# Question 2627: Audit Every Table Row against an Affine Rule [Return to catalogue page 14](https://ocme.evemisslab.com/en/questions/page/14/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261021-r01-mathematical-representation-foundations/#q-20261013-advance-20261021-r01-mathematical-representation-0077) 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. R0004-02627 · q-20261013-advance-20261021-r01-mathematical-representation-0077 ## Original question and givens Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=-9, pairs=[[0, -9], [1, -10], [2, -11]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?). - G001 — Use only the explicitly supplied rule, symbols and finite pair table. Graph appearance supplies no values, and agreement on the listed inputs proves no equivalence on unlisted domains. a=-1, b=-9, pairs=[[0, -9], [1, -10], [2, -11]]. Normalize tuple collections by removing identical tuples and sorting their JSON strings with the source’s en-locale comparison. This is canonical record order, not numerical coordinate order; number any output rows from 0 only after that normalization. Give (a,b,every [x,listed y,a*x+b,match?] row,all rows match?). ## 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-mathematical-representation](https://ocme.evemisslab.com/en/mko/mko-mathematical-representation/) ## Step-by-step derivation S001 · impliesBefore declared symbolic rule or complete finite table After keep the stated domain and representation distinct Conditions: none Use only the declared rule or table;agreement on a finite table alone does not establish function equality on an unlisted domain. Step identifiers and dependenciesRule: representation_scope · Uses: G001 S002 · calculationBefore audit x=0 ax := mul(-1, 0) After audit x=0 ax = 0 Conditions: none Evaluate a·x first;do not replace the rule by a(x+b). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore audit x=0 ax+b := sum(0, -9) After audit x=0 ax+b = -9 Conditions: none Then add the intercept b,fully substituting this input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore listed y=-9; rule ax+b=-9 After agreement=Yes Conditions: none Compare each listed output with the exact rule value;visual proximity is not equality. Step identifiers and dependenciesRule: representation_row_agreement · Uses: G001, S003 S005 · calculationBefore audit x=1 ax := mul(-1, 1) After audit x=1 ax = -1 Conditions: none Evaluate a·x first;do not replace the rule by a(x+b). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore audit x=1 ax+b := sum(-1, -9) After audit x=1 ax+b = -10 Conditions: none Then add the intercept b,fully substituting this input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore listed y=-10; rule ax+b=-10 After agreement=Yes Conditions: none Compare each listed output with the exact rule value;visual proximity is not equality. Step identifiers and dependenciesRule: representation_row_agreement · Uses: G001, S006 S008 · calculationBefore audit x=2 ax := mul(-1, 2) After audit x=2 ax = -2 Conditions: none Evaluate a·x first;do not replace the rule by a(x+b). Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore audit x=2 ax+b := sum(-2, -9) After audit x=2 ax+b = -11 Conditions: none Then add the intercept b,fully substituting this input. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore listed y=-11; rule ax+b=-11 After agreement=Yes Conditions: none Compare each listed output with the exact rule value;visual proximity is not equality. Step identifiers and dependenciesRule: representation_row_agreement · Uses: G001, S009 S011 · impliesBefore [Yes, Yes, Yes] After Yes Conditions: none All listed rows must agree;one counterexample refutes consistency of this table with the rule. Step identifiers and dependenciesRule: all_representation_rows · Uses: G001, S010 ## Reverse checks - 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 — Recompute audit x=2 ax+b with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute audit x=2 ax with exact rational arithmetic, not rounded decimals. - 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 — Recompute audit x=1 ax+b with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute audit x=1 ax with exact rational arithmetic, not rounded decimals. - 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 — Recompute audit x=0 ax+b with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute audit x=0 ax with exact rational arithmetic, not rounded decimals. - 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, -9, [[0, -9, -9, Yes], [1, -10, -10, Yes], [2, -11, -11, Yes]], 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: a550dcdd0666a61f4bfb0391d2cd8ddbffa239cb7bc8a16e044445b389e66f9d - candidate record SHA-256: fab402ca818ab400e30680d3ff97e5b35e6fc2a240c0d6b454d67a389e382df8 - semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56 - question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261021-r01-mathematical-representation-0077.json)