Isochoric Solenoidal Quantum Gravity: Unified Field Theory via Traceless Geometric Flows and Algebraic Regularization
Year: 2026
Type: preprint
Abstract: This monograph establishes an operator-theoretic, non-perturbative formulation of unified field theory denoted Isochoric Solenoidal Quantum Gravity. Spacetime dynamics are formulated on an oriented Riemannian 3-manifold (M³, g) whose spacelike Cauchy slices evolve via the trace-free Ricci shear flow ∂ₜg = -2Ric₀. The solenoidal projection ℙ_TT strictly preserves the Riemannian volume measure (∂ₜdV_g = 0), precluding finite-time metric collapse and coordinate singularities. The infimum of the scalar curvature is unconditionally bounded below by the conformal Yamabe invariant R ≥ R_yamabe > 0, providing an intrinsic geometric floor that replaces phenomenological dark energy fields without introducing an arbitrary cosmological constant. Fundamental matter fields are represented as closed positive Demailly currents T ∈ 𝒟'¹'¹(X). By Siu analyticity, their singular supports decompose into discrete algebraic cycles Z = ∑ cⱼ[Zⱼ] whose topological stability is enforced by the non-perturbative gauge spectral gap Δ_gap > 0 of the compact Lie group G = SU(2). The unified action S_unif[g, A, Ψ] generates simultaneously the traceless Einstein field equations, Yang–Mills field equations with positive mass gap, and solenoidal fluid transport. The master operator identity 𝒯_H = Δ_L + ∇_CS establishes an exact isospectral duality between boundary arithmetic entanglement—governed by the Weil explicit formula, Selberg trace formulas, and Néron–Tate height regulators—and bulk dissipative geometric flow. At astrophysical scales, dark matter phenomenology is derived from the conserved solenoidal vorticity commutator [Ψ, Γ]_μν entering the extended geodesic deviation equation, yielding flat galactic rotation curves and the Baryonic Tully–Fisher relation (v_∞⁴ = G M_baryon a₀) with a₀ = (1/6) c² √(R_yamabe) χ_*. Primordial scale invariance (n_s ≈ 0.965, ℓ₁ ≈ 200) emerges from the Kraichnan–Batchelor inverse enstrophy cascade of the Cotton tensor without inflationary scalar potentials. All 17 foundational theorems and coercivity lemmas are certified in Lean 4 with 0 sorry and 0 unproven axioms, and audited via 14 deterministic passing benchmarks in SIMD-accelerated Rust engines.
Author: Zapata, Richard Santiago
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222649534
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