Dynamic Curry-Howard R: Unified Embedding, Strictness Determination, and Categorical Defects of the Definition Spectrum of Constraint Networks

In a cross-paper theoretical engineering project whose ontology is weighted hypergraphs, the same core concept appears with mutually incompatible definitionsin different papers, constituting a systematic risk. This paper gives a decidable andreproducible unified treatment of seven definition variants of “constraint networks”in a multi-paper series. The seven variants take different values in six dimensions:the domain of the weight function degenerates from the full power set to the hyperedge set or to directed hyperedges; the loss law splits from a global linear budgetinto edgewise monotonicity, linear residual, growth type, and inheritance; the complexity budget splits from absence into node–edge image counting, fiber diameter,and normalized symmetric difference; and the value-domain structure differentiatesfrom a resource monoid into nonnegative real addition, a chain residual lattice,a scaling commutative monoid, and a threshold structure. Under a unified baseCNet⋆, this paper establishes an embedding-spectrum theorem: every variant admits a fully faithful embedding into CNet⋆, its image is explicitly characterized bya closure class, an edge-map type, and a budget shape, the budget translation istermwise closed, and composition laws are transported along the translation. Thewell-formedness of the embeddings depends on an explicit package of value-domainoperation axioms: the chain residual law of the F variant cannot be derived fromgeneral residuated lattices, and in positive integer multiplication under the usualorder the examples 12/6 = 2, 6/4 = 1, and 12/4 = 3 provide counterexamples;the naive supremum extension of the J variant does not preserve the tensor typeor submodularity on general downward-closed families; the normalization factor ofthe M variant must satisfy boundedness and scaling submultiplicativity, otherwisebudget subadditivity fails; and the directional information of the G variant must bepreserved by copy-label encoding. For strictness determination, this paper gives anexplicit family of morphisms that satisfies the global linear budget law while violating all growth-type loss laws, and ninety instances on a parameter grid are verifiedby exact rational simulation. The reverse direction of the class inclusion holds under the value-domain condition µ ⊗ η ≥ µ, and this paper replaces an unverifiedreverse-direction claim by an explicit citation of that condition. For categorical defects, the covering-closure variant lacks equalizers, and an explicit counterexample isgiven; the variant has products but lacks equalizers. For universe-jump safety, in astratified type system, baseline consistency together with the absence of a positiveweight directed cycle containing the jump edge after applying the candidate jump isa sufficient condition for strong normalization of all typable terms after application;necessity holds under the graph-level reading of the Hurkens tower construction,while necessity for general systems is retained as an open item. All computationsuse parametric exact rational simulation and finite-set symbolic checking, and nounpublished data source is used as a theoretical basis.