Paired Statistical Solitons and Torse-Forming Rigidity on Statistical Manifolds

We study paired statistical η-Ricci–Bourguignon almost solitons on a statistical manifold endowed with two dual torsion-free statistical connections. The same potential field is required to satisfy compatible soliton equations for both connections. By adding and subtracting these equations, we obtain identities that separate the Levi–Civita contribution from the cubic tensor associated with the statistical difference tensor. We distinguish the connection-induced symmetric deformation tensor from the ordinary Lie derivative and derive sum and difference identities that isolate the cubic tensor associated with the statistical difference tensor. We prove a spanning-family criterion for full collapse of the dual connections and derive corresponding gradient η-Ricci–Bourguignon and gradient Schouten identities. Explicit affine and Fisher-geometric examples show non-vacuity and exhibit K≠0 together with K(X,·)=0. A Lorentzian perfect-fluid formulation and an exact FLRW-type realization illustrate how the geometric equations enter relativistic models. The mathematical conclusions are separated from model-dependent physical interpretations.