The Legitimacy of Scale Parameter k: From Distorted Reference Frames to Path Selection with Finite Limits

In the main paper(Chen(陈), Q. (2026). From Divergent Series to Transfinite Power Towers: A Generalized Averaging Method Based on "Computable Infinity" and "Continuous Numbers". Zenodo. https://doi.org/10.5281/zenodo.21332964), the decomposition 3=k· s is introduced to solve the transfinite power tower ^∞ x=3. The scale parameter k maps the transfinite target value back to the classical convergent interval s∈[e⁻¹,e]. This construction easily triggers three typical doubts: the parameter is artificially added, it can be arbitrarily applied to any equation, and it lacks fundamental mathematical rationality. Instead of repeating numerical derivations, this paper defends the methodology through three homologous analogies from geometric coordinate transformation, physical path integral and projective geometry: k is not an externally attached coefficient, but an algebraic representation naturally generated after reversible distortion of the value range reference frame. The valid interval k∈[3/e,3e] equals the complete set of coordinate transformations that can regularize transfinite problems into convergent forms. Operations of the same type have been universally recognized in mainstream mathematics and physics, including Feynman path integral and stereographic projection. It can be deduced that the introduction of the scale parameter is a natural corollary of the axiom of coordinate transformation, without logical loopholes caused by artificial ad-hoc construction.