Advancing the unification of probability, curvature, and quantum emergence through Entanglement Compression Theory (ECT).
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Mathematical Foundations of Derived Probability and ECT: Well-Posedness and Gravitational Tests
This paper establishes the formal mathematical basis of Entanglement Compression Theory (ECT). While the Theory of Derived Probability and Entanglement Compression introduced the full physical framework, the Mathematical Foundations paper converts it into an explicitly defined partial differential system—proving existence, uniqueness, and energy conservation for the Primordial Wave Equation (PWE) that governs the Lawrence Universal Wave Function (LUWF).
The core contribution is a complete well-posedness proof in the Sobolev space H¹ for spatial dimensions d ≤ 3 under bounded ∂tℐ. This establishes the LUWF as a mathematically self-consistent foundation for quantum dynamics. From this framework, probability is derived rather than assumed: the Born rule
pi = |⟨φi, Ψ⟩|²follows directly from the energy functional, consistent with Gleason’s and Busch’s theorems.
Sections on compression and curvature show how the balance term
U(x) = T(x)/C(x)defines an effective local light speed c(x)² = U(x), embedding curvature directly in the LUWF substrate and reproducing the weak-field limit of general relativity. This same structure yields measurable departures at strong compression gradients—microarcsecond lensing residuals, Shapiro delay corrections, and residual-density fields mimicking dark matter.
The latter half of the paper formalizes collapse, decoherence, and rebirth as mathematically symmetric processes within the PWE, governed by a threshold energy density near 10⁻²³ J/m³. The “murmuration” section extends this reasoning to collective coherence, introducing braid invariants and survivability bounds that describe how compression-linked domains maintain global order without external synchronization.
Taken together, the Mathematical Foundations paper functions as the rigor spine of ECT: it translates a conceptual unification into a testable, audit-ready mathematical framework with explicit constants, conserved quantities, and falsifiable thresholds.
Full text: Zenodo DOI 10.5281/zenodo.17071135
Author: William Andrew Lawrence — Compression Theory Institute