We present a template for the Promise Constraint Satisfaction Problem (PCSP) which is NP-hard but does not satisfy the current state-of-the-art hardness condition [ACMTCT'21]. We introduce a new "injective" condition based on the smooth version of the layered PCP Theorem and use this new condition to confirm that the problem is indeed NP-hard. In the second part of the article, we establish a dichotomy for Boolean PCSPs defined by templates with polymorphisms in the set of linear threshold functions. The reasoning relies on the new injective condition.

翻译：我们提出了一种承诺约束满足问题（PCSP）的模板，该问题是NP难的，但不满足当前最先进的难度条件[ACMTCT'21]。我们基于分层PCP定理的光滑版本引入了一种新的"内射"条件，并利用这一新条件确认该问题确实是NP难的。在文章的第二部分，我们建立了由具有线性阈值函数集合中多态性的模板定义的布尔PCSP的二分性。这一推理依赖于新的内射条件。