Gravity, Held: On Gravity Acting on Itself Outside a Held Body

Formal Record 17 proved that held edges give gravity wherever it is weak, exactly as Einstein’s theory does at first order, given the form of reading it adopts and one premise: that a keeper’s extent follows its pace. This record takes the static field to every order outside a held body, where gravity acts on itself. This record declares its law for keepers at rest: each keeper’s pace is the fixed point of averaging over its neighbours, with held keepers draining it once at each of their own episodes (Definition 2.1), and at first order it is FR17’s reading of a static source. The operator of that averaging is fixed: in a world of uniformly stretched cells the neighbour graph settles to the Laplacian of the cells’ own metric, with one constant (Proposition 1.1). That the static reading of a slowly varying world settles to that operator, with its source and with pace tending to 1 far away, is stated as Hypothesis H. Its Dirichlet form is proved here for flat metrics (Proposition 1.2), and the rest is stated as H. Given H, outside matter the curvature in the time direction vanishes exactly at every order, R₀₀ = N D²N = 0, and inside held matter at rest it takes Einstein’s value for a pressureless source (Theorem 3.2). The exact four-dimensional form of FR17’s premise, which this record declares, makes the curvature in the space directions vanish too, R_ij = 0, so that outside any static held body all ten vacuum equations of Einstein’s theory hold (Theorem 4.2): the three mixed ones are true of every static field, the six in the space directions are the declared premise, and those six force the one in the time direction through the Bianchi identity, which the static reading, given H, confirms independently. The exact form says more than FR17’s first-order statement and agrees with it at first order (Proposition 4.1). For a round body, one whose field outside it is spherically symmetric, the exterior is then exactly Schwarzschild’s, and the law tying a cell’s extent to its pace comes out as a theorem, not a choice: ψ = 2/(1 + N), so the extent is ℓ = ψ⁻² = ((1 + N)/2)², equal to N at first order (Theorem 5.1). The post-Newtonian parameters are β = γ = 1, which give Mercury’s perihelion advance, the bending and delay of light and every classic test (Corollary 5.2). For bodies that are not round, the vacuum field is held by cells of the right shape (Proposition 6.1). A weight of one over measured distance would put a logarithm into the field and fail at second order; the graph’s own spreading is the weight keepers can read, and it does not.