Gibbs normalizability and maximum-entropy programs with bilinear moment constraints on orthants

Gibbs weights with attractive bilinear interactions need not define probability laws on unbounded orthants. This paper gives a maximum-entropy and partition-function classification of that failure. For densities on $ [0, \infty)^2 $ with prescribed marginal means and product moment $ \mathbb{E}[XY] $, the entropy program is well posed exactly when $ \mathbb{E}[XY]\le \mathbb{E}[X] \mathbb{E}[Y] $. In the attractive case $ \mathbb{E}[XY] > \mathbb{E}[X] \mathbb{E}[Y] $, the entropy supremum equals the means-only product-exponential value but is not attained: The attractive constraint is entropically invisible while destroying normalizability. We then identify the repairs. Self-moment damping of orders $ p, q > 1 $ normalizes every attractive bilinear exponent when $ 1/p+1/q < 1 $ and never normalizes it when $ 1/p+1/q > 1 $. In the critical case $ 1/p+1/q = 1 $, normalizability is decided by an explicit inequality between the damping coefficients, which at $ p = q = 2 $ is the Gaussian condition $ \gamma^2 < 4\alpha\alpha' $; when that inequality holds with equality, the linear multipliers decide, and we classify this remaining boundary completely. Hard-wall confinement also repairs the model: Truncation to $ [0, K]^2 $ restores well-posedness and has sharp entropy deficit $ \Delta(\mu_1^{-1}+\mu_2^{-1})/K+o(K^{-1}) $. In dimension $ n $, quadratic Gibbs normalizability is governed by copositivity of the negated coupling matrix together with the sign of the linear field on the zero cone, in every regime except a degenerate boundary of transverse codimension at least two, which we leave unresolved; the associated decision problem is co-NP-hard, where co-NP is the class of decision problems whose complements belong to nondeterministic polynomial time (NP). The results give checkable criteria for when bilinear maximum-entropy models on orthants are genuinely solvable.