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 (ρP, ℓP, 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 ΩMO/ΩM = √3·π ≈ 5. 4414, matching Planck 2018 at 0. 86σ; and the dark-energy fraction ΩEO = fc ≈ 0. 6869, matching Planck 2018 at 0. 30σ. The closed-form derivations of the structural minimum amplitude Sₘinˢtr = 12 (π²−4) /π⁴ ≈ 0. 7231 (from the diamond Brillouin-zone spectral average; equivalently Sₘin^2, str ≈ 0. 5229) and the geometric maximum amplitude Sₘax^2, 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 Vₑff falls at Sₑq = 0. 7230873678 = 12 (π²−4) /π⁴ to 10-digit precision, and the resulting effective mass mₑff c² = 4. 4373 EP matches the independent linearised-spectrum prediction mᵣadial c² ≈ 4. 44 EP 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 ρ₌₀ₗ, ₀₁ₒ = 2. 18 ρP, removing the GR singularity by the same geometric mechanism that bounds the cosmological bounce, with a structural minimum BH mass Mₘin ≈ 0. 117 mP and a maximum cosmological expansion rate Hₘax ≈ 7. 92 × 10⁴³ s⁻¹ at the Big Bounce. The baryonic regime localises at δS/Sₘin ~ 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 Kc ≈ 6. 71 × 10⁵ (log₁₀ Kc = 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 Kglobal ~ 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.