Harmonic Gravitation: Finite-Edge Einstein Dynamics in Pufferfish Theory

The world is shaped by what can be held. Harmonic Gravitation introduces a finite-Edge formulation of gravity inside Pufferfish Theory (PFT). The paper enforces a strict two-layer contract: upstream, spacetime evolves by ordinary General Relativity; downstream, a finite vantage renders what can be stably held at a chosen resolution (the Edge). The gravitational run is not modified. At fixed Edge, spacetime is governed by standard Einstein–matter dynamics in a well-posed gauge, with covariantly conserved sources. The Edge is not a field and introduces no new degrees of freedom. Instead, it defines a lens: a slice-elliptic Poisson stabilization that renders finite-resolution observables without feeding back into the run. This separation allows Harmonic Gravitation to explain both why Einstein’s equations are forced and where finite-resolution effects belong. Using PFT-native invariants—representation independence, metric-only gravity, locality with second-order equations—the paper shows that in four dimensions the Einstein–Hilbert run is uniquely selected (via Lovelock uniqueness). Matter is treated fields-first; the default Einstein–scalar module guarantees Bianchi compatibility without post-hoc fixes. Finite-Edge visibility is rendered on spacelike slices by harmonic analysis. Energy density on a slice is stabilized before any reporting or lift. A weak-field Newtonian reporting lane then lifts only the stabilized field, yielding finite soft cores with an unmodified 1/r tail. Singular punctures arise only in the zero-Edge idealization, not as finite-resolution observables. The familiar relation E = mc² appears here as a ledger identity: mass is stabilized energy written in mass units, after lensing. The paper also makes explicit where non-closure and curvature can arise in PFT—only from evolving terrain and recomputed readout, not from hidden noncommutation in the gravitational run. Harmonic Gravitation is not a new gravitational law. It is a discipline for keeping dynamics, visibility, and reporting distinct, auditable, and honest at finite resolution.