Answer: No admissible cost distribution exists when the finite list specifies the distribution exactly and \(c_{\max}>0\).
Lemma 3.5 — class: algorithmic
Given a continuum worker pool with task-specific effort costs distributed according to μ on \([0,c_{\max}]\), \(N\), \(b\), \(P_L\), and \(P_H\), and an evaluation oracle for \(F\), choose a bonus \(B ≥ 0\) maximizing the paper's relaxed PA requester objective \((PB)\). Each task draws \(N\) workers from the pool; workers use the paper's threshold effort strategy and the objective uses its stated PA cutoff equation and majority approximation. Assume the PA conditions of Lemma 3.5: \(f\) is twice differentiable with \(f''(c) ≥ 0\), and \(cF(c)\) is convex.
No faithful, nonempty proof task satisfies the supplied requirements. A society represented by a list of \(\tau\) cost types with positive binary-rational masses has an atomic cost distribution; it cannot also have a twice-differentiable density \(f\) or a strictly concave CDF \(F\) on \([0,c_{\max}]\) with \(c_{\max}>0\).
The known-cost model already draws independent worker costs from a continuous distribution. Its relaxed PA objective is claimed concave under the stated density and convexity conditions; the published version retains these assumptions and this result. [Primary source](https://www.ijcai.org/Proceedings/16/Papers/532.pdf)
Consequently, obtaining the requested complexity question requires changing either the mandatory support representation or the smooth-distribution assumptions.
Interpretation and conclusion. The incompatibility is correct if the finite list specifies the cost distribution exactly: each listed mass is the probability of its associated cost. A finite list of realized costs sampled from a continuous distribution would have a different meaning.
Under the exact distribution interpretation, the admissible family is empty. This settles the consistency issue; it does not establish a complexity classification for a revised model.
Proof of incompatibility. Let \(C=c_{\max}>0\), let \(\tau\geq 1\) be finite, and suppose the distribution is specified by costs \(c_i\in[0,C]\) and masses \(p_i\in\mathbb{Q}_{>0}\), with \(\sum_{i=1}^{\tau}p_i=1\). Its probability measure and CDF are
\( \mu=\sum_{i=1}^{\tau}p_i\delta_{c_i} \)
and
\( F(x)=\sum_{i:c_i\leq x}p_i \).
Here \(\delta_c\) denotes the probability measure concentrated at \(c\). Repeated costs cause no difficulty: their masses add.
First, this distribution has no density with respect to Lebesgue measure. Indeed, if an integrable density \(f\) existed, then for every Borel set \(A\) we would have \(\mu(A)=\int_A f(x)\,dx\). Taking \(A=\{c_i\}\) gives
\( \mu(\{c_i\})=\sum_{j:c_j=c_i}p_j>0 \),
whereas \( \int_{\{c_i\}}f(x)\,dx=0 \), because a singleton has Lebesgue measure zero. This is a contradiction. In particular, a twice-differentiable density cannot exist.
Second, the CDF cannot be strictly concave on \([0,C]\). Strict concavity would require, for distinct \(x,y\in[0,C]\) and every \(t\in(0,1)\),
\( F(tx+(1-t)y)>tF(x)+(1-t)F(y) \).
Because the support is finite and \(C>0\), there exists a nonempty open interval inside \((0,C)\) containing no support point. Choose distinct \(x,y\) in that interval. The function \(F\) is constant between them, so
\( F(tx+(1-t)y)=F(x)=F(y)=tF(x)+(1-t)F(y) \).
This contradicts strict concavity. The argument also covers distributions whose atoms occur only at the endpoints.
Thus either smoothness requirement alone rules out the specified finite support distribution. Rational encoding and positivity beyond the existence of an atom are unnecessary for the density contradiction.
What the cited paper establishes. Section 2.2 introduces a cost distribution with a PDF and bounded support, and Assumption 2.1 requires its CDF to be strictly concave. Section 3 treats the distribution as known to the requester. Section 3.2 introduces approximations to the optimization objective, and Lemma 3.5 states concavity for PA under conditions (i) and (ii). Accordingly, that lemma concerns the relaxed objective under its stated assumptions; it does not supply a concavity theorem for a finite support replacement. [Published paper, Sections 2.2 and 3.2](https://www.ijcai.org/Proceedings/16/Papers/532.pdf)
The smooth assumptions themselves are consistent. For example, on \([0,C]\), define
\( f(c)=\frac{3}{2C}-\frac{c}{C^2} \)
and
\( F(c)=\frac{3c}{2C}-\frac{c^2}{2C^2} \).
The density is positive, integrates to one, and is twice differentiable. Moreover,
\( F''(c)=-\frac{1}{C^2}<0 \),
so \(F\) is strictly concave. Also,
\( f''(c)=0 \)
and
\( \frac{d^2}{dc^2}\bigl(cF(c)\bigr)=\frac{3(C-c)}{C^2}\geq 0 \),
so \(cF(c)\) is convex.
If \(C\) is rational, these formulas even have a finite rational description. Consequently, finite *description* is compatible with the smooth assumptions; finite *support* is the obstruction.
A meaningful complexity question therefore requires a revised specification: either permit an encoded continuous distribution, or replace the smooth assumptions with conditions appropriate to atomic distributions. No complexity classification for either revision follows from this incompatibility proof.
⟨structured verdict parsed — rendered above⟩
The prover correctly resolves the consistency question in USER_TASK: a nonempty, finitely supported probability distribution cannot have a Lebesgue density or a strictly concave CDF on a nondegenerate interval. No major or minor mathematical issues were found.
The argument does not establish a complexity classification for a revised continuous model, and the prover explicitly acknowledges this limitation.
USER_TASK. The assumptions \(C>0\), finite \(\tau\geq 1\), \(c_i\in[0,C]\), \(p_i>0\), and \(\sum_i p_i=1\) define a nonempty probability distribution. Distinguishing this representation from a finite sample of a continuous distribution is correct.\[
\mu=\sum_{i=1}^{\tau}p_i\delta_{c_i},
\qquad
F(x)=\sum_{i:c_i\leq x}p_i
\]
are correct. Repeated costs are handled correctly by adding their masses.
\[
\mu(\{c_i\})=\sum_{j:c_j=c_i}p_j>0.
\]
This gives the stated contradiction. The argument applies equally to interior atoms and endpoint atoms. It rules out any Lebesgue density, so it also rules out a twice-differentiable density.
\[
F(tx+(1-t)y)=tF(x)+(1-t)F(y).
\]
This contradicts the strict inequality required by strict concavity. The proof covers singleton support, repeated costs, and support concentrated at the endpoints. No assumption about continuity of the CDF is needed.
\[
F''(c)=-\frac{1}{C^2},\qquad
f''(c)=0,\qquad
(cF(c))''=\frac{3(C-c)}{C^2}\geq 0.
\]
Thus the claims of strict concavity and convexity hold. For rational \(C>0\), the displayed coefficients have finite rational descriptions. This establishes the claimed distinction between finite description and finite support.
No repair is required. The established result is an incompatibility of the supplied requirements. The complexity of either proposed revised specification remains unclassified.
⟨structured verdict parsed — rendered above⟩
Tier: machine-proved, adversarially reviewed by an independent model. No human has checked this proof. Generated from pipeline artifacts by tools/atlas_build.py.