2 If we assume that a discrete quantum gravity theory does have a 4 dimensional continuum regime, and if the theory is local and generally covariant, then the long distance physics will be governed by an effective Lagrangian which is a derivative expansion in which all diffeomorphism in- variant terms are present but higher derivative terms are suppressed by the appropriate powers of the Planckian discreteness length scale, l : where R is the Ricci scalar, a$_{1}$ and a$_{2}$ are dimensionless couplings of order 1, and the dots denote further cur- vature squared terms as well as cubic and higher terms. The coefficient of the leading term, a$_{0}$ , is also naturally of order 1 which would make it 120 orders of magnitude larger than its observed value. However, that would also produce curvature on Planckian scales and so would not be compatible with the assumption of a continuum ap- proximation. In a discrete theory, the question of why the cosmological constant does not take its natural value is the same question as why there is a continuum regime at all and we must look to the fundamental dynamics for its resolution. Assuming there is a resolution and a contin- uum regime exists, locality and general covariance then pretty much guarantee Einstein’s equations due to the natural suppression of the curvature squared and higher terms compared to the Einstein-Hilbert term. So, Lorentz symmetry and discreteness together imply nonlocality, but nonlocality blocks the recovery of general relativity, and if causal sets were incorrigibly nonlocal, this would be fatal. Suppose, however, that the nonlocal- ity were somehow limited to length scales shorter than a certain l$_{k}$ , which could be much larger than the Planckian discreteness scale, l , but yet have remained experimen- tally undetected to date. There is already evidence that this is possible and indeed causal sets admit construc- tions that are local enough to approximate the scalar D’Alembertian operator in 2 dimensional flat spacetime [17, 18]. We add to this evidence here by exhibiting a family of discrete operators that approximate the scalar D’Alembertian in 4 dimensional flat spacetime. Further, both the 2D and 4D operators, when applied to scalar fields on causal sets which are well described by curved spacetimes approximate □ − 1 $_{2}$R , where R is the Ricci scalar curvature. We use this to propose an action for a causal set which is approximately local. We recall that a causal set (or causet ) is a locally finite partial order, i.e. , it is a pair ( C , ⪯ ) where C is a set and ⪯ is a partial order relation on C , which is (i) reflexive: x ⪯ x , (ii) acyclic x ⪯ y ⪯ x ⇒ x = y , and (iii) transitive x y ⪯ z ⇒ x ⪯ z , for all x, y, z ∈ C . Local finiteness is the condition that the cardinality of any order interval is finite, where the (inclusive) order interval between a pair of elements y ⪯ x is defined to be I ( x, y ) := { z ∈ C | y ⪯ z ⪯ x } . We write x ≺ y when x ⪯ y and x = y . We call a relation x ≺ y a link if the order interval I ( x, y ) contains only x and y : they are nearest neighbours. Sprinkling is a way of generating a causet from a d - dimensional Lorentzian manifold ( M , g ). It is a Poisson process of selecting points in M with density ρ so that the expected number of points sprinkled in a region of space- time volume V is ρV . This process generates a causet whose elements are the sprinkled points and whose order is that induced by the manifold’s causal order restricted to the sprinkled points. We say that a causet C is well approximated by a manifold ( M , g ) if it could have been generated, with relatively high probability, by sprinkling into ( M , g ). We propose the following definition of a discrete D’Alembertian, B , on a causet C that is a sprinkling, at density ρ = l − $^{4}$, into 4D Minkowski space M $^{4}$. Let φ : C → R be a real scalar field, then where the sums run over 4 layers L$_{i}$, i = 1 , . . . , 4, and n ( x, y ) := | I ( x, y ) | − 2. So, for example, layer L$_{1}$ is the set of all elements y that are linked to x and as described above, they will be distributed close to a hy- perboloid that asymptotes to the past light cone of x and is proper time l away from x . This sum will not in gen- eral be uniformly convergent if it is over the elements of a sprinkling into infinite M 4 so we introduce an IR cut- off, L >> l , by embedding C in M 4 and summing over the finitely many elements sprinkled in the intersection of the causal past of x and a ball of radius L centred on x . The details of the calculation that shows why 4 layers are necessary in 4D will appear elsewhere, however see [18] for an explanation of why 3 layers are needed in 2D and the conjecture that 4D will require 4 layers. Now let φ be a real test field of compact support on M $^{4}$. If we fix a point x ∈ M 4 (which we always take to be included in C ) and evaluate Bφ ( x ) on a sprinkling into M $^{4}$, its expectation value in this process is given by where ξ := l − $^{4}$V ( x, y ), V ( x, y ) is the volume of the causal interval between x and y and there is an implicit cutoff L , the size of the support of φ , on the integration range. It can be shown that this mean converges, as the discreteness scale is sent to zero, to the continuum D’Alembertian of φ , lim l → 0 ¯ Bφ ( x ) = □ φ ( x ) (5) ¯ Bφ ( x ) := E ( Bφ ( x )) = 4 √$_{6}$$_{l}$ 2 [ − φ ( x ) + 1 l 4 ∫$_{y}$ ∈ J − ( x ) d $^{4}$y φ ( y ) e − $^{ξ}$(1 − 9 ξ + 8 ξ 2 − 4 3 ξ $^{3}$) ] , (4) L$_{i}$ := { y ∈ C : y ≺ x and n ( x, y ) = i − 1 } (3) Bφ ( x ) := 4 √$_{6}$$_{l}$ 2 [ − φ ( x ) +( ∑ y ∈ L 1 − 9 ∑ y ∈ L 2 +16 ∑ y ∈ L 3 − 8 ∑ y ∈ L 4 ) φ ( y ) ] , (2) L eff √ − g ℏ = a$_{0}$l − 4 + a$_{1}$l − $^{2}$R + a$_{2}$R 2 + . . . (1)