What a single discrete axiom does and does not imply: state-graph dimension, a four-dimensional readout, and conditional Lorentzian reconstruction

We give a self-contained reconstruction of four-dimensional Lorentzian geometry and the Einstein equations from a single discrete axiom, and we state precisely what must be added for the reconstruction to go through. The axiom, Z0 ("the equilibrium state never rests"), states that (i) the equilibrium state is not stationary, (ii) the evolution never halts, and (iii) the evolution carries no weight — no probability measure and no selection rule. From Z0 alone we derive a discrete skeleton: paths on a state graph, an integer-valued conserved current, an exact zero-net-charge law, closure with a UV cutoff on path length, and initial-condition refresh. The paper's main structural results are the following. 1. Reconstruction. From the three clauses of Z0, with the all-branch evolution realised on the graph Γ of Z1, the model yields a Lorentzian metric of signature $(1,D-1)$ with D=Γ and dₛ=D analytically exact; the Lovelock premises, and hence the Einstein–Hilbert form Gab+Λ gab=8π G\,Tab; a conserved integer source; and an exact causal support cone with a pulled-front selected speed. The dimension is not a number to be derived but a geometric property of Γ (item 2 below), and the dimensional constants are units rather than parameters (§5.4). 2. The dimension comes from Γ, not from the axioms; given Γ, dₛ=D exactly. We prove that the axiom set admits a model for every number of degrees of freedom m≥2, so no statement depending only on the dimension is derivable — D is a geometric property of Γ, not a value awaiting derivation. Restoring the graph part of Z1 gives, on Γ= ZD, the spectral dimension dₛ=D analytically exact (PD(t)~(2/π t)D/2), together with a dispersion relation, a finite group velocity, microcausality and integer multiplicities. 3. A native readout. The partition that satisfies all five of the corpus's criteria — single-body contextuality, modular type III₁, at least four blocks, the pair-path-sum four-dimensional window, and an exact peak of the pair path sum at $D=4$ — is obtained natively: take the first-return primitive loops f(m)=2\,Catalan(m-1), aggregate them by the mode of the excursion-amplitude distribution, and at $L=14$ one obtains block sizes $[290,82,20,2]$. No free parameter is introduced. 4. The two once-registered "structural obstructions": their standings now separate. ① g-locality (a remote perturbation changes a mid-graph weight ratio, by up to 8.75×10⁻¹) is dissolved: it comes from the k→∞ Perron limit, whereas the original definition is finite k, and at finite k $(L')$ holds (§4.1). ② The rank bound for orbit-size partitions (≤ω(L/2)) still holds, but it constrains only that one way of partitioning, and the amplitude grading of §6.1 does not take that route. 5. The continuum limit: two limits must be separated. The metric limit (spacing a→0 at fixed physical size) satisfies BFT (V/diamᵈ constant) the limit is a torus of dimension d, and this is exactly what the links of §5.11 use; the readout limit (block count K→∞ at fixed substrate) does not satisfy BFT (the family gets fat, dₛ unbounded in ln K) — it is a limit in a coordinate direction. The two are not interchangeable. Curvature comes from the conformal class and volume element on top of the base manifold (g=Ω²δ gives R≠0), which is consistent with a flat base manifold. We do not claim an unconditional derivation of four-dimensional general relativity, and we are explicit about the one place where the readout's rule remains a convention.