poisson_enhancement
The result establishes that the absolute value of the ILG-modified gravitational potential equals the BIT kernel factor times the absolute value of the standard GR Poisson solution. Cosmologists comparing modified gravity models to Newtonian limits would cite this scaling relation when extracting observational predictions. The proof is a short algebraic reduction that unfolds the operator, invokes kernel positivity, and applies ring simplification with absolute-value identities.
claimLet $w(k,a)$ denote the BIT kernel evaluated at wavenumber $k$ and scale $a$. Let $Φ_{ILG}$ be the solution of the kernel-modified Poisson equation and $Φ_{GR} = -4π δρ / k²$ the corresponding GR solution. Then $|Φ_{ILG}| = w(k,a) · |Φ_{GR}|$ whenever $k ≠ 0$.
background
The module defines the modified Poisson equation in Fourier space as a multiplier by the BIT kernel. The kernel itself is a function of wavenumber and scale factor that admits constant, inverse-linear, or exponential forms. KernelParams package the family choice and auxiliary constants needed to instantiate the multiplier. The local theoretical setting is the derivation of elementary stability and scaling bounds that relate the ILG potential directly to its GR counterpart. Upstream results supply the kernel definition and the structural identifications that embed the Poisson operator inside the larger ILG ledger.
proof idea
The proof unfolds the definition of the poisson_operator, applies simp to eliminate the conditional branch under the hypothesis k ≠ 0, invokes the positivity lemma for the kernel, rewrites the numerator via ring algebra to factor out the kernel, and finishes with the absolute-value multiplication rule together with abs_of_pos.
why it matters in Recognition Science
The bound supplies the precise scaling factor that anchors the ILG Poisson solver inside the Recognition Science framework. It quantifies the enhancement of the gravitational potential by the kernel multiplier, thereby connecting the Fourier-space formulation to the self-similar fixed-point and eight-tick octave structures. No downstream theorems are recorded yet, so the result functions as a foundational stability statement for subsequent ILG derivations.
scope and limits
- Does not derive or justify any particular functional form of the kernel.
- Does not treat time-dependent or relativistic extensions of the Poisson equation.
- Does not address vanishing source densities or the homogeneous case.
- Does not extend the bound to vector or tensor gravitational potentials.
formal statement (Lean)
36theorem poisson_enhancement (P : KernelParams) (k a δρ : ℝ) (hk : k ≠ 0) :
37 let Φ_ILG := poisson_operator P k a δρ
proof body
Tactic-mode proof.
38 let Φ_GR := -(4 * Real.pi * δρ) / k^2
39 |Φ_ILG| = kernel P k a * |Φ_GR| := by
40 unfold poisson_operator
41 simp only [if_neg hk]
42 have h_kernel_pos : 0 < kernel P k a := kernel_pos P k a
43 -- Rewrite -(4πw·δρ)/k² as w·(-(4π·δρ)/k²) under absolute value.
44 have h_eq : -(4 * Real.pi * kernel P k a * δρ) / k^2
45 = kernel P k a * (-(4 * Real.pi * δρ) / k^2) := by ring
46 rw [h_eq, abs_mul, abs_of_pos h_kernel_pos]
47
48/-- Coercivity Bound: The modified potential is non-vanishing for any non-vanishing source. -/