A Relational Unification Framework via Type III$_1$ Algebras

This paper presents Oros Fundus, a strictly background-independent relational framework in which macroscopic spacetime and matter emerge as the thermodynamic equilibrium of a purely algebraic phase transition. The framework is governed by a single variational principle, the Coherence Law (δS_vN = δ⟨K_ω⟩), derived as a structural theorem from the KMS condition, Araki relative entropy, and Petz sufficiency within a Type III₁ von Neumann algebra. Via the Noncommutative Geometry Spectral Action Principle, this thermodynamic equilibrium condition is shown to be equivalent to the variation of the Einstein-Hilbert action, yielding the Einstein Field Equations through the heat kernel expansion and inner fluctuation mechanism of the total Dirac operator — without presupposing a background metric. The Weak Equivalence Principle and the Ryu-Takayanagi formula follow as necessary consequences of local causal horizon thermodynamics. The framework derives a parameter-free, monotonically phantom dark energy baseline w(τ) = −1 − (η_opt/2)(1 + e^{−τ}), where η_opt ≈ 0.054 emerges entirely from topological invariants. This baseline is consistent with DESI DR2 indications and makes specific predictions for upcoming surveys. The Hubble (H₀) and S₈ tensions are derived as non-commutative algebraic parallaxes arising from the graded tensor product boundary between the continuous spatial buffer and the discrete internal matter algebra. Particle content emerges from the same algebraic boundary conditions. Three independent derivation paths — the Jordan rank of the exceptional boundary algebra J₃(𝕆), the Jones subfactor index at the D₄ principal graph, and the associativity bound on the internal fiber division algebra — converge on exactly three fermion generations (N_g = 3) and three spatial dimensions. The framework is governed by six foundational axioms whose axiomatic structure maps in a 6-to-6 bijection onto the framework's six algebraic phase transitions. Version note: v1.542 incorporates mathematical and expository strengthening throughout: explicit Proposition for the Central Gravity Bridge (Coherence Law to EFE), GVV CLT state-dependence resolution, intersection subfactor closure, modified luminosity distance formula (Corollary 2a), JB-algebra clarification for the epistemic shadow mapping, lattice QCD reconciliation for the mass gap, and β₀ formalism consolidation within the Spectral Action. Note on Research Cluster: This is the foundational physics manuscript of the Oros Fundus framework. It is supported by two companion papers: Mathematical Corollaries: "The Topologies of Incompleteness" (DOI: 10.5281/zenodo.20031967) Cosmological Analysis: "The DESI Phantom Crossing as a Parameterization Artifact" (DOI: 10.5281/zenodo.19477320)