MATH.Preface:3.3 - Change a construction and develop its theory

MATH.13 establishes how a transformation of the data relates to transformations of admissible candidates and answers. That relation can transfer a solution, constrain a unique answer or expose an impossible choice requirement. MATH.8 constructs the resulting orbit, removes repetitions and separates one orbit from all solutions. MATH.9 constructs a choice compatible with the transformations when the input’s own symmetries permit one.

When candidates satisfy constraints, MATH.10 constructs changes that stay within them and calculates what those changes do to a criterion. The result may be an improving candidate or a necessary condition. A minimum requires the corresponding additional argument. In a physical-action calculation the requested condition can be stationarity.

Symmetry and variation can simplify the same problem while answering different questions. One establishes a relation among transformed problems and solutions; the other investigates admissible changes and their effect on a criterion. Preserve the conclusion supplied by each.

MATH.21 constructs an object through converging or compatible approximations. The intended use selects what convergence must preserve: a finite prefix, function values, an integral or another observation. MATH.20 can provide the necessary error bound; MATH.19 can supply a missing limit or interchange argument. A limit’s existence and an effective way to obtain the requested finite information require their respective constructions.

MATH.22 changes the permitted objects or reasoning by changing assumptions. Follow the affected proof steps and constructions, retaining a conclusion when its justification survives or can be repaired. MATH.18 supplies interpretations between accounts; MATH.6 can separate a claimed consequence from what the new assumptions allow.

MATH.23 turns a variation, obstruction or unexplained regularity into a next mathematical claim and a proving or refuting operation. If equal weighting of group means fails for unequal groups, restricting all groups to equal size abandons the original need. Retaining sums and counts repairs the construction and opens a general question about combinable summaries. Proof construction, countermodels and changed axioms then provide different continuations.