Mathematical Foundations of Derived Probability and ECT: Well-Posedness and Gravitational Tests

First published September 7, 2025 Version 3 — (2025-09-12) Updates: Title adjusted for journal scope. Born Rule derivation embedded (Gleason ≥3D, Busch for 2D). κ_T = 1, κ_S = −1 now shown via coefficient matching. Abstract revised. • Cavity-QED example includes units and uncertainty. Grönwall bound added with named constants. Formatting fixes: proof blocks, lemmas, theorem stack. Version 2 (2025-09-07): This update fixes LaTeX formatting (references, spacing, title/affiliation placement) and corrects scope - changing language that implied global claims so local equivalences and arguments are explicitly marked as local. This mathematical foundations paper isolates and formalizes the mathematics behind the Theory of Derived Probability and Entanglement Compression. It provides the functional-analytic foundation for the Lawrence Universal Wave Function (LUWF) and its governing Primordial Wave Equation, written in the operative formiħ ∂tΨ = −αd ∇²Ψ + β ℂ[Ψ] Ψ. Here the compression operator ℂ[Ψ] is real and multiplicative, specified by a δ-regularized logarithmic form and assumptions that enforce local Lipschitz continuity on H¹. Under bounded ∂tℐ, the initial-value problem is globally well-posed in H¹ for d ≤ 3, with conserved energy functional and continuity law. The companion restates and proves the central result of derived probability: pi = |⟨φi, Ψ⟩|², obtained from the Primordial Wave Equation under the standard measure axioms. It makes explicit the dispersion–compression parameterization, the role of units, and the distinctions between the scalar compression field Cs(x), the operator ℂ[Ψ], and the effective light speed c(x). The curvature link is developed through the compression-balance invariant U(x) = T(x)/C(x) and the eikonal reduction, giving c(x)² = U(x) and null-geodesic propagation in the weak-field closure that recovers the Einstein limit. Every lemma and theorem is reproduced with consistent notation and unit tracking from the main paper, making this document “audit-ready.” The results are not abstract formalities: each one is tied directly to falsifiable predictions. These include light-deflection and lensing residuals, Shapiro delay, cavity-QED coherence windows, neutron-star time dilation, residual-density construction, and cosmic-acceleration closure. For readers, this companion is a self-contained entry point into the mathematical engine of Entanglement Compression Theory. It provides a dense map of the formal structures, operator distinctions, and derivations that connect probability, quantum mechanics, and gravitation in one compressive framework. Readers are encouraged to download the core theory papers for the full narrative and physical development: Theory of Derived Probability and Entanglement Compression: Extending Entanglement Compression Theory (ECT) to Quantum Gravity and Cosmology — https://doi.org/10.5281/zenodo.15786696 The Oscillation Principle — https://doi.org/10.5281/zenodo.17058692