Universal Baseline Law (UBL) and Unified Substrate Theory (UST): Full Mathematical Foundations

This deposit contains two core mathematical documents that define the Universal Baseline Law (UBL) and the Unified Substrate Theory (UST). The first document, Full Math Derivation of the UBL, shows how the √N noise structure emerges from independent randomness, equipartition, fluctuation–dissipation, phase‑space geometry, and cross‑domain stochastic systems. It anchors the noise floor at the Planck scale and defines UBL as the ratio ρ/ρₚ, giving a dimensionless activity level from 0 to 1. The derivation shows that N = 1 corresponds to the Planck floor, where fractional uncertainty reaches its minimum possible value (σ = 1). The second document, Mathematical Foundations of the Unified Substrate Theory (UST), formalizes UST as a field‑based model built on a substrate field ϕ(x,t), an activity density a(x,t), an energy density ρ(x,t), and a normalized UBL field u(x,t). It defines tension, damping, geometric coupling, and stochastic noise, and combines them into a single evolution equation: ∂u/∂t = D∇²u − α(u − u₀) + γβu + ξ(x,t) The document proves fixed‑point solutions, shows how UST reduces to classical physics when u ≪ 1 and N ≫ 1, and shows how it enters the quantum‑gravity regime when u → 1 and N → 1. Together, these documents provide the complete mathematical chain: Start from variance addition and √N noise. Apply it across physical systems. Anchor the scale at the Planck floor. Define UBL as ρ/ρₚ. Build UST as the dynamical field theory that evolves UBL in space and time. This combined DOI serves as the canonical reference for the mathematical foundations of UBL and UST. This work is part of a larger collection of UST documents. The other versions available in the DOI record are not revisions of this document. They are separate papers written for different purposes. Some versions present the full mathematical proofs behind the update rules, others provide a technical physical description of substrate behavior, and others are formal proof papers built around the Universal Balance Laws. Together, these documents form a complete set: a plain‑language booklet, a physical description paper, and full mathematical proof papers, each offering a different perspective on the same underlying theory. If you have questions or want to discuss the work, you can contact me directly at [email protected] github: unified-substrate-theory/Unified-Substrate-Theory: A public research archive containing manuscripts, derivations, diagrams, and supporting materials for Unified Substrate Theory.