A Maximum-Entropy Method for Zero-Skewness Valence GPDs Constrained by Nucleon Electromagnetic Form Factors

We formulate a reduced-profile maximum-entropy method (MEM) framework for constructing constrained zero-skewness valence-quark generalized parton distribution (GPD) transverse profiles from the four nucleon electromagnetic form factors $F_1^p(t)$, $F_1^n(t)$, $F_2^p(t)$, and $F_2^n(t)$. The form-factor sum rules fix only $x$-integrated moments of the GPDs; the forward limit of $H_v^q$ is fixed separately by the valence parton distribution functions, and the normalization of $E_v^q$ by the flavor anomalous magnetic moments. These complementary constraints are combined through the ansatz $H_v^q(x,t)=q_v(x)\exp[t f_H^q(x)]$ and $E_v^q(x,t)=e_v^q(x)\exp[t f_E^q(x)]$, where the positive profile functions encode the $x$-dependent transverse structure. Rather than attempting an unrestricted functional inversion, we use the entropy functional as a regularizing criterion on a low-dimensional positive profile manifold. In the numerical proof-of-concept calculation, a smooth elastic form-factor input and analytic forward distributions are adopted, together with the reduced form $f(x)=0.05+(1-x)^2\exp(c_0+c_1x+c_2x^2)$, which suppresses local modes that elastic moments alone cannot constrain. Within this reduced ansatz, the resulting profiles reproduce the imposed elastic moment constraints, satisfy the forward normalizations after discrete-grid normalization, and give impact-parameter distributions with the expected transverse shrinkage at large $x$. The construction provides a controlled zero-skewness baseline for connecting elastic form-factor constraints to $x$-dependent transverse profiles, and it offers a stable starting point for future analyses incorporating empirical form-factor fits, modern PDF inputs, lattice-QCD generalized form factors, and hard exclusive observables.