We identify a premouse inner model $L[\mathbb{E}]$, such that for any coarsely iterable background universe R modelling $\mathsf{ZFC}$, $L{[\mathbb{E}]}^R$ is a proper class premouse of R inheriting all strong and Woodin cardinals from R. For each ordinal α, $L{[\mathbb{E}]}^R|\alpha$ is $(\omega ,\alpha )$-iterable, via iteration trees which lift to coarse iteration trees on R.

We prove that $(k+1)$-condensation follows from $(k+1)$-solidity and $(k,{\omega }_1+1)$-iterability (that is, roughly, iterability with respect to normal trees). We also prove that a slight weakening of $(k+1)$-condensation follows from $(k,{\omega }_1+1)$-iterability (without the $(k+1)$-solidity hypothesis).

The results depend on the theory of generalizations of bicephali, which we also develop.

我们确定了鼠标前的内部模型 $\mathrm{大号}[\mathbb{Ë}]$，以便对于任何粗略可迭代的背景Universe R建模$\mathsf{零碳燃料}$， $\mathrm{大号}{[\mathbb{Ë}]}^{\mathrm{[R}}$是的正确类premouse [R继承了所有强大和伍丁红衣主教[R 。对于每个序数α，$\mathrm{大号}{[\mathbb{Ë}]}^{\mathrm{[R}}|\alpha$ 是 $（\omega ，\alpha ）$-可迭代，通过迭代树提升到R上的粗略迭代树。

我们证明 $（ķ+\mathrm{1个}）$-冷凝从 $（ķ+\mathrm{1个}）$-坚固性和 $（ķ，{\omega }_{\mathrm{1个}}+\mathrm{1个}）$-可迭代性（即，相对于普通树的可迭代性）。我们还证明，$（ķ+\mathrm{1个}）$-冷凝从 $（ķ，{\omega }_{\mathrm{1个}}+\mathrm{1个}）$可迭代性（无 $（ķ+\mathrm{1个}）$-实体假设）。

结果取决于我们也发展的双头概化理论。