This work addresses two questions that the standard framework of physics treats as inputs rather than problems: why Newton's gravitational constant G has the value it does, and why the reduced Planck constant ħ has the size it does. In the usual picture both are measured and inserted by hand, and the Planck relation that binds them to the speed of light c is left unexplained. We show that if the physical vacuum is modelled as a single relativistic elastic medium — a complex scalar field Ψ = S·e^iθ governed by one Lagrangian, characterised by a stiffness density ρP, a coherence length ℓP, and the causal structure c — then G and ħ are not independent inputs but structural identities of the medium. Linear response of the field fixes G = c²/ (ρP ℓP²), a value confirmed independently by the one-loop Sakharov induced action; canonical quantisation of the phase mode gives ħ = ρP c ℓP⁴. Eliminating ρP between the two returns the Planck relation ℓP = √ (ħG/c³), so a single pair (ρP, ℓP) reproduces both constants at once, to better than 0. 1%, with no parameter fitted to either. Each identity is moreover the unique dimensionally admissible combination of the medium's scales, so the agreement is a forced selection rather than a coincidence. As a physical corollary, the same two identities reproduce the Chandrasekhar mass, unifying quantum degeneracy support and gravitational collapse within one medium. The governing Lagrangian is not postulated but constructed as the most general effective field theory the medium's symmetries allow, unique at the order that governs gravity and quantum coherence. The manuscript states plainly the one degree of freedom that remains calibrated rather than derived — the numerical value of (ρP, ℓP) — and gives a falsifiable forward prediction: a Planck-suppressed correction to the dispersion of gravitational waves. This manuscript is an extracted, self-contained result from the broader PIU-Ψ research programme (Zenodo DOI 10. 5281/zenodo. 21205881). A self-contained Python verification script reproducing every numerical result is included.
Manuel Alberto Celedon Mejia (Thu,) studied this question.