The Linguistic Architecture of Mathematical Thought (Lemma): Demonstrative Injectivity and Referential Mapping in Japanese

Introducing the working concept of a "Language OS" to describe the foundational grammatical and lexical architecture of thought, this lemma redefines the Japanese demonstrative system (ko-so-a). Moving beyond traditional spatial-territory models, the study formalizes demonstratives through the mathematical lens of "injectivity" (one-to-one mapping) between speaker and listener: ko represents the speaker's request for the listener to establish an injective mapping; so denotes mutual injectivity already established between both parties; and a indicates a direct injective mapping into the listener's internal memory space. Through discourse analysis and comparisons with Indo-European demonstratives (noting the predominantly literary status of German jener), this lemma reinforces the reification framework introduced in Part 2 by clarifying how reified objects are uniquely indexed in shared mental space, thereby bridging everyday deixis and the analysis of mathematical symbolization and variable introduction in Part 3