## Appendix B Core Equations for T3 SD  
The basic mathematical definition of a discrete system model in T3 SD is defined as a quintuple [20]:  
\[Z=(S_{Z},I_{Z},O_{Z},N_{Z},R_{Z}), \tag{11}\]  
where \(Z\) is the name of the system, \(S_{Z}\) is the set of its states, \(I_{Z}\) is the set of its inputs, \(O_{Z}\) is the set of its outputs, \(N_{Z}\) is its next state function, and \(R_{Z}\) is its readout function that specifies the outputs for each state. Another layer to this system model must be added for comparisons to system elaborations. This is the functional system design (\(FSD\)). Further clarity will arise from discussion of the metamodel in the next section. For this, the \(FSD\) is provide and defined as a 3-tuple [20]:  
\[FSD=(Z,DS_{Z},\ TS_{Z}), \tag{12}\]  
where \(Z\) is the minimum model of the system, \(DS_{Z}\) is the initial state, and \(TS_{Z}\) is the discrete timescale of the system. A design that implements \(FSD\) is said to be buildable [35]. The buildable system design (\(BSD\)) is used to define an elaboration from \(Z\) and \(FSD\). \(FSD\) is formally defined as [20]:  
\[BSD=(Z@,SCR), \tag{1}\]  
where \(SCR\) is the system coupling recipe used to define connectivity of components, subsystems, and systems; and \(Z@\) is the resultant system from the coupling. The \(Z@\) of \(BSD\) is homomorphically compared to the \(Z\) of \(FSD\) to establish measured elaboration. Further clarity will arise from discussion of the metamodel in the next section. The \(SCR\) is formally defined as [20]:  
\[SCR=(V_{SCR},C_{SCR}), \tag{2}\]  
where \(V_{SCR}\) is the vector systems to be coupled and \(C_{SCR}\) is the system connectivity between outputs and inputs. [PERSON] described the \(SCR\) as \"the mathematical theory of system coupling: how systems can be put together by input/output relationships to create hierarchical models of more complex systems\" [20]. What is not shown in this equation is that \(V_{SCR}\) consists of minimal models consistent with Equation (A1). This is critical for understanding emergent behavior, which was discussed earlier in this article. A design that defines \(BSD\) is said to be implementable [35]. The implementable system design (\(ISD\)) is used to define the real system and provides a homomorphic mapping between the increasingly technology dependent elaborations of the system design. \(ISD\) is formally defined as [20]:  
\[ISD=(Z,DS_{Z},\ TS_{Z},\ Z@,SCR,Z_{S},\ H_{S},\ H_{I},\ H_{O}), \tag{3}\]  
where \(Z\) is the system model; \(DS_{Z}\) is the initial state; \(TS_{Z}\) is the discrete timescale of the system; \(SCR\) is the system coupling recipe used to defined connectivity of components, subsystems, and systems; \(Z@\) is the resultant system from the coupling; \(Z_{S}\) are the components, subcomponents, and modes of the real system; \(\mathrm{H_{S}}\) is a homomorphic mapping between states for the levels of abstraction; \(H_{I}\) is a homomorphic mapping between inputs for the levels of abstraction; and \(H_{O}\) is a homomorphic mapping between outputs for the levels of abstraction. In regard to the use of homomorphism, [PERSON] stated that \"system models must also support... functionality preserving simplification and elaboration. One system model is a homomorphic image of another system model if and only if they have the same functionality (but the homomorphic image system model might be simpler and/or \"smaller\" and the other may be \"larger\" or more elaborated)\" [35]. [PERSON] referred to the last four items in Equation (3) as the \"implementation artifacts\" [20]. Therefore, we have created a new equation to capture this part of the \(ISD\) which is defined as:  
\[IA=(Z_{S},\ H_{S},\ H_{I},\ H_{O}), \tag{4}\]  
where \(Z_{S}\) are the components, subcomponents, and modes of the real system; \(H_{S}\) is a homomorphic mapping between states for the levels of abstraction; \(H_{I}\) is a homomorphic mapping between inputs for the levels of abstraction; and \(H_{O}\) is a homomorphic mapping between outputs for the levels of abstraction.