# Question 13517: Compute a Binomial Square Using Ring Operations, Not Label Arithmetic [Return to catalogue page 68](https://ocme.evemisslab.com/en/questions/page/68/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r01-finite-commutative-ring-foundations/#q-20261014-advance-20261028-r01-finite-commutative-ring-0092) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0026-13517 · q-20261014-advance-20261028-r01-finite-commutative-ring-0092 ## Original question and givens Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Require commutativity. a=1, b=2. Give (a,b,sum=a+b,lhs=sum*sum,a2,ab,b2,rhs=a2+ab+ab+b2,equal?). - G001 — Use the same qualified complete finite unital ring on Fin(8)={0,…,7}, with addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]] and multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 6, 1]] in label order. All unital-ring axioms, including distinct two-sided identities, must already hold. The identity labels need not be 0 and 1. Commutativity is a separate multiplication obligation; even a commutative ring may have nonzero zero divisors and need not be a field. Require commutativity. a=1, b=2. Give (a,b,sum=a+b,lhs=sum*sum,a2,ab,b2,rhs=a2+ab+ab+b2,equal?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-finite-commutative-ring](https://ocme.evemisslab.com/en/mko/mko-finite-commutative-ring/) ## Step-by-step derivation S001 · impliesBefore same parent carrier Fin(8);addition=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 6, 5, 0, 7, 4], [2, 3, 6, 7, 0, 1, 4, 5], [3, 6, 7, 4, 1, 2, 5, 0], [4, 5, 0, 1, 6, 7, 2, 3], [5, 0, 1, 2, 7, 4, 3, 6], [6, 7, 4, 5, 2, 3, 0, 1], [7, 4, 5, 0, 3, 6, 1, 2]];multiplication=[[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 2, 3, 4, 5, 6, 7], [0, 2, 6, 4, 6, 4, 0, 2], [0, 3, 4, 1, 2, 7, 6, 5], [0, 4, 6, 2, 6, 2, 0, 4], [0, 5, 4, 7, 2, 1, 6, 3], [0, 6, 0, 6, 0, 6, 0, 6], [0, 7, 2, 5, 4, 3, 6, 1]] After both complete operation tables typed and closed Conditions: none Bind subsequent ideals or commutativity to this same parent ring.Preserve the complete original tables;carrier labels are not integer arithmetic values. Step identifiers and dependenciesRule: parent_ring_original_complete_tables · Uses: G001 S002 · impliesBefore every additive triple checked;first associativity failure=null;all additive pairs commute=Yes After additive zero candidates=[0];actual zero=0;two-sided additive inverses in label order=[0, 5, 4, 7, 2, 1, 6, 3] Conditions: none Audit additive associativity,commutativity,the two-sided zero and each inverse over the complete carrier.This qualified parent is the problem's premise,checked by complete finite conditions rather than samples. Step identifiers and dependenciesRule: parent_additive_abelian_group_complete_audit · Uses: G001, S001 S003 · impliesBefore every multiplicative triple checked;first associativity failure=null After multiplicative identity candidates=[1];actual one=1 Conditions: none Multiplication here requires associativity and a two-sided one,not commutativity or inverses of every nonzero element. Step identifiers and dependenciesRule: parent_multiplicative_monoid_complete_audit · Uses: G001, S002 S004 · impliesBefore all 512 left and 512 right distributivity triples checked After first left failure=null;first right failure=null;zero=0;one=1;distinct=Yes Conditions: none Audit both distributive obligations completely and separately.Zero and one must be distinct actual elements.Unital alone implies neither a commutative ring,domain nor field. Step identifiers and dependenciesRule: parent_both_distributivities_and_distinct_identities · Uses: G001, S003 S005 · impliesBefore unordered parent elements=[0, 1];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S004 S006 · impliesBefore unordered parent elements=[0, 2];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S005 S007 · impliesBefore unordered parent elements=[0, 3];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S006 S008 · impliesBefore unordered parent elements=[0, 4];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S007 S009 · impliesBefore unordered parent elements=[0, 5];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S008 S010 · impliesBefore unordered parent elements=[0, 6];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S009 S011 · impliesBefore unordered parent elements=[0, 7];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S010 S012 · impliesBefore unordered parent elements=[1, 2];a*b=2;b*a=2 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S011 S013 · impliesBefore unordered parent elements=[1, 3];a*b=3;b*a=3 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S012 S014 · impliesBefore unordered parent elements=[1, 4];a*b=4;b*a=4 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S013 S015 · impliesBefore unordered parent elements=[1, 5];a*b=5;b*a=5 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S014 S016 · impliesBefore unordered parent elements=[1, 6];a*b=6;b*a=6 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S015 S017 · impliesBefore unordered parent elements=[1, 7];a*b=7;b*a=7 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S016 S018 · impliesBefore unordered parent elements=[2, 3];a*b=4;b*a=4 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S017 S019 · impliesBefore unordered parent elements=[2, 4];a*b=6;b*a=6 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S018 S020 · impliesBefore unordered parent elements=[2, 5];a*b=4;b*a=4 