69.6 Verified Solution Set

Verified Solution Set is the final, complete collection of every candidate value that has successfully passed through candidate verification, transformation risk control, and contextual admissibility, assembled together, checked for duplicates, and reported as the definitive answer to the original problem.

All Candidate Branch Review

Reviewing Every Path That Produced a Candidate

This review confirms that every distinct branch of the solving process, such as each sign branch of a square-root method or each factor of a factored equation, has been accounted for and has contributed its resulting candidate to the review.

Why Reviewing Every Branch Matters

Because a solving method can produce more than one candidate through more than one distinct path, confirming that every such path has been included in the review prevents a candidate produced by an overlooked branch from being silently missed.

Duplicate Candidate Removal

Removing Repeated Values from the Collected Candidates

If the same numerical value appears more than once among the collected candidates, perhaps produced by two different branches of the solving process, only a single instance of that value is retained.

Why Duplicate Values Are Consolidated

Because a solution set represents the distinct values that satisfy the original problem, listing the same value more than once would misrepresent the number of genuinely distinct solutions the problem actually has.

Rejected Candidate Exclusion

Removing Every Candidate That Failed Any Verification Step

Every candidate that failed the original-relation test, the transformation risk check, or the contextual admissibility check is formally excluded from the set being assembled.

Why Exclusion Happens Only after Full Verification

Waiting to exclude a candidate until after it has been fully evaluated against every relevant check ensures that no candidate is dismissed prematurely, before it has had a genuine opportunity to be confirmed as admissible.

Accepted Candidate Collection

Gathering Every Fully Confirmed Candidate

Every candidate that has passed all relevant verification steps, and survived both the duplicate removal and rejection exclusion, is gathered together into the collection that will become the final solution set.

Why This Collection Represents the Core of the Final Answer

This collection represents every value that has been confirmed, through every applicable check, to be a genuine and admissible solution, forming the direct basis of the final reported answer.

Empty Verified Set Outcome

Recognizing When No Candidate Survives Verification

If every candidate produced during solving fails at least one verification step, the resulting verified solution set is empty, meaning the original problem has no admissible solution.

Why an Empty Set Is a Legitimate, Meaningful Outcome

An empty verified solution set is not treated as a failure of the solving process itself but as a legitimate and meaningful result, correctly reflecting a situation in which no value both satisfies the original relation and meets every relevant contextual requirement.

Final Verified Solution Statement

Reporting the Completed Set as the Final Answer

The completed verified solution set is presented as the final, definitive answer to the original problem, whether it contains one value, several values, or no values at all.

Why This Statement Marks the True Completion of the Process

Because every value in this set has passed through the entire sequence of collection, testing, risk control, and contextual checking, this final statement represents the true completion of the solving process, distinct from the earlier, merely provisional candidate values first produced by the initial solving steps.