies with position, can the resulting gradient in the "clock rate" (local tempo of time) reproduce the gravitational field equations in the weak field limit?
Background
Consider a discrete spacetime modelled as a 3D cubic lattice with a finite number of cells. The physical cell size L(r) can vary locally. Define the local tempo of time as κ(r)=L_0/L(r), where L_0 is the cell size at infinity (flat spacetime limit). In the weak field limit, one would expect κ(r)≈1+ϕ(r)/c^2, where ϕ is the Newtonian gravitational potential.
For a composite body of height H in a gravitational field, the position dependent cell size means that lower layers of the body experience a higher density of cells (smaller L) than upper layers. Consequently, a particle in the lower layers must traverse more cells per unit proper time, which corresponds to a slower local clock rate. The internal energy of a bound system then acquires a height dependent term:
E(z)=m_0 c^2 κ(z)=m_0 c^2 (1ⓜ+(ϕ(z))/c^2 )=E_0+m_0 ϕ(z).
For a uniform field ϕ(z)=gz, this gives E(z)=E_0+m_0 gz. The force on the body is F=-dE/dz=-m_0 g, and thus a=F/m_0=-g, recovering the equivalence principle and Newton's second law without introducing gravitons or curved spacetime as a fundamental concept — gravity emerges as an internal stress caused by the gradient in the local clock rate.
Generalized field equation
For an arbitrary mass distribution ρ(r), one can write a generalized Poisson type equation for κ:
∇^2 κ(r)=4πG/c^2 ρ(r)" " f(κ),
with a nonlinear function f(κ) that may include a singularity at some critical value κ_"crit" (to prevent cell overcrowding and avoid singularities):
f(κ)=κ/(1-(κ_"crit" /κ)^2 ).
In the weak field limit (κ≈1, f(κ)≈1), this reduces to the standard Poisson equation ∇^2 κ=(4πG/c^2)ρ, and since κ=1+ϕ/c^2, we recover ∇^2 ϕ=4πGρ.
The question
Does this kind of position dependent lattice spacing model — where gravity emerges from a spatial gradient in the local cell size (and thus in the local clock rate) — face any fundamental inconsistencies with known physics? In particular:
Does it violate local Lorentz invariance or the equivalence principle beyond the weak field limit?
Are there known no go theorems or experimental constraints (e.g., from tests of the equivalence principle, gravitational redshift, or binary pulsar observations) that would rule out this mechanism?
How would such a model connect to or differ from established discrete approaches like causal set theory or loop quantum gravity?
I am aware that discrete spacetime models typically struggle with Lorentz violation, but I am specifically interested in whether the position dependent lattice spacing mechanism itself is viable as an effective description of gravity, or whether it inevitably leads to contradictions with observations.