Four distinctions that the plan must preserve

common law is not equality of the two limiting functions. The external quench manuscript asserts convergence of a pair $(A_{n,J},B_{n,J})$ to a pair $(A_g,B_g)$, with one common pair law for the two disorders. It does not assert $B_g(s,t)=A_g(t)$ at every finite waiting time. Its relaxation assertion concerns $s\to \infty$ OAI227Quench (Section 1, thm:main).

{Convergence in $C_{\mathrm{loc}}$ is convergence in law.} Uniform-on-compacts is the topology of the random-function space in the external claim. Without a specified coupling, it is not a pathwise supremum comparison between a Gaussian sample and a Rademacher sample.

proposed limit has an implicit representation. The inspected source writes $B_g$ as a series involving an edge measure, its semigroup, and an entrance density. Thus “the theorem does not write $B_g$” is too strong. The correct distinction is between such a representation and an evaluated formula, a convergence rate, or a finite certificate OAI227Background,OAI227Entrance.

finite certificate does not verify the infinite theorem. Producing the requested finite comparison neither proves nor automatically opens verification of the external process limit. A necessary milestone must not be read as a sufficient verification criterion.