Generalized root model

⚠️ Curated Limitations & Negative Results

• Roots, statistic domains and reference measures must be specified. The signed convention is not the ordinary identity sqrt(x^2)=x on the real line. This record does not assert a general Gaussian higher-order normalization theorem.
• Finite partition is an explicit requirement, not a theorem that every base exponential family permits every finite tensor. Poisson factorial domination and exponential tail restrictions require separate reasoning.
• 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.