| paper | d4-journals-geb-2026-0034 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | no |
Corollary 1 is an existence theorem, and no other named result asserts an algorithm, complexity bound, or hardness result. The support-based consensus is a sensible population description, but studying its computation would add a computational question absent from the anchored result, so bit (a) fails.
fails bit a — no named computational result to mirror
No named result is covered by a computational mirror; the population consensus \(C(\mu)\) restates a collective-choice object, while the preference-lattice characterizations and other comparative-statics results remain outside any computational mirror.
The best positive opening is §3’s collective-choice setting. Imagine many residents choosing among a finite, ordered set of public policies, with \(N\) residents but only \(\tau\) distinct complete preference rankings. Let \(\mu_t\) be the share with ranking \(t\). A direct population version of the paper’s consensus is
\[
C(\mu)=\bigcap_{t:\mu_t>0}X(t),
\]
the policies that every positive-mass type ranks best.
The closest named result is Corollary 1, proved in Appendix G: when \(X\subseteq\mathbb R\) is compact, there exists a monotone social choice function that respects unanimity. That makes the large-population scenario plausible, but Corollary 1 is an existence result about a social choice function, not a complexity result. Its population version also depends only on which types have positive mass; changing their shares without changing the support leaves \(C(\mu)\) unchanged. A mass-sensitive variant, such as choosing the policy with the greatest share of top-ranking supporters, would be a new problem rather than a direct mirror.
So I cannot supply a valid anchor or a continuous problem per anchor under the requested criteria. The paper contains no named result asserting that a problem is in P, NP-hard, FPT, or otherwise computationally classified. Its numbered results concern comparative statics and preference characterizations. The preference set in §4 represents uncertainty over preferences; it is not a distribution of population types. The paper’s continuous mathematics therefore does not itself establish a computational question about a continuous society.
The weakest point is decisive: even the most promising collective-choice connection offers no computational result to continueize, and the direct consensus mirror discards the mass information that defines the continuous population model. A complexity study of computing consensus for supported preference types could be a reasonable extension, but it would be a new research question, not an honest anchor-based case for this paper.
Corollary 1 is the only plausible anchor, and it cannot support a computational mirror. It asserts that a monotone social choice function respecting unanimity exists on the paper’s domain of finite preference profiles. It does not give an algorithm for a specified input or select a particular function to compute. The setting also allows arbitrary compact \(X\) and arbitrary preferences, with no finite encoding or complexity parameters. A continuous-population version would therefore need a computational problem added to the result.
The strongest population reading is to let \(t\) range over complete preference rankings of a finite policy set and give each type mass \(\mu_t\). Then the paper’s consensus becomes
\[
C(\mu)=\bigcap_{t:\,\mu_t>0}X(t).
\]
This is a sensible population description, but the result factors through the support: changing the masses while keeping the positive-mass types fixed changes nothing in Proposition 1 or Corollary 1. That alone does not invalidate a well-posed question. The obstacle is that the paper asks no question about those masses, and its existence theorem supplies no distinguished social choice rule whose computation or manipulation could be studied.
A more ambitious proposal would study how much mass must move to change consensus. But breaking unanimity can take arbitrarily little mass: if a reachable type ranks the consensus option below another option, assigning it any positive mass removes that option from consensus. The infimum transfer cost is then zero. Requiring a target option to become unanimous may instead require removing all mass of every type that opposes it, but that introduces a transfer objective and costs absent from the paper. A mass threshold or a particular selection rule could make a useful new control problem; neither is a mirror of Corollary 1.
The uncertainty results do not supply another anchor. In §3.3, \(P\) is a set of preferences the analyst considers possible, and in §4 the maxmin criterion takes the worst preference in a set. Neither assigns population frequencies. Reading those sets as distributions would continuize uncertainty rather than the population; interpreting them as populations still leaves the paper’s results support-based.
So the proponent’s concession is the strongest point: the paper contains substantial order-theoretic results, but no named computational result to continuize. The honest limit of this negative case is that one could build worthwhile algorithmic or control questions from the paper’s objects. They would be new computational projects, not continuous mirrors of its results.
The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.