Root- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>T</mml:mi> <mml:mover accent="true"> <mml:mi>T</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:math> flows unify 4D duality-invariant electrodyna

We present a single, dimension-independent framework that links four-dimensional duality-invariant nonlinear electrodynamics to two-dimensional integrable sigma models. Both sectors are shown to obey the same firs-order Courant–Hilbert equation, solved by a common generating function and an auxiliary-potential formulation. Within this structure, a discrete <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>φ</a:mi> </a:math> parity acts as a selection rule, organizing deformation series into integer versus fractional powers. Two commuting deformations—an irrelevant parameter <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>λ</c:mi> </c:math> and a marginal parameter <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>γ</e:mi> </e:math> —admit universal flow representations that recover root- <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>T</g:mi> <g:mover accent="true"> <g:mi>T</g:mi> <g:mo stretchy="false">¯</g:mo> </g:mover> </g:math> dynamics and extend them in a controlled way. The construction yields closed-form families (generalized Born-Infeld, logarithmic, and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>q</k:mi> </k:math> deformed) and a new integrable model, all realized in 2D and 4D. These results replace case-by-case analyses with a unified route to solvable nonlinear theories, with immediate relevance to gauge dynamics, string-inspired effective actions, and integrable models.