poisson_operator_full
plain-language theorem explainer
The definition assembles the full causality-bound Poisson operator as the sum of a kernel-independent background term and a perturbation term on which the BIT kernel acts. Cosmologists modeling structure formation under Recognition Science modifications would cite this operator to separate homogeneous expansion from density contrasts. The construction is a direct algebraic sum of the two component operators.
Claim. The full Poisson operator is defined by $P(P, k_0, k, a, bar rho, delta rho) = P_{bg}(a, bar rho) + P_{pert}(P, k_0, k, a, delta rho)$, where $P$ encodes kernel family parameters, $a$ is the scale factor, $bar rho$ the background density, and $delta rho$ the density contrast.
background
The ILG module introduces the modified Poisson equation as a Fourier-space multiplier and establishes basic stability and scaling bounds relative to standard GR. The background operator solves the homogeneous equation and is independent of kernel details, while the perturbation operator applies the BIT kernel family (constant, inverse-one-plus-z, or exponential). Upstream results include the kernel definition from Cosmology.BITKernelFamilies and the independent voxel semantics from ClassicalBridge.Fluids.LNALSemantics, which ensure the operator respects spatially decoupled evolution.
proof idea
The definition is a one-line wrapper that adds the background Poisson operator evaluated at the scale factor and mean density to the perturbation operator that receives the kernel parameters, minimum wavenumber, wavenumber, scale factor, and density contrast.
why it matters
This definition supplies the central object referenced by the CausalityBoundsPoissonCert structure and the two homogeneity theorems that follow it. It completes the Poisson-with-kernel statement targeted by the module, connecting to the Recognition Science forcing chain through J-uniqueness and the phi-ladder. The construction supports causality bounds within the eight-tick octave while leaving open the explicit matching of the kernel to the alpha band.
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