Gravitation as Phase Synchronisation on a Finite Substrate

We derive gravitation in the Finite Ring Cosmology framework from three identifications: mass is the cardinality of a phase-locked cluster of substrate degrees of freedom, distance is decoherence, and time is the shared scale-dilation driving every cell in common. Gravity is then spontaneous synchronisation, and five stated premises yield Newton’s law F = Gm₁m₂/r² with G = ℏc/m_P²: the inverse square from harmonicity in three derived dimensions, the product m₁m₂ from coherent additivity, universal attraction from the single arrow of the drive, and the equivalence principle as an identity. The relativistic completion gives post-Newtonian β = γ = 1: deflection, Shapiro delay, perihelion, and Lense–Thirring at their observed values. On the theory’s exponential branch (the alternative is GR-coincident) the exact static solution is horizonless but operationally black, with a parameter-free shadow 4.6% wider than general relativity predicts (55.7 vs 53.3 µ as for Sgr A^∗, 1σ above current Event Horizon Telescope constraints), a −4.4% ringdown frequency shift, suppressed evaporation, and an area-law entropy recovering the de Sitter entropy ∼ Ω of the substrate. The resolution floor, read as an acceleration, derives the galactic radial-acceleration scale a₀ = cH₀/2π ≈ 1.1×10⁻¹⁰ms⁻² against the fitted 1.2×10⁻¹⁰. The discrete Fierz–Pauli functional is exhibited and exactly gauge-invariant; its second-order evaluation decides the strong-field branch and carries the remaining coefficients.