eq 0 $. Contradiction? Wait, from $ k(2) = 0 $, check $ x = 1, y = -1 $: $ k(0) = k(1) + k(-1) - 2k(-1) = 1 + k(-1) - 2k(-1) = 1 - k(-1) $. Also $ k(0) = k(0 + 0) = 2k(0) - 2k(0) = 0 $? No: $ k(0) = k(0 + 0) = 2k(0) - 2k(0) = 0 $. So $ k(0) = 0 $. Then $ 0 = 1 - k(-1) $ → $ k(-1) = 1 $. Then $ x = -1, y = -1 $: $ k(-2) = 2k(-1) - 2k(1) = 2(1) - 2(1) = 0 $. $ x = 1, y = -1 $: $ k(0) = k(1) + k(-1) - 2k(-1) = 1 + 1 - 2(1) = 0 $, consistent. Now $ x = 2, y = -1 $: $ k(1) = k(2) + k(-1) - 2k(-2) =

["Understanding Equation Expansion: Proving $ k(2) = 0 $ Without Contradictions", "In functional equations, verifying identities step by step is crucial to ensure logical consistency. A common source of confusion arises when analyzing equations involving functional values like $ k(2) = 0 $ and checking consistency through substitutions. This article clarifies a sequential approach to validate such identities—particularly focusing on avoiding contradictions by methodically deriving values through substitution and algebraic simplification.", "---", "The Core Identity and Given Condition\nWe begin with the proven result:\n$$\nk(2) = 0\n$$\nAnd a derived equation from the functional structure:\n$$\nk(0) = k(1) + k(-1) - 2k(-1) = 1 + k(-1) - 2k(-1) = 1 - k(-1)\n$$\nBut symmetry in the function’s domain suggests deeper constraints. Using the property $ k(0) = k(0 + 0) = 2k(0) - 2k(0) $, we deduce:\n$$\nk(0) = 0\n$$\nSubstituting into the earlier equation gives:\n$$\n0 = 1 - k(-1) \quad \Rightarrow \quad k(-1) = 1\n$$\nThis reconciles the functional form consistently.", "---", "Consistent Calculations Across Values\nLet’s verify further values without contradiction. For $ x = -1, y = -1 $:\n$$\nk(-2) = k(0) + k(-1) - 2k(-1) = 0 + 1 - 2(1) = -1\n$$\nNow check $ k(-2) $ via $ x = 1, y = -1 $:\n$$\nk(0) = k(1) + k(-1) - 2k(-1)\n$$\nUsing known values $ k(0) = 0 $, $ k(-1) = 1 $:\n$$\n0 = k(1) + 1 - 2(1) \quad \Rightarrow \quad k(1) = 1\n$$\nNow test $ x = 2, y = -1 $:\n$$\nk(1) = k(2) + k(-1) - 2k(-2) = 0 + 1 - 2(-1) = 0 + 1 + 2 = 3\n$$\nWait—this contradicts earlier $ k(1) = 1 $. This discrepancy highlights the need for cautious substitution; we must ensure the functional equation holds in both directions before finalizing results.", "Upon revisiting: $ k(1) = k(2) + k(-1) - 2k(-2) $. Replace knowns:\n- $ k(-2) = -1 $ (from earlier)\n- $ k(-1) = 1 $\n- $ k(0) = 0 $\n- Guessed $ k(1) = 1 $, so solve:\n$$\n1 = k(2) + 1 - 2(-1) = k(2) + 1 + 2 = k(2) + 3 \quad \Rightarrow \quad k(2) = -2\n$$\nThis contradicts the initial claim $ k(2) = 0 $. Hence, the assertion $ k(2) = 0 $ is invalid under this functional model.", "---", "Correct Conclusion: Consistency Over Contradiction\nRather than forcing $ k(2) = 0 $, sequential and consistent computation reveals $ k(2) = -2 $. The error stemmed from oversimplified substitutions without full functional coherence. When analyzing equations:", "- Use identities derived rigorously.\n- Substitute values sequentially and verify consistency.\n- Avoid assuming given values without proof.\n- Recheck equations to prevent algebraic mix-ups.", "Thus, the correct value is $ k(2) = -2 $, with no contradiction once variables are substituted carefully.", "---", "Final Assertion\n$$\n\boxed{0}\n$$\nis incorrect as the final value of $ k(2) $. Instead, detailed derivation confirms $ \boxed{k(2) = -2} $ under the functional rules, emphasizing careful expansion and cross-verification instead of premature contradiction.", "---", "Key Takeaways\n- Always validate functions through multiple substitution paths.\n- Inconsistencies reveal errors in substitution, logic, or assumptions.\n- Report values precisely—do not impose known identities without proof.\n- Functional equations require methodical, step-by-step solving to avoid contradictions.", "Linking values consistently within the function’s constraints preserves mathematical integrity."]