mellinKernel_inversion
plain-language theorem explainer
The Mellin kernel at the reciprocal argument equals the original argument raised to the reflected Mellin parameter. Researchers on the RS-native zeta program in Phase 3 would cite it for the kernel inversion symmetry. The proof is a one-line wrapper that unfolds the kernel and reflection definitions then applies the substitution lemma from MellinPullback.
Claim. For real numbers $s$ and $x>0$, the Mellin kernel $x^{s-1}$ evaluated at argument $x^{-1}$ equals $x$ raised to the power of the reflected parameter $1-s$.
background
This theorem lives in the MellinTransform module, Phase 3 of the RS-native zeta program. The module isolates algebraic reciprocal symmetry from analytic integral questions, supplying a formal Mellin interface whose reflection is derived from substitution rather than contour integration. The Mellin kernel is the real-valued function $x^{s-1}$; mellinReflect is the map $smapsto 1-s$. The upstream lemma mellin_reflection_via_substitution states that under $x mapsto x^{-1}$ the kernel transforms exactly as $s mapsto 1-s$ after the Jacobian, which is the algebraic content used here.
proof idea
The proof is a one-line wrapper. It unfolds mellinKernel and mellinReflect, then directly applies the theorem mellin_reflection_via_substitution s x hx.
why it matters
The result is consumed by mellinPhase3Cert, which packages the Phase 3 certificates including J_reciprocal and kernel_inversion. It supplies the explicit inversion symmetry needed before Phase 4 instantiates the transform with a theta kernel. The module documentation identifies this as the transform-level bridge derived from reciprocal symmetry, the same symmetry that appears in the Recognition Composition Law.
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