Universal Integrity Principle (PIU-Ψ) — Mathematical Compendium V4.8: A Single Substrate. The Constants of Nature, Derived and Verifiable

What if the constants of nature were not free parameters to be measured, but ratios you could derive — and check, line by line? This is the complete mathematical compendium of the Universal Integrity Principle (PIU-Ψ): the full algebra, unabridged, behind the claim that gravity, light, particles, space, time, and the dark sectors are not separate ingredients but configurations of a single physical substrate — el Pleno — described by one field Ψ = S·eiθ and six axioms. Most "theories of everything" ask for trust. This one shows its work. Every closed form, every coefficient, every number is obtained on the page, and each result is stamped with its epistemic status — derived, strong logical deduction, calibrated, or hypothesis — so a reader never has to guess where proof ends and conjecture begins. What the compendium derives, from three Planck-scale properties (ρ_P, ℓ_P, c) and zero tuned cosmological parameters: Newton's G = c²/(ρ_P ℓ_P²) and Planck's ℏ = ρ_P c ℓ_P⁴, both to better than 0.1% — the "constants" are structural identities, not inputs. The Einstein field equations as the elastic action of the medium (Sakharov-induced), and the Dirac equation as the effective theory of its phase modes. Heisenberg uncertainty with an exact ℏ/2 floor, plus a derived ultraviolet correction (coefficient 3/80, no free parameter); the Born rule as conserved-charge accounting; the Tsirelson bound 2√2 from the lattice's tetrahedral geometry. Maxwell (factor 8/9) and Yang-Mills from phase circulation on a diamond lattice; charge quantization as topology. The proton as a Borromean three-vortex soliton — fractional charge, the 2+1 quark asymmetry, and confinement as geometric consequences — with the Δ–N split predicted to ~90% from a single calibrated scale. The dark-to-baryonic ratio Ω_MO/Ω_M = √3·π ≈ 5.4414 (0.86σ from Planck 2018) and the dark-energy fraction Ω_EO ≈ 0.6869 (0.30σ) — pure geometry, no fitting. No singularities: the Big Bang becomes a finite bounce and every black-hole core freezes at a universal density ρ ≈ 2.18 ρ_P, with a minimum collapse mass M_min ≈ 0.117 m_P. The two structural amplitude bounds arrive as closed-form theorems: S²_min = 12(π²−4)/π⁴ from diamond Brillouin-zone spectral averaging, and S²_max = (8/3)√(2/3) from the Kepler–Hales packing theorem. A formal Bayesian comparison against ΛCDM yields a Bayes factor K_c ≈ 6.71×10⁵ — decisive on the Jeffreys scale. The closing move is the one that lingers: the Lorentz factor that opens relativity reappears, at the very end, as the flow index of the medium's rheology — γ² = 1/(1−v²/c²) is the same dilatant stiffening that regulates the cosmic bounce. The chain bites its own tail. PIU-Ψ does not compete with General Relativity or Quantum Mechanics — it derives why they hold. Both emerge as effective readings of one substrate. The key computations (Brillouin-zone averages, the 1/40 and 8/9 coefficients, the potential minimization, Born and Tsirelson, Lane-Emden, the topological baryon number) are verified symbolically and numerically. Where a result is imported, it is marked; where a thread is open, it is named. The invitation is not to believe it — it is to check it. Single-author independent research. All rigorous criticism welcome: it is, in the end, what turns a proposal into science.