From the Damped Spring to the Cosmos: A Derivation of the Zadel Cosmological Distance Law from the Universal Principle of Damped Oscillation

We demonstrate that cosmological redshift admits a natural description as thelarge-scale manifestation of the universal phenomenon of damped oscillation—thesame exponential amplitude decay observed in every spring, pendulum, electricalcircuit, and decaying quantum state in nature. Treating the propagating photonas an extreme underdamped electromagnetic oscillator subject to a cosmologicaldissipative force characterised by decay rate γ = ln 2 / T, where T is the photon waveenergy half-life, we derive the master wavelength evolution equation f(λ; t) = λ·e+γtfrom four physical axioms via the Cauchy functional equation. The cosmologicalrelaxation time τ = T /ln 2 is identified as the mean photon lifetime, and thelogarithmic ratio ln(z+1)/βθis identified as the cosmological logarithmic decrementof classical damped oscillation theory. Incorporating the generalised vectorial velocitycomposition factor βθ =p(V /C)2 + 1 + 2 cos θ · (V /C)—derived within the Galilean(non-relativistic) approximation valid for V ≪ c—we obtain the Zadel Distance Lawin its definitive preliminary form:D =ξln 2 βθ lnz + 1βθ,where ξ = T C is the ZCP characteristic cosmological length. This paper presentsthe Zadel Cosmological Parameter (ZCP) framework as an analytically tractable toymodel—a minimal, parameter-sparse phenomenological description of cosmologicalredshift—and not as a proposed replacement for the standard ΛCDM model. Appliedto the Pantheon+ Type Ia supernova dataset (1,701 SNe Ia), the law achievesχ2/DOF = 1.089 in a preliminary uncalibrated confrontation; a fully recalibratedBayesian analysis with MB as a free nuisance parameter and the full 1,701 ×1,701 covariance matrix is deferred to Paper C. From the best-fit parameter T =9.925 Gyr, we derive a Universe age of approximately 39.5 Gyr, offering a potentialphenomenological account of the JWST high-redshift galaxy formation anomalywithout additional parameters. Outstanding obligations include a ZCP-nativederivation of the luminosity distance, a self-consistent ZCP recombination epoch and FIRAS spectral distortion budget (Paper B), and a relativistic extension of thekinematic framework.