The Calcium-Lanthanum Singularity Theorem: Exhaustive Discovery and Multi-Tool Cross-Validation of the d-Block and f-Block Regime Switching Centers in the Periodic Table ——First Deployment of the Factor Hierarchy Law in Chemistry, Verification of the Statistical Signature of Regime Factors, and Confirmation of the Chemical χ-Blindness (I)

Abstract

This paper deploys the Factor Hierarchy Law in chemistry for the first time, conducting a presupposition-free regime audit of 118 elements by exhaustively testing 27 candidate factors, 9 explained variables, 7 transformations, and 10 structural break tools, totaling over 1,600 independent tests. The core findings unfold in seven layers. (1) Singularity regime centers: at Z=20 (calcium) and Z=57 (lanthanum), the Chow F under the inv transformation reaches 1.53×10²³ and 3.37×10²³, respectively. The f-block/d-block intensity ratio of 2.20 deviates only 2.2% from (3/2)²=2.25, revealing for the first time the quantitative law F(l) ∝ l² governing regime switching intensity. Ten tools independently validate the two singularity centers, and all Bai-Perron multi-break BIC improvements are positive. (2) Statistical signature of regime factors: for the first time, the statistical criterion distinguishing regime factors from executive factors is precisely defined—suppressed by multicollinearity under strict VIF constraints and breaking into the optimal combination when constraints are relaxed. The complete interaction effect matrix yields 14 significant results out of 27 tests (p<0.05), with zero false positives in synthetic data negative controls (0/4). (3) Boundary concentration law of interaction effects: interaction effects are highly concentrated on regime boundary lines. Four boundary element categories—noble gases, actinides, alkaline earth metals, and post-transition metals—all show significant interactions (p<0.02), whereas transition metals (n=35), the largest category within the transition zone, are not significant (p=0.0841), and lanthanides (n=15) within the f-block interior are equally silent (p=0.9884)—interaction effects concentrate on boundaries, not interiors. (4) Topological uniformity of intrinsic transition bandwidths: the regime transition bandwidths of four intrinsic properties are strictly unified at 3.0 periods (coefficient of variation=0), significantly different from condensed-phase properties (p=0.0202). (5) Chemical χ-blindness law: regime switching signals of d/f orbitals are completely silent under raw metrics (Chow F as low as 1.78) but are activated by inv/cube transformations, with amplification up to 6.8×10²⁰ times. (6) Chemical precursor cascade: the double breaks of first ionization energy at period=2.0 and 5.0 are independently validated by Bai-Perron (BIC improvement=56.1), and the triple breaks of electronegativity on the number axis (Z=54, 72, 86) mark the complete boundary of the f-block regime switch. (7) Absolute regime boundary and institutional boundary: at valence_electrons=2, ten tools yield a coefficient of variation of zero; regression discontinuity design confirms significant intercept jumps for five properties; the density–atomic mass correlation reverses completely between synthetic and natural elements (r=-0.965 vs r=+0.733, p=0.0000). All conclusions are supported by permutation tests (p=0.002), synthetic data negative controls (0/5 real + 0/4 synthetic), and cross-environment reproducibility verification (zero discrepancy across 13 columns on two independent computers). This paper completes the first full deployment of the Factor Hierarchy Law in chemistry, providing an indispensable empirical foundation for the construction of the Regime Periodic Table (RPT) in the second paper. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.