Introduction and the corrected target

The supplied preprint Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four, attributed to OpenAI and dated 24 September 2026, states a finite-volume Gromov–Hausdorff–Prokhorov limit for every fixed $q>4$ OpenAI211. Its graph metric uses all edges, its measure is normalized degree measure, and its stated conclusion forgets the root and FK decoration. No uniformity as $q\downarrow 4$ is asserted.

The proposed extension asks for an operational certificate built from its block-tree machinery while refusing gravitational interpretations not justified by that machinery. This paper completes a mathematically specified version of that extension. It does not independently prove the supplied FK scaling-limit claim.

The probabilistic ingredients have established antecedents: conditioned-tree coding limits BroutinMarckert,Janson, size-biased spine methods LyonsPemantlePeres, and map block decompositions AddarioBerry. The supplied preprint also contains the two-exploration-order truncation argument motivating our offspring-bounded path theorem. We include the local argument with attribution rather than claim a new general branching-process principle.

The operational contribution is the precise choice of channels, their error normalization, explicit finite-size bounds, and the separation of four types of probability or approximation:

Cheung's supplied manuscripts emphasize comparable distinctions between access, complete residuals, deterministic return, destructive readout, and physical identification CheungI,CheungHawk04A,CheungHawk04B, CheungEn07A,CheungEn07B,CheungEn07C. Every finite channel statement used below is stated and proved for the interfaces printed here.