WeakFieldBridge
The WeakFieldBridge structure encodes the weak-field limit of the discrete-to-continuum bridge, where the Newtonian potential satisfies the Poisson equation sourced by mass density with RS-native coupling 4 phi^5. Researchers deriving Newtonian gravity from lattice J-cost models would cite it to match linearized Einstein equations. The definition directly assembles the potential function, density function, and exact Poisson relation without additional lemmas.
claimIn the weak-field limit where the metric satisfies $g_{00} = -1 + 2Phi$ with $|Phi| ll 1$, the structure consists of a potential function $Phi: mathbb{R}^4 to mathbb{R}$, a density function $rho: mathbb{R}^4 to mathbb{R}$, and the Poisson equation $nabla^2 Phi(x) = 4 phi^5 rho(x)$, which follows from the linearized Einstein equations with coupling $kappa = 8 phi^5$.
background
The DiscreteBridge module maps the J-cost lattice of Recognition Science to continuum curvature via the path lattice J-cost to quadratic defect to lattice Laplacian to continuum $nabla^2$ to Ricci scalar to Einstein tensor. The local setting assumes the flat Minkowski limit as base case with D=3 spatial dimensions from the forcing chain, and packages nonlinear Regge convergence as an explicit hypothesis rather than an axiom. Upstream results include the Chain structure supplying minimal recognition relations and Constants.G giving the gravitational constant in RS-native form $G = lambda_rec^2 c^3 / (pi hbar)$.
proof idea
As a structure definition it directly assembles the three components: the potential map, the density map, and the universal quantification enforcing the Poisson relation with coefficient (1/2) * 8 * phi^5. No lemmas are applied and no tactics are used; the definition itself encodes the bridge from lattice Laplacian to Newtonian gravity.
why it matters in Recognition Science
This definition supplies the explicit weak-field Poisson relation required by the discrete-to-continuum bridge and supports DiscreteContinuumBridge together with flat_limit_chain. It realizes the coupling kappa = 8 phi^5 from T5 J-uniqueness and T6 phi fixed point in the forcing chain, closing the Tier 1 proved path from lattice defects to Newtonian gravity while leaving nonlinear Regge convergence as the open Tier 3 hypothesis.
scope and limits
- Does not establish existence or uniqueness of solutions to the Poisson equation.
- Does not address the strong-field or nonlinear regime of general relativity.
- Does not derive the value of phi or lattice spacing from further axioms.
- Does not include time-dependent gravitational waves or relativistic corrections.
formal statement (Lean)
107structure WeakFieldBridge where
108 potential : (Fin 4 → ℝ) → ℝ
109 density : (Fin 4 → ℝ) → ℝ
110 poisson : ∀ x, laplacian potential x = (1/2 : ℝ) * 8 * phi ^ 5 * density x
111
112/-! ## §4 Coupling Constant Chain -/
113
114/-- The Einstein coupling κ = 8πG/c⁴ = 8φ⁵ in RS-native units.
115 This is derived in `Constants.lean` and `ZeroParameterGravity.lean`. -/