Source clarifications and corrections

Source: the official NeurIPS 2024 paper.

Post-Theorem 3.7 proof-location sentence

• The paragraph attributes a linearly infeasible pairwise-majority-consistent (PMC) ranking to Appendix A.6 → that example belongs to Appendix B. Appendix A.6’s strict-majority relation has a directed cycle among its positive-axis candidates and a reverse cycle among its negative-axis candidates, so no transitive PMC ranking exists there.

• The formalization therefore records the source sentence as a proof-location error. It proves the actual Appendix-A.6 fact that no PMC ranking exists, and it proves the Appendix-B infeasibility example under its own Appendix-B source owner. Theorem 3.7 itself is unchanged.

Lemmas 3.4–3.5: the perturbation seam

• Lemma 3.4’s weak zero-perturbation inclusion → a strict cone for every minimizer. Here OPT(0) is the zero-perturbation minimizer set and r the candidate score.

• At epsilon = 0, each copied candidate used in the proof has the same feature vector as its original. The paper defines the region R_{c' over c} by the weak comparison r(c') >= r(c). Consequently, the printed Lemma 3.4 inclusion OPT(0) subset R_{c' over c} is true for every parameter and supplies no strict margin from the boundary. It therefore cannot by itself justify the open-set stability step printed in Lemma 3.5.

• The source’s coordinate bounds give theta_1>A3>0 and theta_2<A4 for every core minimizer theta. Choosing delta>0 with delta A4<A3 yields theta_1>0 and delta theta_2<theta_1, ranking each original above its copy for every positive perturbation.

• Lemma 3.5’s open-set inference → a strict infimum gap on the closed half-space where the copy weakly outranks its original, for all sufficiently small epsilon>0. Coercivity, uniform continuity on a compact set, and an exterior lower bound establish the gap. Theorem 3.1’s hypotheses and impossibility conclusion do not change.

Theorem 3.1 positive-input branch: restricted-infimum sign

• inf_{y <= 0} loss(y) < inf_y loss(y) → inf_{y <= 0} loss(y) > inf_y loss(y). The preceding proof gives loss at least loss(0) on nonpositive inputs and a strictly better positive input. “Lower bounded by loss(0) from above” correspondingly means “bounded below by loss(0).”

Appendix A.1: one-sided derivatives

• Ordinary derivative data → one-sided convex derivatives for the four-point argument supporting Lemma 3.2; ordinary differentiability at the selected points is unnecessary.
• The later abbreviation -2w,-w,w,2w → the originally selected points -w,-w/2,w/2,w, which the displayed threshold calculation uses.

Six-candidate feasibility witness

• Footnote 7’s parameter (1,1) → (delta,2) for the ranking a>a'>b>b'>c'>c. With the printed feature x_{c'}=(-epsilon,delta epsilon) and 0<delta<1, (1,1) gives c>c'; the replacement gives the required strict order on the source construction’s parameter domain.

Rule-level tie conventions

• The LCPO, Copeland, and leximax-plurality correspondences → selectors with profile-independent candidate or ranking tie keys in the checked rule-level results. A single-valued selector need not admit such a key; necessity of this restriction for the broader claims is not established. LCPO means leximax Copeland subject to Pareto optimality.
• Theorem C.5’s “without loss of generality” first-profile output → an explicit premise that the Pareto-Kemeny rule returns the source ranking v1 there. The reduction from every selector to that local case remains a formalization gap; necessity of the local choice is unknown.