We present a consistent and complete description of the coupling to matter in the Teleparallel Equivalent to General Relativity (TEGR) theory built from a Cartan connection, as we proposed in previous works. A first theorem allows us to obtain parallel transport from the Cartan connection into a proper Ehresmann connection, while a second ensures to link the TEGR-Cartan connection to the Ehresmann one-form that contains the Levi-Civita connection. This yields a coupling to matter in agreement with observations and the Equivalence Principle. As the fundamental fields proceed from the Cartan connection, if one insists on interpreting TEGR as a gauge theory of translations, such translation gauge field can be extracted from the consistent theory presented. However, this would entail a fundamental change in the structures known for gauge theory and a split between gauge field and connection is imperative. The willingness to take such a step is left to the reader.

中文翻译：

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伸缩平行重力作为量规理论：使用Cartan连接的物质耦合

正如我们在先前的工作中所提出的，我们在从Cartan连接建立的Teleparallel等效广义相对论（TEGR）理论中给出了对物质耦合的一致而完整的描述。第一个定理使我们能够从Cartan连接获得并行传输到适当的Ehresmann连接，而第二个定理则确保将TEGR-Cartan连接与包含Levi-Civita连接的Ehresmann单形式链接。这产生了与观察和等效原理一致的物质耦合。由于基本领域是从卡丹关系出发的，因此如果人们坚持将TEGR解释为翻译规范理论，那么可以从提出的一致理论中提取这种翻译规范领域。然而，这将导致量规理论已知的结构发生根本变化，并且必须在量规场和连接之间进行拆分。采取这种步骤的意愿留给了读者。