Poisson generalized root model

⚠️ Curated Limitations & Negative Results

• The all-finite-parameter result depends on factorial weights, finite dimension and finite maximum order. It does not justify dropping the factorials, using an infinite graph or taking unbounded parameter limits.
• The conditional normalizer is generally an infinite sum. Existence does not certify a numerical cutoff or make exp(eta_1s) its general conditional mean.
• Tensors are super-symmetric and zero on repeated indices. Full ordered-index contraction counts a distinct unordered ell-variable term ell! times; coefficients must be translated consistently when switching to once-counted clique notation. Separately fitted node regressions do not automatically agree on compatible joint tensors.
• The general family includes all 1<=ell<=j<=k terms. The paper’s strongest-interaction subfamily retains only ell=j and omits lower-degree terms; its experiments do not evaluate every member of the general family. Nonzero higher-order interactions do not require nonzero lower-order interactions.
• Base-family naming does not make all full conditionals or marginals ordinary Poisson or exponential laws. An interaction coefficient is not a universal marginal-correlation or causal-effect measure.
• For T=x, the printed radial expression omits the z^(p-1) Jacobian needed for a continuous L1-radius law under x=z*v. A bounded directional domain also does not alone establish integrability from pointwise finite ray integrals. Count directions require their own discrete measure. Direct joint domination is used for the Poisson and strict exponential regimes.
• The numerical section requires qualification: after Equation (15), E[x^a] must use exp(M(a)-M(0)), not the printed subtraction of M(1). The first row of Equation (14) requires exponent 1/j+1 for covariance with x, not the printed 1/j+1/2. Linear bounds must retain the base log measure B; inclusive count-interval probabilities need F(b)-F(a-1), and zero atoms need explicit handling. These are curator checks of displayed expressions, not a reproduced implementation defect or an author-issued correction.
• Nodewise l1 fitting, approximate conditional normalizers and Newton-like updates are estimation procedures separate from the model definition. A fixed number of approximation intervals gives no universal accuracy guarantee. No author code, empirical fit, optimizer benchmark or graph-recovery experiment was reproduced.
• The reported k=3 Poisson word-count examples rank weighted tuples from Classic3 abstracts and sampled Grolier articles. They are qualitative context, not evidence of held-out prediction, causal discovery or universal graph recovery. The proposed low-rank and staged parameter reductions are not established evaluated variants in this source.