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Question 30675: Both Affine Composition Orders with Slope-First Coefficients

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0060-30675 · q-20261011-advance-20261015-r04-composition-0075

Original question and givens

Total Q functions f(x)=(23)x+(-1), 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)=(23)x+(-1), 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001

S002 · implies

Before

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 dependencies

Rule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: R60_RAW_SAFE_rational_pair_NONZERO_den_SIGN_normalized · Uses: G001, S003

S005 · calculation

Before

Affine GF ORIGINAL c times a slope := mul(-2, 23)

After

Affine GF ORIGINAL c times a slope = -46

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Affine GF ORIGINAL c times b := mul(-2, -1)

After

Affine GF ORIGINAL c times b = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Affine GF ORIGINAL constant cb plus d := sum(2, 1)

After

Affine GF ORIGINAL constant cb 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Affine FG ORIGINAL a times c slope := mul(23, -2)

After

Affine FG ORIGINAL a times c slope = -46

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Affine FG ORIGINAL a times d := mul(23, 1)

After

Affine FG ORIGINAL a times d = 23

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Affine FG ORIGINAL constant ad plus b := sum(23, -1)

After

Affine FG ORIGINAL constant ad plus b = 22

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Affine ORIGINAL f input0 multiply := mul(23, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Affine ORIGINAL f input0 plus b := sum(0, -1)

After

Affine ORIGINAL f input0 plus b = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Affine ORIGINAL g AFTER f input0 multiply := mul(-2, -1)

After

Affine ORIGINAL g AFTER f input0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Affine ORIGINAL g AFTER f input0 plus d := sum(2, 1)

After

Affine ORIGINAL g AFTER f input0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Affine ORIGINAL f AFTER g input0 multiply := mul(23, 1)

After

Affine ORIGINAL f AFTER g input0 multiply = 23

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Affine ORIGINAL f AFTER g input0 plus b := sum(23, -1)

After

Affine ORIGINAL f AFTER g input0 plus b = 22

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Affine reverse GF coefficient input0 product := mul(-46, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Affine reverse GF coefficient input0 sum := sum(0, 3)

After

Affine reverse GF coefficient input0 sum = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Affine reverse FG coefficient input0 product := mul(-46, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Affine reverse FG coefficient input0 sum := sum(0, 22)

After

Affine reverse FG coefficient input0 sum = 22

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Affine ORIGINAL f input1 multiply := mul(23, 1/2)

After

Affine ORIGINAL f input1 multiply = 23/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Affine ORIGINAL f input1 plus b := sum(23/2, -1)

After

Affine ORIGINAL f input1 plus b = 21/2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Affine ORIGINAL g AFTER f input1 multiply := mul(-2, 21/2)

After

Affine ORIGINAL g AFTER f input1 multiply = -21

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

Affine ORIGINAL g AFTER f input1 plus d := sum(-21, 1)

After

Affine ORIGINAL g AFTER f input1 plus d = -20

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Affine ORIGINAL f AFTER g input1 multiply := mul(23, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Affine ORIGINAL f AFTER g input1 plus b := sum(0, -1)

After

Affine ORIGINAL f AFTER g input1 plus b = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Affine reverse GF coefficient input1 product := mul(-46, 1/2)

After

Affine reverse GF coefficient input1 product = -23

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Affine reverse GF coefficient input1 sum := sum(-23, 3)

After

Affine reverse GF coefficient input1 sum = -20

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Affine reverse FG coefficient input1 product := mul(-46, 1/2)

After

Affine reverse FG coefficient input1 product = -23

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Affine reverse FG coefficient input1 sum := sum(-23, 22)

After

Affine reverse FG coefficient input1 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Affine ORIGINAL f input2 multiply := mul(23, -1)

After

Affine ORIGINAL f input2 multiply = -23

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Affine ORIGINAL f input2 plus b := sum(-23, -1)

After

Affine ORIGINAL f input2 plus b = -24

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Affine ORIGINAL g AFTER f input2 multiply := mul(-2, -24)

After

Affine ORIGINAL g AFTER f input2 multiply = 48

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

Affine ORIGINAL g AFTER f input2 plus d := sum(48, 1)

After

Affine ORIGINAL g AFTER f input2 plus d = 49

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Affine ORIGINAL f AFTER g input2 multiply := mul(23, 3)

After

Affine ORIGINAL f AFTER g input2 multiply = 69

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

Affine ORIGINAL f AFTER g input2 plus b := sum(69, -1)

After

Affine ORIGINAL f AFTER g input2 plus b = 68

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Affine reverse GF coefficient input2 product := mul(-46, -1)

After

Affine reverse GF coefficient input2 product = 46

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

Affine reverse GF coefficient input2 sum := sum(46, 3)

After

Affine reverse GF coefficient input2 sum = 49

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

Affine reverse FG coefficient input2 product := mul(-46, -1)

After

Affine reverse FG coefficient input2 product = 46

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

Affine reverse FG coefficient input2 sum := sum(46, 22)

After

Affine reverse FG coefficient input2 sum = 68

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

Affine ORIGINAL GF constant minus FG constant := sub(3, 22)

After

Affine ORIGINAL GF constant minus FG constant = -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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

source roles = [23, -1, -2, 1, [0/1, 1/2, -1/1]]

After

derived = [[-46, 3], [-46, 22], [3/1, -20/1, 49/1], [22/1, -1/1, 68/1], -19]

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 dependencies

Rule: 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

([-46, 3], [-46, 22], [3/1, -20/1, 49/1], [22/1, -1/1, 68/1], -19)

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: d7fedb23f935b28f56837374f0dd5a3c97892282ca61ab7efdb91d609257d382
  • candidate record SHA-256: ab4b7b75c0108539779f0285d709c8419c84d9d15dad3e24fd132e9cb1b81a73
  • semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf
  • question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON