kernel
plain-language theorem explainer
The declaration defines the ILG kernel w(k,a) as 1 plus C times the alpha power of max(0.01, a over k tau0). Continuum-limit and equilibrium-response modelers cite this expression when assembling Duhamel integrals or response coefficients. It is realized as a direct noncomputable definition of the closed-form formula.
Claim. The ILG kernel is the function $w(k,a) = 1 + C (max(0.01, a/(k τ₀)))^α$, where α = (1 - 1/φ)/2 is the exponent, C the amplitude, and τ₀ the reference time scale supplied by the KernelParams bundle.
background
KernelParams is the structure that packages the ILG exponent α = (1 - 1/φ)/2, amplitude C, and reference time τ₀ together with the side conditions τ₀ > 0 and α ≥ 0. The module states that this kernel formalizes Infra-Luminous Gravity via the explicit formula w(k,a) = 1 + C (a/(k τ₀))^α, with the max safeguard inserted for numerical safety. Upstream, τ₀ is supplied by Constants.tau0 as the fundamental tick duration in RS-native units; the same module also imports the BIT kernel families for comparison.
proof idea
The declaration is a direct noncomputable definition that simply transcribes the mathematical expression, inserting the max(0.01, ·) guard on the ratio before raising to the power α.
why it matters
The kernel appears inside UniversalResponseCert (via jcostHessianCert) and supplies the integrand for DuhamelKernelDominatedConvergenceAt and duhamelKernelIntegral in the 2-D continuum limit. It therefore supplies the concrete operator needed to close the ILG side of the Recognition Science forcing chain (T5 J-uniqueness through the self-similar fixed point) and to connect the CPM coercivity constants to the eight-tick octave.
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