PIU-Ψ: the Principle of Universal Integrity — corpus, mathematical compendium and verification notebook (V5.1)

One statement. One Lagrangian. And from there, the rest. The universe is energy: finite at every point and conserved. From that single statement, and from a single dynamical object —the Lagrangian F1 over the complex field Ψ = S·e^(iθ)— this programme derives, in cascade, gravity, quantum mechanics, electromagnetism, the structure of matter and the content of the universe. Not as analogies: as successive evaluations of the same Lagrangian in different regimes, with the full algebra in view at every step. The constants stop being inputs. G = c²/(ρ_P ℓ_P²) and ħ = ρ_P c ℓ_P⁴ follow from the dynamics of F1 —the statics of the modulus in one case, the quantisation of the phase rotor in the other— and return unit coefficient, with not a single fitted parameter. Gravitation is weak because the medium is rigid; the quantum of action is a mechanical property of the medium, not an axiom of nature. What emerges, and from where. Newton's three laws as the low-energy limit, together with the origin of what Newton left primitive: what mass is, where G comes from, why inertia exists. Einstein's equations as the effective elastic action of the medium. The uncertainty relation with an exact ħ/2 floor and its derived ultraviolet correction. Maxwell and Yang-Mills as the circulation of the phase over the lattice. Particles as topological knots, with spin, fractional charge and confinement as geometric consequences. And a native thermodynamics: the equilibrium occupation and the 2π²/45 coefficient of the Stefan-Boltzmann law are obtained by counting configurations of the eigenmodes of F1, with no quantum statistics assumed. In cosmology, with zero free parameters. The dark-sector ratios —√3π between the two populations, Ω_DE = f_c and Ω_m = 1 − f_c— fall within 1σ of Planck. The equation of state w = −1 is exact in the ground state. The singularity, of the Big Bang and of the black hole, does not occur: the medium is too rigid to permit it. And the whole can be refuted. The programme does not merely state its results: it states which measurement would break them. Measuring ρ_P or ℓ_P by a route passing through neither gravitation nor quantisation exposes the product of the two coefficients; measuring both, each branch separately. To this are added bounds a datum can violate —v_GW = c, w ≥ −1, the age of the universe measured in ticks of its own rate— and an executable verification notebook that recomputes 99 quantities from their first-principles formulas, with no preloaded values. Every claim carries its label. Derived, strong logical deduction, calibrated with the anchor declared. The open ends are inventoried with their severity, not concealed: what is missing is named where it is missing. The package. Corpus (429 pp ES / 425 pp EN), setting out the programme and classifying each result · Mathematical Compendium (106 / 107 pp), with the complete derivations, the intermediate steps and the dimensional analysis · Synthesis of results (25 pp) · Reproducible Python verification notebook. Equivalent Spanish and English editions, LaTeX sources included. Preprint. The invitation is to check it.