Simpler Short Extenders Forcing - arbitrary gap

Our aim is to present a version of short extenders forcing which splits between cardinals above and below the critical cardinal and allows to blow up the power arbitrary higher. 1 The Main Preparation Forcing The definition of the forcing below follows the ideas of [1], [2], [3], [4]. We extend the method of [4] to gaps up to κ. The forcing adds a certain kind of a simplified morass with linear limits with a gap up to κ. The object added by a parallel forcing in [4] adds Velleman’s [5] simplified morass with linear limits of the gap 1. We do not know if there exists an analog of Velleman morass for higher gaps. A generic set for the forcing below probably may serve as such morass. Fix two cardinals κ and θ such that κ < θ and θ is regular. Definition 1.1 The set P ′ consists of all sequences of triples. 〈〈A0τ , A , C 〉 | τ ∈ s〉