Abstract

We develop an operational marking theory for critical plane Galton–Watson trees conditioned on their total size, assuming only finite offspring variance. Fixed scalar labels bounded by offspring number satisfy a uniform ancestral-path law of large numbers. For finite-dimensional channel marks, we define a genuine path-averaged channel and prove convergence in half-diamond distance to its size-biased spine average, uniformly over paths of macroscopic depth. Explicit finite-size probability bounds retain the cost of size conditioning and an exactly defined height tail.

A second construction gives an actual serial channel: offspring-bounded nonnegative labels determine weak depolarizing steps with strength $N^{-1/2}$. Their compositions admit a uniform half-diamond comparison with a depth-dependent channel, and their preorder error profiles converge to an explicit functional of Brownian excursion. These are designed finite operational models, not dynamics inferred from a metric scaling limit.

We prove a three-correlation qubit certificate with separate tree, sampling, calibration, and implementation allowances. We also give a conditional transfer to weighted planar-map block decompositions, including the exact corner-to-degree sampling identity and a local proof of the endpoint criterion for finite offspring variance.

The supplied OpenAI preprint of 24 September 2026 is recorded as the source of a claimed $q>4$ Fortuin–Kasteleyn map limit, not used as an unverified theorem premise. Its application to the present framework requires the printed analytic and model-identification conditions. Counterexamples show that metric data, selected entropy, and path-label concentration do not determine quantum recovery, entanglement, or a radiation history. In particular, the Brownian CRT has Hausdorff dimension two, not one; its tree structure does not make it a two-dimensional manifold or a gravitational surface. No spacetime emergence, holographic reconstruction, Page curve, or characteristic quantum coefficient is constructed.