The Universal Law of Connection and the Universal Law of Contrast: Constitutional Structure, Mathematical Equivalence, and Physical Consequences

The Universal Law of Connection and the Universal Law of Contrast address physical existence throughout this universe, without exemptions by spatial scale, field, material composition, or physical configuration. Contrast states that physical existence is intrinsically differentiated: objective distinctions belong to its constitution rather than originating in observer-assigned names. Connection states that this differentiated constitution is one relationally connected whole: every division into two nonempty portions has a crossing constitutive relation. We give a precise one-present hyperrelational representation and establish an equivalence between universal multiscale Contrast (MC), connectedness (UC), the all-cut condition (CUT), and the absence of nonempty proper closed components (CLOSED), under the explicit component-coverage condition (CC). Connectedness is inherited by every complete, relation-preserving constituted scale; absolute constitutional isolation is excluded throughout the resulting structure. We then place the exact theorems alongside independently specified physical-domain constructions: nonfactorizable admissibility, source–environment constraints, grounded gravitational Green fields, joint gauge–matter reconstruction, symmetry-compatible differentiation, Standard Model consistency conditions, and nonadditive interaction operators. A physical record can omit constitution without abolishing it, while all physically grounded information is attributable to the complete configuration. Each theorem's mathematical hypotheses and each physical demonstration's domain are explicitly separated from the laws' universal physical assertion. The paper gives a unified and independently readable scientific statement, with the complete formal proof record and verification programs retained in its versioned technical archive.