Emergent Quantum–Cosmological Conjugacy from Relativistic Information Transport

We derive a quantum conjugacy between spacetime diffusivity and inertial mass from relativistic information-transport kinematics. Two Lorentz-invariant laws—(i)~an invariant-time gauge for timelike segments, \( ds=c\,dt \), and (ii)~diffusive evolution \( d\epsilon/ds=c- \)yield a first-order action whose canonical quantization gives \( [\hat\epsilon,\hat m]=i\hbar \) and the emergent Cosmological Uncertainty Principle~(CUP), \( \Delta\epsilon\,\Delta m\ge\hbar/2 \). Independence across coherence cells of size \( \ell_{\rm coh} \) amplifies the bound to \( \Delta\epsilon\,\Delta m\ge(\hbar/2)\,N_{\rm eff} \) with \( N_{\rm eff}=D/\ell_{\rm coh} \), extending quantum uncertainty to cosmic baselines. A single area-diffusion parameter provides an operational unification of Planck and Hubble times across \( {\sim}\,61 \) orders of magnitude. Applied to black-hole horizons, the CUP reproduces Hawking's temperature exactly. for de~Sitter space, a naive 1/H correlation window overshoots by a factor \( \pi \), while KMS/Unruh calibration (\( \tau=\pi/H \)) recovers the standard Gibbons–Hawking result \( T_{\rm dS}=\hbar H/(2\pi k_B) \). Unlike generalised or extended uncertainty principles that deform the position--momentum commutator, the CUP introduces a new conjugate pair (\( \epsilon,m \)) while leaving the Heisenberg sector intact. These results position CUP as an emergent, testable quantum--informational constraint on cosmological observables rather than an added axiom.