B.5:4.2 - Perform the contribution that is missing
The following are common alternatives, not an exhaustive classification or a sequence to complete.
| Missing result | Useful operation | What the result can establish |
|---|---|---|
| An object, procedure or structure with desired properties | Construct it from stated elements and permitted operations; test examples and counterexamples; prove the relevant properties or expose the obstruction. | A construction, existence or impossibility result under its mathematical premises. |
| A consequence or a mathematical justification | Make premises explicit and derive the consequence. Identify the step that carries the argument and the conditions on which it depends. | The implication or theorem within the stated domain and inference rules. |
| An explanation of an anomaly, opportunity or probe result | Use B.5.2 to generate serious rival conjectures and compare their explanatory fit, constraints and prospects for criticism. | A qualified explanatory conjecture, with its grounds and unresolved rivals. |
| A better description of an insufficiently understood phenomenon | Make a purposeful observation or exploratory measurement; vary a condition and inspect what becomes distinguishable. | Observations and possible regularities that can generate or change questions. |
| An empirical consequence of a conjecture | Derive the expected contrast, obtain relevant observations and compare the actual result with it. | Bounded corroboration, a discrepancy or a qualified basis for rejecting or revising the claim. |
Recover a construction before trying to execute it. Use this continuation when the source names a desired object or property but you cannot yet obtain the result needed by the question. B.5.RC expands the method below with worked cases and explanations of shared prerequisites, alternative constructions and missing operations.
- Identify the starting objects or data that are available. State what must be produced and which property the receiving use needs.
- Find the source’s operations for producing or combining those objects. For each needed operation, recover its inputs, application conditions and output. Keep a statement that an object exists with certain properties as an existence claim; seek a way to obtain an instance when the next use requires one.
- Work backward from the desired result to the required intermediate results and starting inputs. Preserve joint dependencies and alternative ways where the source supplies them. A missing operation is a question for the source or the relevant specialist; a rule you propose is an addition to be tried and justified.
- Work a small instance from the available inputs, applying each recovered operation when its conditions hold. If the notation prevents the operation, use A.6.3.RT to prepare and compare a more usable expression. For a mathematical account, use its formation and equality rules; identify where a proposed identification changes the operations or property being used.
- Establish the needed property by the appropriate argument or test. Return the construction and what follows under its premises, or the input, rule or unsupported transition that still prevents the result. Stop when this supplies the receiving use.
For example, a stand specification calls for a portable display. Its assembly rules allow a compatible upright to be joined to a base and a compatible panel to be attached to that upright. Those rules yield the assembly order and the connections to check. Portability remains a requirement to check on the resulting design. If the supplied panel does not fit, the next contribution is a compatible panel, an adapter with its connection rules, or another assembly design. The construction result here is the assembly design; whether the erected stand is stable needs its physical-design argument.
Construction and deduction can work together: an auxiliary object can make a proof possible, and a theorem can suggest a new construction. Novelty does not belong exclusively to one inference type.
For a hypothesis-led test, derive the consequences needed to interpret the test. Check whether those consequences conflict with retained premises, and follow their implications through the part of the model relevant to the question. Return inconsistent premises for revision before relying on their joint prediction. Keep the prediction, observations, measurement conditions and inference recoverable before treating the outcome as corroboration. A simulation establishes a result of the simulated model; applying it to the physical target needs a supported model–world correspondence.
An exploratory finding can supply a hypothesis. Its subsequent empirical assessment must account for how that hypothesis was obtained and for the dependence or selection involved. Use the domain’s appropriate design and inference; merely relabeling the same data as a later test does not create independent evidence.
A bounded result may be sufficient at any of these contributions. Assurance belongs to the particular claim and receiving use under B.3 and B.3.3, including the applicable domain proof or validation obligations.