| paper | d4-journals-scw-2001-0036 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | no |
The paper reports structural facts about whether particular QMV rules can be represented by a single weighted rule, plus voting-power calculations and evaluations. It contains no numbered result asserting an algorithm, hardness, or other computational claim that could anchor a continuous-population question.
fails bit a — no named computational result to mirror
No result is covered: the structural findings about the \(N_{15}\), \(N_{27}\), and \(N'_{27}\) rules and the voting-power evaluations lack a qualifying computational anchor.
This paper supplies no qualifying anchor for a positive continuization case. It contains no numbered Theorem, Lemma, Corollary, or Proposition asserting a computational-complexity result such as tractability or hardness. Its calculations of voting power are reported in tables, using external software.
The paper does prove structural facts: §2.2 shows that \(N15\) cannot be represented by a single weighted rule; §3.2 shows that \(N27\) can; and §4.1 shows that \(N'27\) cannot. But these are not named computational-complexity results of the kind your anchor rule requires. The cited Lemma 3.3.12 is from another work, not this paper.
There is also a substantive obstacle to turning those findings into a convincing population mirror. The coalitions and pivotality the paper studies are among 15 or 27 member-state representatives. Citizens’ population sizes enter the fairness analysis as constituency sizes; they are not the agents whose yes/no choices form the winning coalitions. A model with many repeated-type jurisdictions might be worth studying, but it would be a new problem, not a continuous version anchored in this paper’s named results. I would not propose it as one of the required mirror questions.
The appended case offers no affirmative anchor to defeat. It argues the negative itself: the paper has no theorem, lemma, corollary, or proposition about computational complexity. Its strongest results are structural—whether these particular rules can be represented by a single weighted rule—and its power-index calculations concern fixed councils of 15 or 27 members. There is no complexity problem, algorithm, or hardness result here for a continuous mirror to classify.
The best rescue would be to study councils with many jurisdictions sharing complete types, where a type includes its voting weight and population. One could ask how Penrose or Banzhaf power, majority deficit, or quota design behaves as those type counts grow. That could be a sensible problem for some federations. But it is not anchored in this paper: its coalition members are state representatives, while citizens’ population sizes enter as constituency data and a fairness proxy. Continuizing citizens leaves the council’s coalition game finite; replacing the council with a continuum of jurisdictions changes the institution. Individual pivotality then needs a limiting normalization or interpretation of its own. Those choices might support a new asymptotic voting-power project, but they do not turn this paper’s fixed-instance evaluation into a computational result about continuous societies.
So the negative case is strong for this paper as a ChoCo mirror: there is no named computational result to mirror, and the plausible extension asks a different question. The universal claim is weaker. I cannot rule out a worthwhile continuous study of large, repeated-type councils; I would classify that as a new project rather than a mirror of this paper.
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.