Physics of Causality Gravity as the response of a causal order: from a Lorentz-invariant substrate to the Einstein-Hilbert stiness of the Benincasa-Dowker action

Causal Physics takes the causal order of events, not spacetime geometry, as the primitive of physics. We formulate it so that every step can be confronted with a pre-registered numerical test, and we report the outcome of these tests. Substrate. The tests decide the nature of the substrate. A discrete substrate with a finite number of neighbours per event and a rest frame is not supported; we adopt a Lorentz-invariant causal set, a Poisson sprinkling in which geometry is read from order and number, which the tests support. Matter is a flow of events that carries its own four-velocity, and the action of a flow is its proper time as counted by the order. Continued as a longest chain of the order, such a history falls with the Newtonian acceleration. In a model extended by two assumptions (the flow carries an orientation that aligns with its causal past), a topological defect placed by hand is locked and keeps a cost; it is a candidate for rest mass. Gravity as the response of the order. Near matter the order changes: clocks slow down, causal cones narrow and events become denser. A fixed order gives the scalar (Nordström) class γ=−1, cones that only narrow give γ=1/ds, and an order that conserves its causal throughput gives γ=2/(ds−1), which is γ=1 exactly in three spatial dimensions. In 3+1 dimensions this throughput principle is the linearised coupling of general relativity to the full energy-momentum tensor; on 3+1 sprinklings the throughput principle reads γ=1.000±0.045. A substrate rule in which every flow narrows the cone in its own rest frame and capacity deficits add reproduces this coupling in the static response of 3+1 sprinklings to moving flows: two counter-streaming beams gravitate with the predicted active mass (2.40±0.05 against 2.36, alternatives 1.56 and below), the cones become anisotropic and a single beam tilts them. The rule from the action of the order. With proper time counted by the order, the substrate rule is the linear response of an order whose action has the Einstein-Hilbert stiffness. We compute the expectation value of the smeared Benincasa-Dowker action of a causal set to second order in a static perturbation, in 3+1, by deterministic quadrature with nothing from general relativity put in. The Einstein-Hilbert stiffness emerges in the conformal sector (6.0003 against 6), in the mixed sector (7.983 against 8), and in the optical sector (2.002 against 2). With all three coefficients γ=1 follows from the action of the order (0.997 to 1.000). The discreteness corrects the stiffness non-analytically, 1−K^≃(kℓk)2[4.0ln⁡(1/kℓk)+0.9], and adds a non-local term that grows with the age T of the static field, −0.10(kℓk)3T/ℓk, which we predicted analytically before computing it. Predictions. Beyond the leading order the sectors differ, and each weakens with the age T of a static field: linearly in the conformal and mixed sectors, faster than linearly in the optical sector within the computed range. The age terms are not suppressed like the local corrections: if the age of a static field is cosmological, a non-locality scale at the level of existing bounds, ℓk∼10−19 m, places the breakdown of the static response at tens of micrometres or above, within reach of torsion balances. The asymptotic form of the optical age term, which fixes the far-field law of a point mass, is the next computation. Beyond the linear order the theory is completed by general relativity, to be derived thermodynamically; strong-field predictions of a scalar field equation are therefore not predictions of the theory. Every result quoted here is reproducible with the open Causality Lab.