| 1 | MATH.16 - Choose a Mathematical Construction from Its Required Maps (Universal Property) | Usable, evolving | universal property; product; coproduct; pullback; function object; currying. What maps should a new object support, and how can that requirement select a construction? | MATH.2 for quotients; MATH.5 for generator extensions; MATH.7 for reversible representations; C.29 for interpretation in another subject. |
| 2 | MATH.17 - Construct Mathematical Spaces of Operations and Operations on Them | Usable, evolving | operations as objects; admissible functions; closure; higher-order operation; transformation law. How can a rule be constructed, combined or changed while retaining the consequence its use needs? | MATH.16 for function objects; MATH.1/.2 for retained steps or identification; MATH.18 for interpretation between accounts; C.29 for a working-method application. |
| 3 | MATH.1 - Build a Mathematical Structure of Composable Paths | Usable, evolving | generators; paths; endpoints; identity; associativity. Which elementary steps can be composed? When do different sequences need to remain distinct? | MATH.2 when paths will be identified; C.29 when the construction represents another subject. |
| 4 | MATH.2 - Treat Objects as the Same While Preserving Operations (Quotient) | Usable, evolving | quotient; congruence; equivalence; partial operation; refinement. Can these objects be treated as the same without losing a later operation or result? | MATH.1 when paths first need construction; MATH.6 for a counterexample to the proposed identification. |
| 5 | MATH.5 - Extend a Generator Assignment While Preserving Operations (Homomorphism) | Usable, evolving | generators; relations; homomorphism; free structure; extension. How can a choice on generators determine an operation-preserving map on everything they generate? | MATH.1 for paths; MATH.2 when an extension must descend to a quotient. |
| 6 | MATH.7 - Transport a Mathematical Structure Through a Bijection | Usable, evolving | bijection; transport; inverse map; isomorphism; domain of operation. How can a useful operation, law and answer be carried through a change of representation? | MATH.2 when identification is proposed instead of a bijection; C.29 for interpretation in another subject. |
| 7 | MATH.18 - Compare Mathematical Accounts through Interpretations | Usable, evolving | interpretation; primitive operation; preservation; reflection; round trip; equivalence. Which constructions, equations and maps transfer between two mathematical descriptions, and what can be recovered? | MATH.5 for generated interpretations; MATH.7 for bijective transport; MATH.2 for quotients; MATH.17 for transformations of operations. |