| 1 | MATH.11 - Construct an Invariant from Transformation Rules | Usable, evolving | invariant; transformation rule; preservation equation; coefficient; reachability. How can a preserved expression be constructed and used to obtain a formula or exclude a target? | MATH.2 for identification; MATH.7 for transport; C.29 for a consequence about the modeled subject. |
| 2 | MATH.13 - Derive a Consequence from a Symmetry | Usable, evolving | symmetry; uniqueness; fixed point; orbit; conservation; numerical update. What does a transformation preserve, and which conclusion actually follows from that symmetry? | MATH.8 for the full orbit construction; MATH.9 for compatible choice; MATH.10 for admissible variation. |
| 3 | MATH.8 - Generate a Solution Family by Symmetry | Usable, evolving | group action; solution orbit; stabilizer; repetitions; representatives. Which solutions can be generated from one solution, and how much of the solution set does this cover? | MATH.13 for an unresolved symmetry consequence; MATH.9 when one compatible representative is required. |
| 4 | MATH.9 - Determine Whether and How a Choice Rule Can Respect Symmetry | Usable, evolving | equivariant choice; stabilizer; symmetry obstruction; additional data. Can one allowed answer be chosen consistently with symmetry? What can replace an impossible choice? | MATH.8 for solution orbits; MATH.13 for an earlier consequence or uniqueness question. |
| 5 | MATH.10 - Improve a Mathematical Candidate or Derive a Necessary Condition by Admissible Variation | Usable, evolving | admissible variation; improving change; stationary point; boundary minimum; constraint; first variation. Which change is allowed, can it improve the candidate, and what condition follows? Does that condition establish an optimum? | B.5.RA if the variational argument needs recovery; C.29 when a mathematical variation represents a subject change. |
| 6 | MATH.21 - Construct an Object through Convergent Approximations | Stable | limit; completeness; compatible approximation; uniform convergence; error control. How can finite approximations construct an object while retaining the next operation? | MATH.20 for bounds; MATH.19 for convergence and interchange arguments; MATH.2 for classes of representations; computational methods for effective obtaining. |
| 7 | MATH.22 - Change Axioms and Trace Their Consequences | Stable | axiom change; theory; interpretation; model; independence; proof repair. Which constructions and consequences survive when assumptions change? | MATH.18 for interpretations; MATH.19 for replacement proofs; MATH.6 for countermodels; MATH.23 for a further conjecture. |
| 8 | MATH.23 - Develop a Conjecture by Changing a Construction | Stable | conjecture; construction variation; proof and refutation; generalization; next problem. How can a change or obstruction yield a precise useful claim and an attainable next operation? | MATH.19 for proof construction; MATH.6 for refutation; MATH.22 for theory change; B.5.QD/C.40.CD for continued inquiry. |