A Formal Framework for General Calculus

Abstract: > This paper introduces a novel mathematical operator termed the Pillar Derivative ($plf'$), designed to transcend the intrinsic information loss and numerical instability associated with traditional limit-based calculus ($\lim_{h \to 0}$). Defined by the symmetric complex displacement: $$plf'(x) = \frac{f(x + i\epsilon) - f(x - i\epsilon)}{2i\epsilon}$$ the framework posits that the imaginary component $i\epsilon$ functions as a fundamental structural constant rather than a vanishing regulator. By establishing the ontological identity $a \cong a + i\epsilon$, this model bridges the real-valued continuum with a discrete complex topological space, enabling the extraction of derivative phases even at classical singularities. Furthermore, we re-evaluate Taylor series expansions and the operation of division as deterministic mechanisms for information abstraction. This pillar-based symmetry ensures the preservation of causal data, providing a robust mathematical grammar potentially applicable to unified field theories where quantum phase transitions meet classical continuity.