# Question 30661: Both Affine Composition Orders with Slope-First Coefficients [Return to catalogue page 154](https://ocme.evemisslab.com/en/questions/page/154/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r04-composition-foundations/#q-20261011-advance-20261015-r04-composition-0061) 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. R0060-30661 · q-20261011-advance-20261015-r04-composition-0061 ## Original question and givens Total Q functions f(x)=(9)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant). - G001 — Total Q functions f(x)=(9)x+(0), g(x)=(-2)x+(1); ordered inputs=[0/1, 1/2, -1/1]. Give (g∘f [slope, constant], f∘g [slope, constant], three values for g∘f, three values for f∘g, g∘f constant minus f∘g constant). ## 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-composition](https://ocme.evemisslab.com/en/mko/mko-composition/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density15-composition-noncommuting-affine, density15_composition_noncommuting_affine, finite_explicit_composition_protocol] After derived = [[GF_SLOPE_constant, FG_SLOPE_constant, GF_values, FG_values, SIGNED_constant_difference]] Conditions: none density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status. Step identifiers and dependenciesRule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001 S002 · impliesBefore source roles = [Composition original rational input0, [0, 1]] After derived = [0/1] Conditions: none Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1. Step identifiers and dependenciesRule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S001 S003 · impliesBefore source roles = [Composition original rational input1, [1, 2]] After derived = [1/2] Conditions: none Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1. Step identifiers and dependenciesRule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S002 S004 · impliesBefore source roles = [Composition original rational input2, [-1, 1]] After derived = [-1/1] Conditions: none Both RAW pair components are SAFE and the denominator is nonzero.Negative denominators normalize sign/gcd;derived fractions are not SAFE/1e6-capped.Integer fractions retain/1. Step identifiers and dependenciesRule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S003 S005 · calculationBefore Affine GF ORIGINAL c times a slope := mul(-2, 9) After Affine GF ORIGINAL c times a slope = -18 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Affine GF ORIGINAL c times b := mul(-2, 0) After Affine GF ORIGINAL c times b = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Affine GF ORIGINAL constant cb plus d := sum(0, 1) After Affine GF ORIGINAL constant cb plus d = 1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Affine FG ORIGINAL a times c slope := mul(9, -2) After Affine FG ORIGINAL a times c slope = -18 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Affine FG ORIGINAL a times d := mul(9, 1) After Affine FG ORIGINAL a times d = 9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Affine FG ORIGINAL constant ad plus b := sum(9, 0) After Affine FG ORIGINAL constant ad plus b = 9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Affine ORIGINAL f input0 multiply := mul(9, 0) After Affine ORIGINAL f input0 multiply = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Affine ORIGINAL f input0 plus b := sum(0, 0) After Affine ORIGINAL f input0 plus b = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Affine ORIGINAL g AFTER f input0 multiply := mul(-2, 0) After Affine ORIGINAL g AFTER f input0 multiply = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Affine ORIGINAL g AFTER f input0 plus d := sum(0, 1) After Affine ORIGINAL g AFTER f input0 plus d = 1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Affine ORIGINAL g input0 multiply := mul(-2, 0) After Affine ORIGINAL g input0 multiply = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Affine ORIGINAL g input0 plus d := sum(0, 1) After Affine ORIGINAL g input0 plus d = 1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Affine ORIGINAL f AFTER g input0 multiply := mul(9, 1) After Affine ORIGINAL f AFTER g input0 multiply = 9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Affine ORIGINAL f AFTER g input0 plus b := sum(9, 0) After Affine ORIGINAL f AFTER g input0 plus b = 9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Affine reverse GF coefficient input0 product := mul(-18, 0) After Affine reverse GF coefficient input0 product = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Affine reverse GF coefficient input0 sum := sum(0, 1) After Affine reverse GF coefficient input0 sum = 1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Affine reverse FG coefficient input0 product := mul(-18, 0) After Affine reverse FG coefficient input0 product = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Affine reverse FG coefficient input0 sum := sum(0, 9) After Affine reverse FG coefficient input0 sum = 9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Affine ORIGINAL f input1 multiply := mul(9, 1/2) After Affine ORIGINAL f input1 multiply = 9/2 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Affine ORIGINAL f input1 plus b := sum(9/2, 0) After Affine ORIGINAL f input1 plus b = 9/2 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Affine ORIGINAL g AFTER f input1 multiply := mul(-2, 9/2) After Affine ORIGINAL g AFTER f input1 multiply = -9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Affine ORIGINAL g AFTER f input1 plus d := sum(-9, 1) After Affine ORIGINAL g AFTER f input1 plus d = -8 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Affine ORIGINAL g input1 multiply := mul(-2, 1/2) After Affine ORIGINAL g input1 multiply = -1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Affine ORIGINAL g input1 plus d := sum(-1, 1) After Affine ORIGINAL g input1 plus d = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Affine ORIGINAL f AFTER g input1 multiply := mul(9, 0) After Affine ORIGINAL f AFTER g input1 multiply = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Affine ORIGINAL f AFTER g input1 plus b := sum(0, 0) After Affine ORIGINAL f AFTER g input1 plus b = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Affine reverse GF coefficient input1 product := mul(-18, 1/2) After Affine reverse GF coefficient input1 product = -9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Affine reverse GF coefficient input1 sum := sum(-9, 1) After Affine reverse GF coefficient input1 sum = -8 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Affine reverse FG coefficient input1 product := mul(-18, 1/2) After Affine reverse FG coefficient input1 product = -9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Affine reverse FG coefficient input1 sum := sum(-9, 9) After Affine reverse FG coefficient input1 sum = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Affine ORIGINAL f input2 multiply := mul(9, -1) After Affine ORIGINAL f input2 multiply = -9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Affine ORIGINAL f input2 plus b := sum(-9, 0) After Affine ORIGINAL