The constants of nature as dressed pure numbers of a single elastic medium: Newton's constant and the quantum of action as a proof of scaling

Newton's constant G and the reduced Planck constant ħ are measured, not explained: standard physics inserts them by hand and leaves unexplained why they have the values they do and why the Planck relation ℓP = √ (ħG/c³) binds them to the speed of light. This paper tests a single structural claim about what such constants are. The claim is that a physical medium is described internally by dimensionless pure numbers, each derived from one Lagrangian whose coefficients are all fixed by the geometry of the medium—none free, none fitted to data—and each fixing a behaviour of the medium, and that a dimensional constant is one of those pure numbers dressed in human units by a single final act of scaling. Units, and the one calibrated number the theory carries, enter the physics at exactly one place: the last step. The claim is proved in the most fundamental sector available, treating the vacuum as one real relativistic elastic medium with an internal structure, described by a complex scalar field Ψ = S·e^iθ with a stiffness density ρP, a coherence length ℓP, and the causal structure c. Solving the modulus sector by two independent routes—classical linear response and the one-loop Sakharov induced action—gives the gravitational identity G = c²/ (ρP ℓP²) ; canonical quantisation of the phase rotor gives the action identity ħ = ρP c ℓP⁴. These are two distinct pure numbers, a compliance and an angular inertia, derived in ratios before any unit is introduced. Eliminating ρP between them returns the Planck relation as an algebraic consequence, so that one scaling act—anchoring the pair (ρP, ℓP) —dresses both constants at once, reproducing G and ħ to better than 0. 1% with no parameter fitted to either, and yielding the Chandrasekhar mass as a corollary that needs both. This is what makes the result more than the truism that constants carry units: a units convention could dress one number, but it could not make one anchoring land correctly and simultaneously in gravity and in quantum coherence, sectors that share no scale, nor produce the Planck relation as a by-product. The anchoring is forced—the calibrated pair simply is the medium's stiffness and grain—so the agreement is not circular but refutable: with the prefactors derived and no second parameter to adjust, the two monomials had every opportunity to miss the measured values and did not. The same pattern is then exhibited where the yardstick is never laid: the medium's dimensionless dissipation floor η/s, derived from the same Lagrangian, whose universality across twenty-one orders of magnitude in temperature—from the quark–gluon plasma to ultracold Fermi gases—no unit convention could produce. The one degree of freedom that remains calibrated rather than derived is stated precisely, with a falsifiable prediction for the dispersion of gravitational waves. This paper extracts the fundamental-constants sector of the broader PIU-Ψ programme (Principio de Integridad Universal) ; the complete corpus is deposited separately (DOI 10. 5281/zenodo. 21205881) and an overview is available at https: //piuniversal. com. A self-contained Python verification script (standard library only) accompanies the paper and reproduces the two structural identities, the dissipation-floor number and its symbolic cancellation of all dimensional constants, the dimensional-uniqueness counts, and the enumerative Bayesian corroboration.