Two-dimensional sigma models that are causal and classically integrable can share the same leading T·T̄ interaction yet respond in sharply different ways to strong fields. An exact reconstruction study shows that agreement at leading order does not fix the full nonlinear behaviour of the model: the freedom left after matching the leading term is quantifiable and physically relevant.

The work addresses the reconstruction of exact autonomous stress-tensor generators for one-scale Courant-Hilbert families in two dimensions. The key of the method is to separate the invariant stress scale from a signed trace ratio, which reduces the reconstruction to a scalar inversion on each invertible physical branch. The resulting profile determines both the deformation law and the relation between local energy density and pressure. A regular Legendre transformation reproduces the same profile in the auxiliary potential formulation.

The comparison between generalized Born-Infeld and logarithmic models illustrates the difference. Both share the leading interaction, but in the former the pressure saturates while in the latter it grows without bound. Their exact generators are respectively quadratic and nonpolynomial, the latter involving the W₋₁ branch of the Lambert W function. The flows commute locally with the root-T·T̄ flow within a common smooth branch and retain a common classical Lax form.

The significance of the result lies in delimiting how much nonlinear freedom survives once the leading deformation term is fixed. In effective field theories and in the phenomenology of T·T̄-like deformations, two models indistinguishable at leading order can differ drastically in strong-field regimes, with consequences for pressure saturation and for the structure of solutions. Exact reconstruction thus offers a tool to classify integrable continuations that were previously treated as essentially equivalent.