# Question 33959: An Affine Residual Equals Absolute Slope Times Forward Error [Return to catalogue page 170](https://ocme.evemisslab.com/en/questions/page/170/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations/#q-20261015-pulse-20261119-r05-0009) 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. R0066-33959 · q-20261015-pulse-20261119-r05-0009 ## Original question and givens Use nonzero slope=-2, intercept=9, approximation=4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). - G001 — Use nonzero slope=-2, intercept=9, approximation=4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction). ## 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-numerical-approximation-error](https://ocme.evemisslab.com/en/mko/mko-numerical-approximation-error/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod50-numerr-affine-residual-forward, [approximation, domain, intercept, operation, slope]] After derived = [numerr-affine-residual-forward, Exact_numerical_error_density] Conditions: none THIS source uses LOCAL-hyphen operations with exact keys/domain.Keep source field order and numeric/Boolean/string roles;integer rational texts omit/1.Original questions,English,MKOs,ASTs and statuses remain unchanged. Step identifiers and dependenciesRule: R66_EXACT_source_bank_keys_local_operation_domain · Uses: G001 S002 · impliesBefore source roles = [error ORIGINAL slope, 100000000, -2, 1] After derived = [[-2, 1], -2] Conditions: none Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S001 S003 · calculationBefore error ORIGINAL slope ORIGINAL numerator over denominator := div(-2, 1) After error ORIGINAL slope ORIGINAL numerator over denominator = -2 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [error ORIGINAL intercept, 100000000, 9, 1] After derived = [[9, 1], 9] Conditions: none Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S003 S005 · calculationBefore error ORIGINAL intercept ORIGINAL numerator over denominator := div(9, 1) After error ORIGINAL intercept ORIGINAL numerator over denominator = 9 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [error ORIGINAL approximation, 100000000, 4, 1] After derived = [[4, 1], 4] Conditions: none Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S005 S007 · calculationBefore error ORIGINAL approximation ORIGINAL numerator over denominator := div(4, 1) After error ORIGINAL approximation ORIGINAL numerator over denominator = 4 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [Affine exact root SOURCE naive div, 100000000, -9, -2] After derived = [[9, 2], 9/2] Conditions: none Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S007 S009 · calculationBefore Affine exact root SOURCE naive div ORIGINAL numerator over denominator := div(-9, -2) After Affine exact root SOURCE naive div ORIGINAL numerator over denominator = 9/2 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Affine exact root ORIGINAL rational div := div(-9, -2) After Affine exact root ORIGINAL rational div = 9/2 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [Affine residual original slope times approximation SOURCE naive mul, 100000000, -8, 1] After derived = [[-8, 1], -8] Conditions: none Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S010 S012 · calculationBefore Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator := div(-8, 1) After Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator = -8 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Affine residual original slope times approximation ORIGINAL rational mul := mul(-2, 4) After Affine residual original slope times approximation ORIGINAL rational mul = -8 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore source roles = [Affine residual add original intercept SOURCE naive sum, 100000000, 1, 1] After derived = [[1, 1], 1] Conditions: none Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S013 S015 · calculationBefore Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator := div(1, 1) After Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator = 1 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Affine residual add original intercept ORIGINAL rational sum := sum(-8, 9) After Affine residual add original intercept ORIGINAL rational sum = 1 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [Affine forward original approximation minus exact root SOURCE naive sub, 100000000, -1, 2] After derived = [[-1, 2], -1/2] Conditions: none Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S016 S018 · calculationBefore Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator := div(-1, 2) After Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator = -1/2 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Affine forward original approximation minus exact root ORIGINAL rational sub := sub(4, 9/2) After Affine forward original approximation minus exact root ORIGINAL rational sub = -1/2 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore source roles = [Affine absolute slope times forward error SOURCE naive mul, 100000000, 2, 2] After derived = [[1, 1], 1] Conditions: none Bound the SOURCE PRE-reduction numerator/denominator and require a nonzero denominator BEFORE sign/gcd normalization.MOD50 uses100million with NAIVE operations,not prior cross-cancellation;this Wave4 mean uses one million.Out-of-envelope is a source-slice hold,not mathematical falsity. Step identifiers and dependenciesRule: R66_SOURCE_PREreduction_components_then_sign_gcd · Uses: G001, S019 S021 · calculationBefore Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator := div(2, 2) After Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator = 1 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Affine absolute slope times forward error ORIGINAL rational mul := mul(2, 1/2) After Affine absolute slope times forward error ORIGINAL rational mul = 1 Conditions: none Calculate exactly from THIS step’s ORIGINAL ordered operands.Retain the SOURCE rational pre-reduction envelope separately;do not use floating values or silently widen a different bank. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore source roles = [-2, 9, 4] After derived = [9/2, 1, 1, -1/2, 1/2, 1, Yes] Conditions: none For m≠0 first solve the TRUE root−b/m.Calculate |m*xhat+b| and |xhat−root| SEPARATELY.Since m*root+b=0,residual=|m|*forward.Occasional numerical equality does not identify the definitions;the supplied model gives the true root. Step identifiers and dependenciesRule: Error_AFFINE_exact_root_residual_forward_scale_identity · Uses: G001, S022 ## Reverse checks - S023 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S022 · premise_and_result_audit — Recompute Affine absolute slope times forward error ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S019 · premise_and_result_audit — Recompute Affine forward original approximation minus exact root ORIGINAL rational sub with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S016 · premise_and_result_audit — Recompute Affine residual add original intercept ORIGINAL rational sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S013 · premise_and_result_audit — Recompute Affine residual original slope times approximation ORIGINAL rational mul with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S010 · premise_and_result_audit — Recompute Affine exact root ORIGINAL rational div with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Affine exact root SOURCE naive div ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S007 · premise_and_result_audit — Recompute error ORIGINAL approximation ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S005 · premise_and_result_audit — Recompute error ORIGINAL intercept ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S003 · premise_and_result_audit — Recompute error ORIGINAL slope ORIGINAL numerator over denominator with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. - S001 · premise_and_result_audit — Reverse-audit THIS step against its ORIGINAL ordered roles and complete premises.Accept the conclusion only under these conditions;do not invert a one-way implication or call a finite check a general formal proof. ## Result (-2, 9, 4, 9/2, 1, 1/2, 1, Yes, residual_and_forward_error_are_distinct_quantities) 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: 00c1ab9ea8bd7f75c398b974689203308a0f1cdabef471f3ad3855ead1bda567 - candidate record SHA-256: 073d20f12f17416c779f51fdcd17103064d124ded374c6c9939d41ae5358c683 - semantic review receipt SHA-256: 99b4f114b2ec93164b6aa5681e0107033a240e7a3f0a8988633ba65320ede267 - question-first witness SHA-256: d9bd3a1940da8a73f790ee3fe460c9a8523791bdbe72c8e0304e3740b66710cc - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r05-0009.json)