We determine the full nim-value structure of additive subtraction games in the {\em primitive quadratic} regime. The problem appears in Winning Ways by Berlekamp et al. in 1982; it includes a closed formula, involving Beatty-type {\em bracket expressions} on rational moduli, for determining the {\mathscr P}-positions, but to the best of our knowledge, a complete proof of this claim has not yet appeared in the literature; Miklós and Post (2024) established outcome-periodicity, but without reference to that closed formula. The {\em primitive quadratic} case captures the source of the quadratic complexity of the problem, a claim supported by recent research in the dual setting of sink subtraction by Bhagat et al. This study focuses on a number theoretic solution involving the classical closed formula, and we establish that each nim-value sequence resides on a linear shift of the classical {\mathscr P}-positions.

翻译：我们确定了原始二次情况下加法减法游戏的完整尼姆值结构。该问题出现在1982年Berlekamp等人的《胜负之道》中，书中包含一个涉及有理模数上的比蒂型括号表达式的闭式公式，用于判定P-位置，但据我们所知，该论断的完整证明至今尚未在文献中出现；Miklós与Post（2024）建立了结果周期性，但未涉及该闭式公式。原始二次情形揭示了问题二次复杂性的根源，这一论断得到了Bhagat等人在汇减法对偶设定中最新研究的支持。本研究聚焦于涉及经典闭式公式的数论解，并证明每个尼姆值序列均位于经典P-位置的线性偏移上。