Zulfia Programming: From Classical KKT Conditions to a Single Structural Identity ∆ = 0 (Revised Version)

Constrained optimization is classically written with inequality and equality functionsg(x) ≤ 0, h(x) = 0 and characterized by Lagrange multipliers and the Karush–Kuhn–Tucker (KKT) conditions. We formalize Zulfia Programming (ZP), a single-identityform of the same problem: a structural component ∆(x, s), built with squared slackvariables, vanishes exactly at the valid configurations (the Zulfia identity condition,ZIC). One proposition summarizes the relation to the classical theory:• The ZIC reproduces classical feasibility exactly, and local minimizers correspond;• When ∆ has full-rank Jacobian, which is the linear independence constraint qual-ification (LICQ), first-order optimality in ZP is KKT stationarity with comple-mentary slackness, with the multipliers appearing as the coefficients of D∆⊤;• The sign conditions μ ≥ 0 are not part of first-order ZP stationarity and comefrom a second-order condition along the slack directions.Two worked examples and a comparison table follow. ZP re-expresses the classicaltheory in one identity; it does not remove multipliers or constraint qualifications.