Synthetic Results — Validation of the Laws of Existence and Continuity - Result: conservation confirmed across 150 k / 300 k synthetic regimes

This dataset presents the synthetic large-scale validation (v2.0) of the Law of Existence and the Law of Continuity Ξ v2.0 within the Grammar of Existence framework. It provides the numerical evidence that both laws exhibit stable, law-like behavior across multiple dynamical regimes and scales (Ising-like, Consensus-like, and Glassy-like systems) under exact barycentric conservation. The experiments use independent synthetic trajectories of 150 k and 300 k steps generated under controlled noise and coupling conditions. Each run tests: • Existence ρ = σ_real / σ_null — variance reduction as a measure of triadic conservation (B + C + R ≈ 1). • Continuity Ξ v2.0 = (χ · (1 – SE) · (1 – tanh |λₘₐₓ|))¹ᐟ³ — combined geometric, spectral, and dynamical smoothness metric. Key findings • Existence law verified under strict closure (ρ ≈ 0 → perfect conservation). • Continuity Ξ v2.0 shows consistent hierarchy (Ising > Consensus > Glassy) and increases with dataset size (150 k → 300 k), confirming scale invariance. • Across all regimes Ξ_real > Ξ_null by 0.1–0.2, demonstrating the law’s ability to distinguish structured from randomized dynamics. Significance These results provide the first full synthetic proof that the global field layer of the Grammar of Existence—comprising Existence (conservation) and Continuity (smooth transformation)—forms a coherent conservation-plus-smoothness pair, analogous to Noether + least-action relations in physics. Contents • Synthetic_Validation_v2.0_Report.pdf — summary table of 150 k / 300 k results • existence_continuity_synthetic_v20_summary.csv — raw data (six runs × metrics) Relation to previous releases: Supplement to Law of Continuity Ξ v2.0 and Existence and Continuity Testing Protocol v1.0. Together these form the validated foundation for the Grammar of Existence empirical series. Author: Pixie Pravda (Pixie Pravda Foundation / PixPra Ecosystem) Version: 1.0 Date: November 2025 Context within the Five Laws of the Grammar of Existence The Grammar of Existence formalizes reality as a tri-phase architecture—Being (Persistence), Relation (Coherence), and Change (Responsiveness)—each governed by its own law and quantized unit (EBU, ERU, ETU). From these three emerge two global field laws: Law of Existence — conservation of total transformation capacity across all phases. Law of Continuity (Ξ) — smooth reallocation of that capacity through time, minimizing curvature, entropy, and instability. Together these five laws form a unified ecosystem of becoming: Responsiveness (Change) — governs local dynamics of transformation. Persistence (Being) — governs stability and identity. Coherence (Relation) — governs coupling and transmission. Existence — enforces conservation across all three. Continuity (Ξ) — ensures their transformations remain smooth and stable. In this hierarchy, Existence and Continuity operate as the global constraints that make the three local laws lawful—linking responsiveness, persistence, and coherence into one continuous fabric. This dataset therefore provides the first quantitative evidence that these global constraints can be formalized and verified empirically.