This deposit provides the rigorous mathematical specification, formal algebraic proofs, and python verification codes for Operator 1: Local Graph Expansion Operator (G_n to n+1), which serves as the foundational structural cornerstone of the Status-Relational Entropy (SRE) Dynamics framework. The Local Graph Expansion Operator dynamically injects a single new vertex and expands the frontier coupling parameters without collapsing or distorting the read-only historical matrix topologies, successfully achieving a boundary-free algebraic system base. Core Algebraic Achievements:- Theorem 1 (Inductive Limit Ring): Constructs the full-historical multivariate polynomial ring via the inductive limit of commutative rings along canonical embedding morphisms, fully insulating parameter naming domains across scaling rounds.- Theorem 2 (Morphic Analysis): Formally proves the strict injectivity of the expansion operator to guarantee lossless historical preservation, while non-surjectivity limits the output's image strictly to konstant realized blocks.- Theorem 3 (Universal Diagonal Invariant Theorem): Proves that the 2-step formal walking diagonal polynomial of the newly injected edge collapses unconditionally to the real scalar invariant n+1 under any binary mapping, completely independent of individual variable assignment strategies.