Answer Space = Operation(Problem Space)
Every problem supplies a Problem Space. The learner chooses an invariant-preserving Operation that transforms that evidence into a proven Answer Space.
Problem Space
Raw input, constraints, candidates, structure, and rules supplied by the problem.
Operation
The reasoning shape that transforms the problem.
Answer Space
The required result together with its proof of validity.
Proof
Initialization, maintenance, termination, and correctness.
Operations CCF learners derive
Scan + Recorder
Two-Pointer Convergence
Decision Tree Search
Frontier Expansion
Range Reduction
Priority Recorder
Transformation Law in action
Target Element
Problem Space: unsorted integer array + target.
Operation: Direct Scan.
Answer Space: first matching index or -1.
Proof: Every candidate is checked one at a time; a returned index satisfies candidate == target; -1 follows Scope Exhaustion.
Two Sum
Problem Space: integer array + target sum.
Operation: Scan + Recorder.
Answer Space: two different indices.
Proof: The Recorder contains only prior Frames, so the same element is not reused.
Sliding Window
Problem Space: contiguous candidate region.
Operation: Expand + Repair.
Answer Space: best valid window.
Proof: Expand to gather evidence, repair when invalid, and commit only valid Frames.
Dynamic Programming
Problem Space: repeated subproblems.
Operation: State Recorder.
Answer Space: optimal or counted result.
Proof: Each state records a reusable sub-answer and the recurrence preserves correctness.
Why this matters
Traditional learning asks, “What pattern is this?” CCF asks, “What Operation transforms this Problem Space into a proven Answer Space?”
Every Case File is the Transformation Law in action. The Reasoning Pipeline identifies the Problem Space, selects the Operation, builds the Answer Space, and proves the route. The Problem Simulator lets learners perform that transformation step by step.