Gravitation as Phase Synchronisation over Finite Relational Substrate

We derive gravitation on the finite relational substrate from three identifications: mass is the cardinality of a phase-locked cluster, distance is decoherence, and time is the observer’s scale-dilation, which no embedded shell shares exactly. A field has no proper ideals, so every massive object is a non-ideal embedded shell whose frame drifts; gravity is the relaxation of that forced drift. We obtain Newton’s law with G = ℏc/m_P² and the equivalence principle as an identity; the relativistic completion gives post-Newtonian β = γ = 1 and the classical tests at their observed values; and the radiative sector, made conservative by the quarter-turn conjugate momentum, gives two helicity-±2 gravitons propagating at c. The exact static solution is horizonless but operationally black, with a parameter-free shadow 4.6% wider than general relativity (1σ above current Event Horizon Telescope limits), a −4.4% ringdown shift, and an area-law entropy recovering the Bekenstein–Hawking ¹/₄ and the de Sitter entropy. The resolution floor fixes the galactic acceleration scale a₀ = cH₀/2π and the full radial acceleration relation as a registration crossover with no fitted function; the primordial spectrum is the scale-invariant (n_s = 1) invariant of the drive. Every order-one constant is determined, leaving the solid angle 4π the one recurring number: the construction is parameter-free. Every exact claim is verified in finite-field or cyclotomic arithmetic, the continuum entering only as a labelled degenerate idealisation.