f input2 plus b = -9 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Affine ORIGINAL g AFTER f input2 multiply := mul(-2, -9) After Affine ORIGINAL g AFTER f input2 multiply = 18 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Affine ORIGINAL g AFTER f input2 plus d := sum(18, 1) After Affine ORIGINAL g AFTER f input2 plus d = 19 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Affine ORIGINAL g input2 multiply := mul(-2, -1) After Affine ORIGINAL g input2 multiply = 2 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore Affine ORIGINAL g input2 plus d := sum(2, 1) After Affine ORIGINAL g input2 plus d = 3 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Affine ORIGINAL f AFTER g input2 multiply := mul(9, 3) After Affine ORIGINAL f AFTER g input2 multiply = 27 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Affine ORIGINAL f AFTER g input2 plus b := sum(27, 0) After Affine ORIGINAL f AFTER g input2 plus b = 27 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Affine reverse GF coefficient input2 product := mul(-18, -1) After Affine reverse GF coefficient input2 product = 18 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Affine reverse GF coefficient input2 sum := sum(18, 1) After Affine reverse GF coefficient input2 sum = 19 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore Affine reverse FG coefficient input2 product := mul(-18, -1) After Affine reverse FG coefficient input2 product = 18 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Affine reverse FG coefficient input2 sum := sum(18, 9) After Affine reverse FG coefficient input2 sum = 27 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Affine ORIGINAL GF constant minus FG constant := sub(1, 9) After Affine ORIGINAL GF constant minus FG constant = -8 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · impliesBefore source roles = [9, 0, -2, 1, [0/1, 1/2, -1/1]] After derived = [[-18, 1], [-18, 9], [1/1, -8/1, 19/1], [9/1, 0/1, 27/1], -8] Conditions: none Substitute twice in the WHOLE Q-affine laws:g(f(x))=(ca)x+(cb+d),f(g(x))=(ac)x+(ad+b).Pairs are[SLOPE,constant],NOT ascending powers.Commuting/constant instances are legal;three samples do not prove arbitrary functions commute. Step identifiers and dependenciesRule: Composition_TOTAL_Q_affine_SUBSTITUTE_each_order_SLOPE_FIRST · Uses: G001, S047 ## Reverse checks - S048 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S047 · premise_and_result_audit — Recompute Affine ORIGINAL GF constant minus FG constant with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Affine reverse FG coefficient input2 sum with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Affine reverse FG coefficient input2 product with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Affine reverse GF coefficient input2 sum with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Affine reverse GF coefficient input2 product with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Affine ORIGINAL f AFTER g input2 plus b with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Affine ORIGINAL f AFTER g input2 multiply with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Affine ORIGINAL g input2 plus d with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Affine ORIGINAL g input2 multiply with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Affine ORIGINAL g AFTER f input2 plus d with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Affine ORIGINAL g AFTER f input2 multiply with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Affine ORIGINAL f input2 plus b with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Affine ORIGINAL f input2 multiply with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Affine reverse FG coefficient input1 sum with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Affine reverse FG coefficient input1 product with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Affine reverse GF coefficient input1 sum with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Affine reverse GF coefficient input1 product with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Affine ORIGINAL f AFTER g input1 plus b with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Affine ORIGINAL f AFTER g input1 multiply with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Affine ORIGINAL g input1 plus d with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Affine ORIGINAL g input1 multiply with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Affine ORIGINAL g AFTER f input1 plus d with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Affine ORIGINAL g AFTER f input1 multiply with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Affine ORIGINAL f input1 plus b with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Affine ORIGINAL f input1 multiply with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Affine reverse FG coefficient input0 sum with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Affine reverse FG coefficient input0 product with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Affine reverse GF coefficient input0 sum with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Affine reverse GF coefficient input0 product with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Affine ORIGINAL f AFTER g input0 plus b with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Affine ORIGINAL f AFTER g input0 multiply with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Affine ORIGINAL g input0 plus d with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Affine ORIGINAL g input0 multiply with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Affine ORIGINAL g AFTER f input0 plus d with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Affine ORIGINAL g AFTER f input0 multiply with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Affine ORIGINAL f input0 plus b with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Affine ORIGINAL f input0 multiply with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Affine FG ORIGINAL constant ad plus b with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Affine FG ORIGINAL a times d with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Affine FG ORIGINAL a times c slope with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Affine GF ORIGINAL constant cb plus d with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Affine GF ORIGINAL c times b with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Affine GF ORIGINAL c times a slope with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S002 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. ## Result ([-18, 1], [-18, 9], [1/1, -8/1, 19/1], [9/1, 0/1, 27/1], -8) 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: 63066055bf852086da93f2708b9d5616cb61a7f4e2bd520743ffcf20ecbc077d - candidate record SHA-256: c26af5bfca69fb69760e1219ac0d80b7391c5a02ea870ce909008eb1596818c3 - semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf - question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r04-composition-0061.json)