Entanglement Compression Theory (ECT) begins from a structural claim: observable reality is not built first from particles, probabilities, or spacetime geometry, but from persistent relational structure under compression. What appears inside spacetime as waves, particles, forces, curvature, and probability is treated as the staged emergence of compression-constrained recoverable structure. ECT is being developed as a closed-framework program that begins from the necessity of ordered dependence, understood here as the minimal structural form of causation, for persistent structured identity and aims to carry that dependency chain through compression dynamics, scalar-content closure, local probability recovery, spacetime emergence, tensor formalism, and gravitational response. The intended endpoint is a framework that reaches the Einstein field-equation level, or a controlled Einstein-extension, without importing spacetime geometry, probability, or gravitational dynamics as unexplained primitives. This paper presents a dependency-ordered mathematical foundation for the current core of ECT, organized around persistent structured identity, finite recurrent stability, weak oscillatory form, LUWF/PWE realization, closed scalar content, boundary erasure, local scalarization, Born-form recovery, and spacetime-emergence setup. The central claim is bounded: ECT admits a closed regime-bound mathematical core through local Born-form recovery and a controlled handoff to dimensional representation, provided each layer is kept distinct and no later structure is used retroactively to prove an earlier one. Within that bounded claim, ECT offers a plausible structural account of the origin of probability, not merely a rule for calculating probabilities. Probability becomes meaningful when deterministic recoverability fails at a boundary, leaving unresolved alternatives under a shared boundary-readable residue. Numerical probability is recovered only after surviving scalar content is locally scalarized, normalized, and interpreted inside a declared probability-predictive regime. The framework begins with persistence. A structured identity cannot persist by label, observer assignment, hidden memory, or imposed naming convention. It must persist by recoverable relational distinction. That requirement gives ordered dependence. Under admissible perturbation, refinement, recombination, transport, loss, forcing, and non-collapse, recoverable identity requires finite recurrent stability. Persistent refinable identity then requires weak oscillatory form in the restricted structural sense of bounded recurrent relational correction, not as an assumed physical wave ontology. The ECT realization layer is supplied by the Lawrence Universal Wave Function (LUWF), the Primordial Wave Equation (PWE), the compression response, LUWF realization profiles, and the Lawrence Amplitude Functional Form (LaFF). In the alphad-beta convention, CPsi is dimensionless and beta CPsi is energy-valued. Under the PWE with real multiplicative compression, rho = |Psi|² satisfies continuity and supplies the admitted local nonnegative conserved scalar density in the closed ECT scalar-content regime, up to positive scale unless another PWE-local conservation law is explicitly derived. Probability is not introduced as primitive. Recoverability-relevant boundary erasure supplies pre-numerical probability-status: unresolved scalar-neutral alternatives under a shared boundary-readable residue. Numerical probability enters only after admissible carrier domains Dᵢ are declared and scalar residues sᵢ = integral over Dᵢ of |Psi|² dx are formed. If the carrier domains partition a finite nonzero sector, normalization gives wᵢ = sᵢ / sumⱼ sⱼ. Inside a declared local probability-predictive regime satisfying measure/outcome realization conditions, wᵢ may be interpreted as pᵢ. In Hilbert-channel realization, this becomes pᵢ = ||Pᵢ Psi||² =, and for rank-one channels pᵢ = ||². Thus Born form is recovered locally, not globally, as the Hilbert-channel expression of normalized surviving scalar residue after deterministic recoverability has failed at the boundary and after the relevant outcome branches have been declared. The spacetime branch is stated as a handoff, not as a completed gravitational theory. Recoverable recurrent structure plus compression-stabilized scalar content may be treated as eligible for candidate dimensional representation once a parameterization rule is declared. Tensor formalism, effective metric response, Einstein-limit recovery, deterministic quantum gravity, cosmology, and observational tests remain downstream modules. The result is a closed core framework for ECT: structural persistence, deterministic realization, closed scalar content, locally interpreted probability, Born-form recovery, and spacetime-emergence setup are placed in a single dependency-locked architecture. Version note This version supersedes Version 5 of Mathematical Foundations of Derived Probability and ECT: Well-Posedness and Gravitational Tests. Version 5 remains a transitional mathematical-foundation version that consolidated the LUWF/PWE realization layer, compression-response notation, well-posedness discussion, scalar continuity, and weak-field geometric handoff. Its direct derived-probability architecture has been superseded here by a dependency-ordered treatment in which boundary erasure supplies pre-numerical probability-status, closed ECT scalar content supplies the surviving scalar density rho = |Psi|², local scalarization supplies normalized scalar residues, and Hilbert-channel realization supplies Born form. Readers should treat this Version 6 paper as the controlling mathematical foundations formulation.
William Andrew Lawrence (Tue,) studied this question.