Emergent Stress-Energy from Relational Gradients in a Pre-Geometric Substrate

This work presents a computationally verified mechanism for the emergence of stress-energy from a pre-geometric relational substrate. Starting from a finite partially ordered set (O, ⪯, Ω), we demonstrate that physically consistent stress-energy arises from relational gradients of the coherence functional Ω, without introducing matter fields a priori. Two competing mechanisms are tested: dependence on defect amplitude φ = Ω_max − Ω, and dependence on relational gradients ∇_O Ω. A decisive computational experiment shows that stress-energy is governed by gradients, not amplitude. A linear-gradient configuration provides a critical test: despite having a defect amplitude comparable to a Gaussian profile (φ_max ≈ 0.500), it produces a stress-energy norm seven times smaller. This rules out amplitude-based mechanisms. The emergent stress-energy tensor satisfies:‖Tμν(Ω)‖ ∝ max|∇_O Ω|² and the associated metric deformation obeys:Δ_g^int − Δ_g^ext ∝ − max|∇_O Ω|² with Pearson correlation r = −0.942. Across all configurations, the spectral dimension remains invariant (d_spec = 2.0000), demonstrating that dimensionality is determined by causal order and decoupled from the matter sector. All results are fully reproducible and supported by the accompanying figure. This work is part of the research program “Law of Continuity of Being (LCS)”.