Exact-Sequence Stability and Ambient Realizations for $C4^{\ast}$-Modules

The theory of C4*-modules is presently dominated by decomposition methods, but it lacks a systematic closure theory. In particular, it is not known in general whether the C4* property is preserved under extensions, kernels, cokernels, or short exact sequences. This is a structural difficulty, since C4*-type conditions are governed by summand behavior and comparison of submodules, and such data are not automatically respected by exact sequences. This paper develops an exact-sequence framework for C4*-modules and strongly C4*-modules. It identifies explicit hypotheses under which these classes are stable under split extensions, admissible kernels, admissible cokernels, and short exact extensions. The paper also separates positive and negative directions: closure results are established under summand-lifting and factor-control assumptions, while converse results show that these hypotheses cannot in general be removed. This produces concrete obstruction patterns for extension stability and for passage to submodules and factor modules. A further aim is categorical. Natural ambient settings are identified in which C4*-modules form an extension-closed subcategory, or at least a relative exact class appropriate to summand-sensitive module theory. Finally, concrete ambient verification theorems are proved: semisimple right modules over any ring provide a canonical exact environment, and over a semisimple artinian ring this extends to the full module category.