We prove the True-KL$_0$ property for a parametric family of heterogeneous scoring rules arising in scored elicitation mechanisms (AI oversight, forecasting competitions, expert surveys). A $d$-dimensional agent with private type $M>1$ reports to a principal who evaluates via a power-$p$ pseudospherical scoring rule, $p \in (d,d+1)$; $M$ captures the agent's information quality relative to a reference. An exact formula $G(M,M') = -R(M,p,d) U(M|M)$ shows DSIC unconditionally: honest reporting maximises expected score for every $M>1$, without distributional assumptions. True-KL$_0$, the property $R(M,p,d)<1$ for all $M>1$, $d \in \{2,3,4\}$, $p \in (d,d+1)$, gives an explicit gain-magnitude bound: the best misreport is always worse than the honest score itself. Two structural tools drive the proof: (i) a substitution $y=(x+1)/(x-1)$ rewrites the loss integral $I_L$ as $\int_1^M F(y)(M^2-y^2)^{d/2} dy$ with $M$-independent weight $F(y)>0$, isolating all $M$-dependence in a single convex factor; (ii) Prekopa's theorem on log-concavity preservation establishes that $I_L$ is log-concave in $M$, the key step in the unimodality proof for $R$. For $d=2$ the log-concavity proof is fully algebraic. For $d \in \{3,4\}$ the Prekopa argument (analytic, covering $M \le M_{cut}(d,p) \le 20$) combines with a certified high-precision numerical step on the residual region $M \in [M_{cut}, 20]$, closed by a large-$M$ asymptotic for $M>20$. We also characterise the dimensional boundary: True-KL$_0$ holds unconditionally for all $p \in (d,d+1)$ when $d \le 4$, but fails above a critical threshold $p_{crit}(d) \in (d,d+1)$ for $d \ge 5$; for $d=5$ we locate $p_{crit}(5) \in (5.5718, 5.5750)$ via high-precision mpmath evaluation (half-width 0.0016, not interval-certified).

翻译：我们证明了在记分启发机制（人工智能监督、预测竞赛、专家调查）中产生的一类参数化异质评分规则具有True-KL₀性质。一个具有私有类型$M>1$的$d$维智能体向主理人报告，主理人通过幂指数为$p \in (d,d+1)$的伪球面评分规则进行评估；$M$表示智能体相对于参考信息质量的信息质量。精确公式$G(M,M') = -R(M,p,d) U(M|M)$表明无条件下满足DSIC：对于所有$M>1$，诚实报告使期望得分最大化，无需分布假设。True-KL₀性质（即对所有$M>1$、$d \in \{2,3,4\}$和$p \in (d,d+1)$有$R(M,p,d)<1$）给出了明确的增益幅度上界：最优虚假报告始终劣于诚实得分本身。证明依赖两个结构工具：(i) 代入$y=(x+1)/(x-1)$将损失积分$I_L$改写为$\int_1^M F(y)(M^2-y^2)^{d/2} dy$，其中权重$F(y)>0$与$M$无关，将所有$M$依赖性隔离到单个凸因子中；(ii) Prekopa对数凹性保持定理确立$I_L$关于$M$的对数凹性，这是证明$R$单峰性的关键步骤。对于$d=2$，对数凹性证明完全是代数的。对于$d \in \{3,4\}$，Prekopa论证（解析方法，覆盖$M \le M_{cut}(d,p) \le 20$）结合了残差区域$M \in [M_{cut}, 20]$上的经过认证的高精度数值步骤，并通过$M>20$的大$M$渐近性给出最终结果。我们还刻画了维数边界：当$d \le 4$时，True-KL₀对所有$p \in (d,d+1)$无条件成立；但当$d \ge 5$时，在临界阈值$p_{crit}(d) \in (d,d+1)$以上失效；对于$d=5$，我们通过高精度mpmath计算定位$p_{crit}(5) \in (5.5718, 5.5750)$（半宽0.0016，未经区间认证）。