The convergence argument required for the weighted-Euclidean Proposition 1 has been checked for a concrete iid joint model of voter weights and ideal points, for both response models. In the finite coordinate model, raw Model A maximizers exist pointwise by compactness and continuity; the Euclidean projection is constructed from C1; and the Definition 2 social utility is defined as the expected sampled utility and proved equivalent to negative population cost.
For the general finite weighted multi-block raw maximization problem, the formalization uses the Borel water-filling rule
[ s_k={d_k,a_k}, ]
where d_k is the current block-to-ideal distance,
a_k is the normalized nonnegative weight, and
\lambda is the maximum over the finitely many candidate
saturated-block multipliers. The response moves distance
s_k toward the ideal on each block. Finite maxima,
arithmetic operations, and square root make this rule Borel. Its proof
of exactness separates two cases: if uncapped positive weight remains,
the amounts exhaust the query’s squared radius and a finite
Cauchy–Schwarz argument proves optimality; otherwise all positive-weight
blocks are at their ideals and the sampled cost is zero.
Thus the concrete iid Definition 2 model has a canonical measurable exact Model A execution, rather than an abstract response-selection input. Applying the resulting Proposition 1 endpoint to a particular paper instance still requires the displayed C1/C2 geometry and ideal-support facts for that instance; this is ordinary source-model instantiation, not a tie-breaking or stochastic-convergence gap.
The Appendix C.6 event estimate used by Proposition 1 follows directly from C3, as explained below.
In the first part of Appendix C.6 Lemma 4, the exceptional event is
[ B(x,r)={(w,z): k, xk-zk_2<r}, ]
where k ranges over component coordinate blocks. The
printed proof then treats this as the full-vector event
\{\lVert x-z\rVert_2<r\}. With more than one block the
events are not equal: an ideal point can be close on one block while far
away on another.
The bound actually used in the subsequent convergence argument nevertheless holds. Every nonempty block has a coordinate, and
[ xk-zk_2<r |x_i-z_i|<r ik. ]
Consequently,
[ B(x,r){z:i, |x_i-z_i|<r}. ]
The C1/C2/C3 bounded-density argument bounds the right-hand
coordinate-slab event by C r. Since this uses only the
ideal-point marginal, it applies to the joint voter law without
requiring weights and ideals to be independent within a voter. Thus the
required exceptional-event probability remains linear in r,
which is sufficient for the summable descent-error estimate.
This is a minor correction to the printed lemma proof, not a caveat on the main-text Proposition 1.