alpha_kernel
alpha_kernel encodes the ILG kernel exponent α = (1 - φ^{-1})/2 ≈ 0.191. Gravity modelers cite it when writing the Fourier modification w_ker(k) = 1 + C (k_0/k)^α in the spatial kernel. The definition is a direct one-line algebraic assignment using the golden ratio.
claimThe kernel exponent is defined by $α := (1 - φ^{-1})/2$.
background
The ILG Spatial-Kernel module writes the Fourier-space modification of the Newton-Poisson relation as w_ker(k) = 1 + C · (k_0/k)^α. Here α = (1 - φ^{-1})/2 matches the value already fixed in alphaLock, while the module derives the complementary amplitude C = φ^{-2} from the half-rung budget identity J(φ) + φ^{-2} = 1/2. This identity follows from φ² = φ + 1 alone and treats J(φ) = φ - 3/2 as the cost penalty for crossing one rung. Upstream cost definitions from MultiplicativeRecognizerL4 and ObserverForcing supply the J-cost appearing in the identity, while rung functions from anchor policies label the steps on the mass ladder.
proof idea
The declaration is a one-line definition that directly assigns the algebraic expression (1 - 1/phi)/2.
why it matters in Recognition Science
This supplies the exponent α required by the ILG kernel when the module derives C = φ^{-2} via the half-rung budget identity. It resolves the prior ambiguity between C = φ^{-2} and C = φ^{-3/2} by enabling the structural complementarity J(φ) + C = 1/2. The value is consistent with J-uniqueness and the phi fixed point in the forcing chain, and is cross-referenced from GravityParameters.alpha_gravity.
scope and limits
- Does not prove α from the forcing chain steps.
- Does not derive or state the amplitude C.
- Does not apply the kernel to specific wave numbers.
- Does not connect to mass formulas or Berry thresholds.
formal statement (Lean)
88def alpha_kernel : ℝ := (1 - 1 / phi) / 2
proof body
Definition body.
89
90/-- The first-rung J-cost penalty (recurs across direct detection,
91 BIT cosmic-aging, biofilm quorum, etc.). -/