AI 中文总结
研究PCA双标图解释的特定声明可接受性，提出以科学声明为评估单位，当满足目标对齐、谱可识别性和投影充分性等条件声明才合理，给出相关诊断方法和真值场景，为PCA双标图声明的合理性判断提供正式程序。
AI 中文摘要
在主成分分析（PCA）之前通常会常规应用单位方差标准化，这用相关几何取代了协方差几何。由此产生的双标图可能计算正确，但无法支持对原始尺度现象的科学解释。因此，我们将科学声明而非分解本身作为方法评估的单位。一个PCA双标图声明只有在声明的科学目标证明所分析的算子合理、所调用的轴或不变子空间在规定水平上可识别、并且显示的投影在实质合理的容差内保持预先指定的关系时，才在表示上是合理的；任何一个条件不满足都会使该声明对于所述解释不合理。一种投影器公式为重复特征值块产生基不变诊断，并且残差 - 格拉姆界量化来自省略坐标的成对误差。六个受控总体场景提供了精确的表示真值，包括零目标与任意投影角度的关联、全空间60度关系坍缩到0度以及不可识别的命名轴。其贡献并非再次提醒缩放很重要：它确立了计算正确性对于科学可解释性是必要但不充分的，并提供了一个正式程序来保留、重新表述、限定或拒绝一个PCA双标图声明。补充材料中提供了一个实际数据示例和自助扩展。
英文摘要
Unit-variance standardisation is often applied routinely before principal component analysis (PCA), although it replaces covariance geometry by correlation geometry. A resulting biplot may be computed correctly yet fail to support a scientific interpretation about the original-scale phenomenon. We therefore make the scientific statement, rather than the decomposition alone, the unit of methodological assessment. A PCA-biplot claim is representationally well posed only when the declared scientific target justifies the operator analysed, the invoked axis or invariant subspace is identifiable at the stated level, and the displayed projection preserves the prespecified relationships within a substantively justified tolerance; failure of any condition makes the claim ill posed for the stated interpretation. A projector formulation yields basis-invariant diagnoses for repeated-eigenvalue blocks, and a residual-Gram bound quantifies pairwise error from omitted coordinates. Six controlled population scenarios provide exact representational truth, including zero target association with arbitrary projected angles, collapse of a full-space 60-degree relationship to 0 degrees, and non-identifiable named axes. The contribution is not another reminder that scaling matters: it establishes that computational correctness is necessary but not sufficient for scientific interpretability and provides a formal procedure for retaining, reformulating, qualifying, or rejecting a PCA-biplot statement. A real-data illustration and bootstrap extension are provided as supplementary material.
Comments19 pages, 3 figures, 3 tables. Supplementary materials and R/Python scripts are included as ancillary files. Complete reproducibility package: https://doi.org/10.5281/zenodo.21361296
