Constant-Time O(1) AST Reduction and Native Delegate Compilation in RICIS-III

Traditional Computer Algebra Systems (CAS) and dynamic expression interpreters evaluate critical points and mathematical singularities via runtime limit approximations, Taylor series expansions, or recursive L'H\^opital routines. These procedures incur an unresolvable computational bottleneck: dynamic tree-traversal complexity of $O(N)$ alongside runtime branching hazards and undefined IEEE-754 states (\texttt{NaN}, division-by-zero traps). This paper presents a formal engineering proof of how the Recursive Indexed Calculus of Identity and Singularity (RICIS-III v7.9) enables strict constant-time $O(1)$ symbolic Abstract Syntax Tree (AST) reduction and native machine delegate compilation. By enforcing Absolute Continuity ($L_0$), the Identity Principle ($L_1$), Safety Protocols ($SP_1$--$SP_5$), Protocol $P_1$, and the geometric realization of Axiom $A_6$ ($S_F \boxtimes I_G \to R(F,G) \xrightarrow{\mu} F \cdot G$), all indeterminate nodes are eliminated during symbolic pre-compilation. The resulting pruned AST compiles into straight-line native execution blocks (such as .NET CLR dynamic delegates or LLVM IR) operating in deterministic $O(1)$ clock cycles per tick with zero runtime branching.