The Universal Integrity Principle (PIU) - A Unified Theoretical Framework (V32.1.4)

The Universal Integrity Principle (PIU) presents a unified theoretical framework in which gravity, quantum mechanics, baryonic matter, dark matter, and dark energy emerge as effective regimes of a single ontological substrate, the Pleno, governed by a single Lagrangian density F1. Six structural problems of contemporary physics — the GR–QM incompatibility, singularities, the nature of the dark sector, the cosmological constant problem, the Hubble tension, and the gauge hierarchy — are addressed simultaneously from a common substrate hypothesis. From three Planckian inputs (ρ_P, ℓ_P, c) and one unifying postulate, the framework derives the Newtonian gravitational constant G (via Sakharov-induced elastic superposition) and the reduced Planck constant ℏ (via Madelung–Bohm canonical quantisation) to better than 0.1%, and produces two structural cosmological predictions with no fitted parameters: the ratio of dark to baryonic matter Ω_MO/Ω_M = √3·π ≈ 5.4414, matching Planck 2018 at 0.86σ; and the dark-energy fraction Ω_EO = f_c ≈ 0.6869, matching Planck 2018 at 0.30σ.The closed-form derivations of the structural minimum amplitude S_min^str = 12(π²−4)/π⁴ ≈ 0.7231 (from the diamond Brillouin-zone spectral average; equivalently S_min2,str ≈ 0.5229) and the geometric maximum amplitude S_max2,geom = (8/3)√(2/3) (from the Kepler–Hales packing theorem) fix the structural reduction of the dimensionless parameters of the framework to derived theorems.A symbolic verification of the F1 tree-level reduction to the non-relativistic Gross–Pitaevskii equation confirms exact convergence: the dynamical minimum of the canonical V_eff falls at S_eq = 0.7230873678 = 12(π²−4)/π⁴ to 10-digit precision, and the resulting effective mass m_eff c² = 4.4373 E_P matches the independent linearised-spectrum prediction m_radial c² ≈ 4.44 E_P from the tripartite Pleno mode-content to 0.06%, providing internal cross-validation of the corpus at Planckian precision.The black-hole interior saturates to the Kepler–Hales density ρmax,abs = 2.18 ρ_P, removing the GR singularity by the same geometric mechanism that bounds the cosmological bounce, with a structural minimum BH mass M_min ≈ 0.117 m_P and a maximum cosmological expansion rate H_max ≈ 7.92 × 10⁴³ s⁻¹ at the Big Bounce. The baryonic regime localises at δS/S_min ~ 10⁻⁷⁹ in S-space, structurally justifying the operational separation of gravitational, quantum and cosmological scales.Bayesian model comparison against ΛCDM, with explicit computational bounding of the algebraic null space over the alphabet {1, 2, 3, 4, π, e} up to Kolmogorov complexity N = 5, yields a cosmological Bayes factor K_c ≈ 6.71 × 10⁵ (log₁₀ K_c = 5.83): decisive on the Jeffreys scale by approximately 3.8 orders of magnitude over the critical threshold for the cosmological sector alone. Extension to the full structural prediction set including G and ℏ yields a global factor K_global ~ 10¹¹, roughly 9 orders of magnitude over the Jeffreys threshold.Twenty quantitative predictions are catalogued, each with explicit falsifiability criteria targeting LISA, Fermi-LAT/CTA, FCC-hh, CMB-S4, LiteBIRD, DESI, BICEP/Keck, and Einstein Telescope over the 2027–2040 horizon.