C_kernel_competing
The definition introduces the competing amplitude C' = φ^{-3/2} as a benchmark value drawn from an earlier three-channel argument. It is cited in downstream comparisons that establish its violation of the half-rung budget identity when added to J(φ). The definition is a direct real-power expression using the golden ratio.
claimThe competing amplitude is given by the real number $C' = ϕ^{-3/2}$.
background
The ILG spatial kernel modifies the Newton-Poisson relation in Fourier space as w_ker(k) = 1 + C · (k_0/k)^α with α = (1 - φ^{-1})/2. The module derives the structurally forced amplitude C = φ^{-2} from the half-rung budget identity J(φ) + C = 1/2, which follows from φ² = φ + 1 alone. Two prior accounts differ: the entropy paper gives C = φ^{-2} via three-channel factorization, while Gravity_From_Recognition gives C = φ^{-3/2}.
proof idea
One-line definition that directly assigns the real power phi raised to -3/2.
why it matters in Recognition Science
It supplies the benchmark value used by C_competing_gt_C_kernel and C_competing_violates_budget to show that φ^{-3/2} exceeds the half-rung budget when paired with J(φ). This contrast selects the forced amplitude C = φ^{-2} (equivalently 2 - φ) that satisfies the RCL-derived identity and lies in the observed band (0.380, 0.385).
scope and limits
- Does not derive C' from the Recognition Composition Law.
- Does not assert that C' satisfies the half-rung budget identity.
- Does not supply positivity or band membership proofs for C'.
formal statement (Lean)
195def C_kernel_competing : ℝ := phi ^ (-(3 / 2 : ℝ))
proof body
Definition body.
196
197/-- `C'` is positive. -/