| paper | d4-journals-mss-2021-0007 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | no |
Theorem 3.7 and the paper’s other named results concern representation size and structure, not a computational problem. The proposed two-bloc mass model changes what representation dimension means and supplies no computational task anchored to a result in the paper.
fails bit a — no named computational result to mirror
No result has a continuous computational mirror; the dimension and Boolean-dimension bounds and constructions remain unmirrored.
This paper contains no named computational-complexity result of the required kind, so I can’t give an honest anchored case for a continuous computational mirror.
The closest result is Theorem 3.7, proved in this paper: for every \(n\ge4\) divisible by \(4\), there exists a complete simple game on \(n\) voters with exactly two voter types whose dimension is at least
\[
A(n/2,4;n/4)\ge {n/2\choose n/4}\frac{2}{n}\ge \frac{4\cdot 2^{n/2}}{n^2}.
\]
That is an exponential lower bound on the representation dimension of certain games. It does not establish a complexity classification for a computational problem, such as finding the dimension or deciding whether it exceeds a threshold. The other named results likewise give representation bounds or constructions, rather than algorithmic complexity results.
A continuum version of the representation question might be worth posing separately, but it would not be a mirror anchored to a qualifying result in this paper.
The proponent raised no qualifying anchor, so there is no computational result to defeat. Theorem 3.7 is the closest candidate: it proves an exponential lower bound on the dimension of a complete simple game with two voter types. But dimension here is a representation measure—the minimum number of weighted games whose intersection represents the rule—not the worst-case complexity of a computational problem. The paper does not study an algorithm for finding that dimension, deciding whether it exceeds a threshold, or solving an intervention problem. The other results are likewise representation bounds and structural constructions.
The best attempt at a mirror is to take two nonatomic voter blocs and extend the family from Theorem 3.7 to mass vectors \((x,y)\), with winning condition \(x\ge 1/4\) or \(x+y\ge 1/2\). That does make the population continuous. But it changes what dimension measures. A weighted game with one weight per bloc is a halfspace in \((x,y)\); an intersection of such games is convex, while the extended winning region is not. To retain the paper’s intersection-dimension question, one would need to let weighted representations distinguish individuals inside each bloc. That restores the atomized coalition structure and the identity-level choices used by the coding and trading-transform argument. Allowing arbitrary weight functions on a nonatomic population instead creates a new function-space representation problem, with no computational question or encoding supplied by this paper.
There is no valid objection that the paper lacks a plausible high-multiplicity regime: its two-type construction is positive evidence for one. That makes a related continuous geometric study conceivable. The negative case is narrower, but stronger for ChoCo’s stated remit: the paper’s central results do not give a computational problem to continuize, and the most natural mass-level extension changes the representation question rather than mirroring it. I can defend excluding this paper as a target for the programme; I cannot honestly establish the universal claim that no worthwhile continuous question could grow from it.
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.