Synthesis of Analytical Operators Through a Structural Invariance Hypothesis

This paper introduces a novel, non-iterative analytical framework for higher-order differentiation and integration, based on a structural invariance hypothesis. Bypassing the computational complexity of classical iterative methods, this approach yields closed-form synthesis for n-th order operators, revealing a profound and exact connection with enumerative combinatorics. We demonstrate that the mathematical architecture spontaneously generates Stirling numbers of the first kind in explicit forms, while Stirling numbers of the second kind govern the determination of primitives. A major result of this research emerges in the transition to autonomous relations: combinatorial complexity strictly converges toward a principle of Unitary Invariance (), regardless of the derivative order, allowing a structural reduction to an Invariant Core. Furthermore, the reveals the existence of "Calibration Thresholds" capable of canceling asymptotic divergences. Finally, the synthesis isolates novel integer sequences within compressed autonomous forms, notably characterized by a rigid architecture featuring a single prime number surrounded by even coefficients. This method provides an asymptotically equivalent, closed-form "shortcut" in infinitesimal analysis, seamlessly bridging the continuous domain with discrete logic.