| paper | d4-conf-sagt-2020-0021 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
The opponent’s point is decisive under the stated gate: Theorems 1.1 and 1.2 analyze expected growth, while Theorem 2.1 gives a geometric characterization; none asserts a computation or hardness result. A cohort-mass version of the growth process is statable and plausibly recognizable, but unanimity depends only on support, so mass rescales admission counts without changing the transitions. The paper therefore has no named computational result to anchor a green mirror.
fails bit a — no named computational result to mirror
The cohort extension would reinterpret the expected-growth bounds in Theorems 1.1 and 1.2 as bounds on expected mass, scaled by cohort size; Theorem 2.1 still describes support geometry. These results remain analytic rather than computational.
The strongest honest positive case is that consensus voting has a natural high-multiplicity spatial model. A voter’s location \(x\in K\) is a complete type: it determines how that voter compares every pair of candidate locations. A society is a measure \(\mu\) over locations, with \(\mu(A)\) the fraction of members in region \(A\). With two candidates, unanimous acceptance means the support of \(\mu\) lies in one candidate’s Voronoi cell; equivalently, its convex hull lies there, as in Theorem 2.1.
One plausible regime is a large membership organization that recruits in cohorts of many near-identical members from a relatively small set of locations or spatial preference types. Then the number of people can greatly exceed the number of types, and a mass-based society is sensible. A corresponding problem would be Continuous Consensus Growth: given an initial rational measure \(\mu_0\) on spatial types, a candidate-location distribution, a cohort mass \(q\), and a horizon \(T\), determine the expected total membership mass after \(T\) rounds. Each round draws two candidate locations; admit one only if all mass in the current society prefers it, then add mass \(q\) of its type. A solution could be an exact expression or an approximation to the expectation with a stated error guarantee.
That is a credible continuous-population question, but it is not an anchor-backed mirror of a named computational result in this paper. The paper contains no named computational result of the kind required by the prompt: none of its numbered theorems, lemmas, corollaries, or propositions establishes a complexity classification or an algorithm for a computational problem. Theorems 1.1 and 1.2 are proved here and bound expected group size—respectively, between \(\ln T\) and \(\ln T\cdot\ln\ln T\) on the unit square, and by \(\Theta(T^{1/8})\) on the unit ball. Theorem 2.1 is a geometric characterization of unanimous winner determination. These are substantial mathematical results, but they do not meet the stated anchor requirement. I therefore cannot honestly name a continuous problem “per anchor” or assign the required tractable/hardness-transfer/continuum-specific classification.
The soft spot in the positive case is also substantive: the paper counts accepted individuals, while the cohort model changes each admission into a positive mass of members. And if the continuous society has support throughout the whole domain, unanimity can prevent any admission. So the model is plausible for a cohort-based regime, but its connection to the paper’s growth bounds is limited. The positive case is a promising adjacent question; it does not satisfy the requested anchor-based case.
The proponent supplies no qualifying anchor: Theorems 1.1 and 1.2 bound expected group size, and Theorem 2.1 characterizes when consensus selects a candidate. None is a complexity result or an algorithm for a computational problem. Calling “Continuous Consensus Growth” a mirror does not fill that gap; it proposes a new stochastic-growth question.
The strongest version of that proposal still runs into the mechanism’s dependence on support. By Theorem 2.1, consensus requires \(\operatorname{supp}(\mu)\) to lie in one candidate’s Voronoi cell. The masses do not enter. If each accepted applicant brings a cohort of mass \(q>0\) at that applicant’s location \(x\), the update is \(\mu'=\mu+q\delta_x\). Its support changes exactly as the original group changes, whatever \(q\) is. The expected cohort mass is just \(q\) times the expected number of admissions, plus the initial mass. That is a rescaling of the paper’s process, not a population relaxation whose mass proportions shape the dynamics.
Making the population diffuse gives a different failure mode. If its support is all of \(K\), a generic pair of candidates has no Voronoi cell containing that support, so unanimity admits nobody. A small positive mass at a distant location can also veto a candidate just as completely as a large mass. Making mass affect the vote would require relaxing unanimity, for example to a weighted threshold; that changes the mechanism behind the paper’s geometric characterization and growth bounds.
One could study the complexity of expected growth for a finite spatial support and a finite candidate distribution. That may be worthwhile in its own right, but its state is the support geometry; the type masses still do not affect transitions. I cannot rule out a useful continuous-seed or stochastic-computation project in every possible regime. The negative case is strongest and credible for the proposed mirror: the paper’s results are analytic, and its unanimity rule makes population mass a poor continuous variable for the dynamics they analyze.
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.