发表机构
Immanuel Kant Federal University（伊曼努尔·康德联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究从纯向量空间的单个Q度量基元出发，无几何公理地推导勾股定理与平方项的代数起源，解释了量子玻恩定则，证明规范不变的唯一可解释度量为平方模且与数域无关。
AI 中文摘要
为何是平方？我们提出了一种无需几何公理的勾股定理及其核心平方项的推导方式，在纯粹的向量空间框架内确立了二者的代数起源。诸如（直）角、旋转、内积、正交性等概念也以逻辑建构的形式涌现，而非被当作预设前提。这些概念由平方项所必需，而由此产生的理论又典范地源自单个定义基元——向量的（正实数量化的）不变量Q度量。这为欧几里得几何提供了代数层面的核心依据。同样重要的是，这些发现还典范地解释了复模平方p=|𝔞|²——即量子玻恩定则（Born rule），并指出了哪些量具备被定量解释的资格。线性结构及其自同构具有刚性：明确定义的可解释量，在规范变换Q→常数×Q下，唯一的规范不变度量就是Q=||·||²；这一结论与数域是实数域ℝ还是复数域ℂ无关。
英文摘要
Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, $\textit{canonically}$ stems from a $\textit{single}$ definitional primitive $-$ the ($\mathbb R^{\vcenter{\hbox{$\scriptscriptstyle+$}}}\!$-quantitative) invariant $\mathcal Q$-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared $p = |\mathfrak a|^2$ $-$ the quantum Born rule $-$ and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the $\textit{well-defined}$ interpretable turns out to be, up to gauge $\mathcal Q {\,\to\,} \mathrm{const} {\,\vcenter{\hbox{$\scriptstyle\times$}}\,} \mathcal Q$, the unique gauge-invariant measure $\mathcal Q=|\hspace{-0.18em}| {\cdot}{\cdot}{\cdot} |\hspace{-0.18em}|^2$; independently of the field $\mathbb R$ or $\mathbb C$.
CommentsLaTeX 6 pages, no figures (v2: minor improvements of text)
