Admissible Image Geometry for Network Motifs in Modal Triplet Theory Conditional Correlations, Sprout Bounds, and What Projection Alone Does Not Imply

Abstract

Projection can discard information without selecting a network geometry. This paper develops a rigorous Modal Triplet Theory (MTT) framework in which observable motif constraints are determined from the actual image R = F(A), where A is a declared admissible source domain and F includes projection, network extraction, and motif measurement. Compatibility constraints are properties of R, not consequences of noninvertibility by itself. We give an exact recombination test for support nonfactorization, a graph-of-a-map sufficient condition, and a regular-value description of local image geometry. A covering-map counterexample proves that failure of a continuous section is compatible with a full product image. We also separate three distinct objects: an embedded physical network, a constrained motif support, and a statistical conditional-dependence graph. For a declared local reserve model, we prove a conditional square-root branch bound and show that near saturation the dominant channel remains nearly straight and the branch nearly orthogonal. The estimate is stable under a relative remainder bound. This proves the result inside the stated positive quadratic model; it does not derive the model, its coefficients, a branch-creation law, or universal empirical prevalence. Those require a selected network source, Hessian, extraction pipeline, and probability or growth dynamics. The result is a falsifiable conditional encoding rather than a universal theory of network morphology.