Unified Field Formula 𝓕_𝓐 = d𝓐 + 𝓐 ∧ 𝓐 with 𝓐 = ω ⊕ A What Einstein and Bohr were debating: double projection of a deeper apparatus

This paper formulates the double projection that allows Einstein's unified-field program and the Einstein–Bohr boundary to be read as exterior manifestations of a common underlying apparatus: the Sistema Vectorial SV. The curvature formula 𝓕𝓐 = d𝓐 + 𝓐 ∧ 𝓐 with 𝓐 = ω ⊕ A is presented as the exterior geometric projection of the SV governing equation 𝓔★TODO,SV(ΓU;τ) ≡ 𝔘unifSV = 0. Dynamics are supplied by a sectorial variational action, the quantum boundary is controlled by the factual class 𝕴F, the correlator C(δ)=−cosδ is derived as the unique minimal irreducible real character of S1, and the relation between the gravito-electromagnetic regime and the quantum factual regime is decided by a ternary co-closure verdict. Explicit demonstration is given, by formal reduction with theorems and proofs, of the structural closure of Einstein's historical unification program in its six identifiable attempts (absolute parallelism, classical and non-abelian Kaluza-Klein, Einstein-Cartan with torsion, non-metric affine connections, Einstein-Straus non-symmetric theory) and the structural dissolution of the Einstein–Bohr dispute in its three historical fronts (the 1935 EPR incompleteness objection, the Solvay conferences Bohrian complementarity, the 1927 Heisenberg indeterminacy). A bidirectional translation operator 𝓣 with explicitly identified structural asymmetry articulates both historical formulations as projections of the same apparatus without axiomatic juxtaposition. The non-symmetric Einstein-Straus theory is treated to all orders in φ, with explicit algebraic expansion through fourth order, formal induction for n ≥ 5 based on Cayley-Hamilton invariants of the antisymmetric operator, and term-by-term projection onto the unified apparatus. The parameters (𝓥, Φ, μ, η, λ) of the quantum factual class 𝕴F are derived from minimum-structure principles, not chosen by convention. Observed relative frequencies under projection to the binary alphabet are obtained structurally from the unique correlator CSV(δ) = −cos δ and match cos²(δ/2) quantitatively over the singlet-type angular regime; full extension to arbitrary two-qubit and n-partite states is established via the canonical Pauli decomposition of the density operator on ℂ²⊗n, recovering the Born rule as an operative derived law rather than a foundational probabilistic axiom. Three independent operational falsifiability criteria establish the structural testability of the apparatus. The appendix develops a full operator-level closure of lateral programs, with tables, examples and reproducible checks. --- Note: Canonical material source on GitHub: https://github.com/juantoniolloretegea/SV-matematica-semantica/tree/main/documentos/adendas/matematica-fisica-factual-contemporanea-sv/formula-de-campo-unificado-conexion-curvatura-einstein-bohr-doble-proyeccion. Supplementary reproducible laboratory deposit on Zenodo: https://doi.org/10.5281/zenodo.20041095. These canonical sources are provided to facilitate direct consultation of the living textual record, reproducible laboratory verification and reader-side translation through standard browser translation tools when required.