MATH.Preface:3 - Solution - Connect constructions through what they supply

MATH.Preface:3.1 - Form objects, operations and representations

MATH.16 starts one step earlier, when several constructions seem plausible. Describe the maps the new object must support, then compare arbitrary allowed ways of supplying or processing its data. The resulting universal property distinguishes, for example, carrying both components from accepting either input, and arbitrary pairs from compatible pairs. It can also specify an object that represents a prepared function. A known construction can then supply the object and its maps.

MATH.1 starts with permitted elementary connections and constructs finite paths, identities and composition. Retaining a path can preserve the order and history that its final effect forgets. MATH.5 starts with assigned values for generators and obtains values for their composites while preserving the operations.

When several descriptions should count as the same input, MATH.2 tests whether the required operation is independent of the representative. For a partially available operation, its availability can matter as much as its output. A failed test supplies a distinction to restore.

MATH.7 addresses a reversible change of representation. It constructs the operations in the new representation and carries results back. Renaming the elements while keeping an unsuitable operation can change the problem; the transported operation supplies the repair.

MATH.17 makes the rules themselves available for construction and change. Select allowable operations, determine whether their composition stays allowable, then construct an operation that transforms them. Its required law follows the intended use: repeating a composite and repeating its stages separately can yield different answers.

MATH.18 compares descriptions with different primitives, allowed maps or equality. Construct the needed interpretations, establish what transfers, and inspect the return. A partial interpretation can be enough for one consequence; equivalence requires the corresponding comparisons in both directions.

These methods can be used separately. They also connect: form expressions from generators, interpret their operations, identify descriptions that preserve the desired answer, then choose a convenient representation for calculation. When the rule or the whole account changes, use MATH.17 or MATH.18 to construct and examine that change.

MATH.Preface:3.2 - Obtain an argument and the object it supports

MATH.19 constructs an argument when the premises and conclusion are known but the connecting steps are missing. Work backward to a sufficient claim and forward to available consequences, then prove a lemma joining them. An unsuccessful induction can require an extra parameter or a stronger intermediate statement. B.5.RA instead helps recover an argument already supplied in another description.

MATH.4 obtains a witness by following the construction of a finite input. A step may require a stronger intermediate result or another parameter. MATH.12 recovers functions, pairs, projections and branch information from proof steps, including non-inductive steps. It distinguishes the operations that produce data from a proof that some suitable data exists.

MATH.6 constructs a case in which the assumptions hold and the proposed conclusion fails. Such a case can reveal a missing premise, a misplaced quantifier or an overly narrow search. MATH.11 instead solves for a function preserved by the allowed transformations. Its value can exclude a target or determine an accumulated quantity. Equal values leave any required reachability construction to be supplied.

MATH.20 obtains a useful comparison before the whole unknown is available. A feasible path bounds a shortest length from above; inequalities covering all paths can bound it from below. The method constructs the comparison, propagates its direction through the needed operations and tightens the part that leaves a consequential gap. It can return enough for the next decision without completing an optimization.

An argument and an obtaining procedure can support each other. Given a finite list, a terminating test and a proof that some listed element passes, testing the entries obtains a witness. If the searched range becomes infinite, the finite-search argument must be reconsidered. The available result may remain a logical conclusion or a procedure for each finite portion.

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.

MATH.Preface:3.4 - Return a construction to further work

The connecting rule is simple: name what one construction returns and what the next one uses. A set of solutions, one chosen solution, a proof of existence and an executable selection support different continuations. Enter at a contribution already available and stop when the requested mathematical result has been supplied.

For a question about another subject, use FPF C.29 to establish the correspondence through which the mathematical result answers that question. C.29.2 develops a missing computational formulation; C.29.3 connects a computation with the arrangement that prepares its inputs, performs it and exposes an interpretable result. B.5.MPC coordinates mathematical, physical and computational reasoning when their contributions must be developed together.

MATH.Preface:3.5 - Use this contribution within the wider repertoire

The Foundational Thinking Suite Reference explains where this mathematical contribution connects with model formulation, physical premises, computation, notation and Method Engineering. A result such as a function object or an interpreted path becomes useful through what the next operation can do with it. Its mathematical laws alone leave the subject interpretation and execution conditions to their respective methods.

The twenty bodies connect formation and interpretation with proof, comparison and continued theory development. MATH.17 makes operations available for change, MATH.18 compares interpretations, MATH.19 builds a missing proof, and MATH.20/.21 connect justified bounds with approximation. MATH.22/.23 develop changed theories and useful conjectures. Start with the contribution needed next and read the methods that supply its missing inputs.

For example, changing the order of a read and an update can preserve the final stored value but change the reading used by a later decision. MATH.1/.5 supply sequences and their interpretation, and MATH.2 tests the proposed identification. Method Engineering uses that distinction when deciding how work may be rearranged. The mathematical construction and its use in the working method remain separately inspectable.

MATH.Preface:3.6 - Constituent actions in ongoing work

A symbolic rewrite can be part of proving a lemma while that lemma’s proof is part of proving a larger claim. The required domain constrains the rewrite at that same moment: cancelling a factor is permissible only under the relevant algebraic conditions, and excluding zero would change a claim that is meant to include it. Knowing the symbols and the theorem goal can leave the intermediate reasoning unavailable. Recover or obtain that reasoning rather than treat the smaller calculation as proof of the whole.

FPF B.1.5.EW helps recover these constituent–whole connections; B.1.5.RS examines a proposed replacement. Use the parts of the vertical that can change the present result. A Method described here can require additional capability, available support and compatible resources at other grains.