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S019 S021 · impliesBefore unordered parent elements=[2, 6];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S020 S022 · impliesBefore unordered parent elements=[2, 7];a*b=2;b*a=2 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S021 S023 · impliesBefore unordered parent elements=[3, 4];a*b=2;b*a=2 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S022 S024 · impliesBefore unordered parent elements=[3, 5];a*b=7;b*a=7 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S023 S025 · impliesBefore unordered parent elements=[3, 6];a*b=6;b*a=6 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S024 S026 · impliesBefore unordered parent elements=[3, 7];a*b=5;b*a=5 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S025 S027 · impliesBefore unordered parent elements=[4, 5];a*b=2;b*a=2 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S026 S028 · impliesBefore unordered parent elements=[4, 6];a*b=0;b*a=0 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S027 S029 · impliesBefore unordered parent elements=[4, 7];a*b=4;b*a=4 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S028 S030 · impliesBefore unordered parent elements=[5, 6];a*b=6;b*a=6 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S029 S031 · impliesBefore unordered parent elements=[5, 7];a*b=3;b*a=3 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S030 S032 · impliesBefore unordered parent elements=[6, 7];a*b=6;b*a=6 After commute=Yes Conditions: none Compare both products for every distinct pair;diagonals agree automatically.One equal pair cannot certify whole-ring commutativity. Step identifiers and dependenciesRule: whole_ring_multiplicative_commutativity_pair · Uses: G001, S031 S033 · impliesBefore all multiplicative noncommuting records=[] After parent ring commutative=Yes;zero divisors not excluded Conditions: none This adds multiplicative commutativity only.It excludes neither zero divisors nor lack of multiplicative inverses. Step identifiers and dependenciesRule: commutative_ring_extra_obligation_not_field · Uses: G001, S032 S034 · impliesBefore a times b:operation=multiplication;left label=1;right label=2;table row=[0, 1, 2, 3, 4, 5, 6, 7] After result label=2 Conditions: none Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels. Step identifiers and dependenciesRule: parent_table_operation_lookup · Uses: G001, S033 S035 · impliesBefore binomial sum:operation=addition;left label=1;right label=2;table row=[1, 2, 3, 6, 5, 0, 7, 4] After result label=3 Conditions: none Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels. Step identifiers and dependenciesRule: parent_table_operation_lookup · Uses: G001, S034 S036 · impliesBefore square of sum:operation=multiplication;left label=3;right label=3;table row=[0, 3, 4, 1, 2, 7, 6, 5] After result label=1 Conditions: none Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels. Step identifiers and dependenciesRule: parent_table_operation_lookup · Uses: G001, S035 S037 · impliesBefore a square:operation=multiplication;left label=1;right label=1;table row=[0, 1, 2, 3, 4, 5, 6, 7] After result label=1 Conditions: none Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels. Step identifiers and dependenciesRule: parent_table_operation_lookup · Uses: G001, S036 S038 · impliesBefore b square:operation=multiplication;left label=2;right label=2;table row=[0, 2, 6, 4, 6, 4, 0, 2] After result label=6 Conditions: none Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels. Step identifiers and dependenciesRule: parent_table_operation_lookup · Uses: G001, S037 S039 · impliesBefore a square plus ab:operation=addition;left label=1;right label=2;table row=[1, 2, 3, 6, 5, 0, 7, 4] After result label=3 Conditions: none Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels. Step identifiers and dependenciesRule: parent_table_operation_lookup · Uses: G001, S038 S040 · impliesBefore add second ab:operation=addition;left label=3;right label=2;table row=[3, 6, 7, 4, 1, 2, 5, 0] After result label=7 Conditions: none Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels. Step identifiers and dependenciesRule: parent_table_operation_lookup · Uses: G001, S039 S041 · impliesBefore add b square:operation=addition;left label=7;right label=6;table row=[7, 4, 5, 0, 3, 6, 1, 2] After result label=1 Conditions: none Look up the declared order in the original operation table.The result remains a carrier label,not the integer sum or product of labels. Step identifiers and dependenciesRule: parent_table_operation_lookup · Uses: G001, S040 S042 · impliesBefore sum label=3;lhs=1;a2=1;ab=2;b2=6;rhs=1 After (a+b)^2=a2+ab+ab+b2 using both distributive laws and ba=ab Conditions: none Expand by both distributive laws to a²+ab+ba+b²,then use qualified commutativity to replace ba by ab.Add the two ab terms with ring addition,not by multiplying with a label named2.Zero divisors do not invalidate this identity. Step identifiers and dependenciesRule: binomial_two_copies_not_integer_label2 · Uses: G001, S041 ## Reverse checks - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (1, 2, 3, 1, 1, 2, 6, 1, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2414cb0336e9fbd0d0a966c1f073521a81f9563decf54df8bbe34f97e2fb62b4 - candidate record SHA-256: e44b1f3be9c7e5cc04aed5dfe44e385bf25c2bd09cfbdd7e7984d6c8223902cc - semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798 - question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r01-finite-commutative-ring-0092.json)