We analyze, using the mathematical formulation of Merge within the Strong Minimalist Thesis framework, a set of linguistic phenomena, including head-to-head movement, phrasal affixes and syntactic cliticization, verb-particle alternation, and operator-variable phenomena. These are often regarded as problematic, as violations of the Extension Condition. We show that, in fact, all of these phenomena can be explained without involving any EC violation. We first show that derivations using Sideward Merge are possible for all of these cases: these respect EC, though they involve some amount of optimality violations, with respect to Resource Restrictions cost functions, andthe amount of violation differs among these cases. We show that all the cases that involve large optimality violations can be derived in alternative ways involving neither EC nor the use of SM. The main remaining case (head-to-head movement) only involves SM with minimal violations of optimality (near equilibrium fluctuations). We analyze explicitly also the cases of multiple wh-fronting, clusters of clitics in Romance languages and possessor agreement construction in Korean, and how an explanation of these phenomena based on SM can be made compatible with the colored operad generators for phases and theta roles. We also show that the EC condition has a clear algebraic meaning in the mathematical formulation of Merge and is therefore an intrinsic structural algebraic constraint of the model, rather than an additional assumption. We also show that the minimal optimality violating SM plays a structural role in the Markovian properties of Merge, and we compare different optimality conditions coming from Minimal Search and from Resource Restriction in terms of their effect on the dynamics of the Hopf algebra Markov chain, in a simple explicit example.

翻译：我们基于强最简论框架内合并操作的数学表述，分析了一系列语言现象，包括核心至核心移位、短语性词缀与句法附着化、动词-小品词交替以及算子-变量现象。这些现象常被视为问题性的，即对扩展条件的违反。我们证明，实际上所有这些现象均可在不涉及任何扩展条件违反的情况下得到解释。首先，我们表明侧向合并推导适用于所有这些情况：这些推导遵循扩展条件，尽管涉及一定程度的资源限制成本函数最优性违反，且违反程度在不同案例中有所差异。我们证明，所有涉及较大最优性违反的案例均可通过既不违反扩展条件也不使用侧向合并的替代方式推导。剩余的主要案例（核心至核心移位）仅涉及具有最小最优性违反（近平衡涨落）的侧向合并。我们亦明确分析了多重wh-前移、罗曼语中的附着词聚类以及韩语中的领属者一致结构，并阐释了基于侧向合并对这些现象的解释如何与相位及θ角色的着色运算元生成器相兼容。我们还证明，扩展条件在合并的数学表述中具有清晰的代数意义，因此是该模型固有的结构性代数约束，而非额外假设。此外，我们揭示了最小违反最优性的侧向合并在合并的马尔可夫性质中发挥结构性作用，并通过一个简单显式示例，比较了源自最小搜索与资源限制的不同最优性条件对霍普夫代数马尔可夫链动力学的影响。