kernelAtRefK_eq
plain-language theorem explainer
kernelAtRefK P a coincides with the general kernel evaluated at unit wave number. ILG kernel workers cite the identity when reducing reference-scale expressions in the w(k,a) model. The term proof is a single simp that unfolds the two kernel definitions and discharges via one_mul.
Claim. Let $P$ be a KernelParams bundle (with exponent $α$, amplitude $C$, reference scale $τ_0$) and let $a ∈ ℝ$. Then the reference-scale kernel equals the general kernel at wave number 1: kernelAtRefK$(P,a)$ = kernel$(P,1,a)$.
background
The ILG kernel is $w(k,a) = 1 + C · (a/(k τ_0))^α$ with $α = (1-1/φ)/2$. KernelParams is the structure holding alpha, C, tau0 together with the positivity hypothesis 0 < tau0. The module states that the kernel reduces to 1 at the reference scale a = k τ_0 and records monotonicity and coercivity links. The upstream kernel definition (from BITKernelFamilies) supplies the general functional form used here; one_mul supplies the arithmetic reduction that appears in the simp set.
proof idea
One-line simp wrapper. The tactic unfolds kernelAtRefK and kernel, then applies the arithmetic theorem one_mul to finish the equality.
why it matters
The lemma sits in the basic-properties section immediately after the kernel definitions and supplies the reference-scale case needed for later statements such as kernel_pos and kernel_ge_one. It directly implements the module claim that the kernel reduces to 1 at a = k τ_0. In the Recognition framework it supports the ILG exponent derived from self-similarity (T5–T6) and the eight-tick octave structure.
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