# Question 28012: Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct [Return to catalogue page 141](https://ocme.evemisslab.com/en/questions/page/141/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r01-newton-method-foundations/#q-20261015-pulse-20261119-r01-0087) 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. R0055-28012 · q-20261015-pulse-20261119-r01-0087 ## Original question and givens Use mode=residual_forward_confusion, complete model={"slope":13,"intercept":-12,"approximation":14}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation). - G001 — Use mode=residual_forward_confusion, complete model={"slope":13,"intercept":-12,"approximation":14}. Give (mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation). ## 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-newton-method](https://ocme.evemisslab.com/en/mko/mko-newton-method/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod50-newton-boundary, newton-boundary] After derived = [[mode, slope, exact_root, absolute_residual, forward_error, residual_equals_forward_error, scale_relation]] Conditions: none List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged. Step identifiers and dependenciesRule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Newton residual ORIGINAL affine root := div(12, 13) After Newton residual ORIGINAL affine root = 12/13 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Newton residual ORIGINAL affine product := mul(13, 14) After Newton residual ORIGINAL affine product = 182 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Newton residual ORIGINAL affine value := sum(182, -12) After Newton residual ORIGINAL affine value = 170 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Newton residual ORIGINAL approximation minus root := sub(14, 12/13) After Newton residual ORIGINAL approximation minus root = 170/13 Conditions: none Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [13, -12, 14, 12/13] After derived = [170, 170/13, No] Conditions: none f(x)=m(x-root),so residual=|m|*forward.The equality flag is an INSTANCE result;both can be0.Do not identify residual with forward error universally. Step identifiers and dependenciesRule: Newton_residual_FORWARD_distinct_quantities_actual_affine_scale · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. - S005 · premise_and_result_audit — Recompute Newton residual ORIGINAL approximation minus root with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Newton residual ORIGINAL affine value with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Newton residual ORIGINAL affine product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Newton residual ORIGINAL affine root with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. ## Result (residual_forward_confusion, 13, 12/13, 170, 170/13, No, residual_equals_abs_slope_times_forward_error_not_forward_error_itself) 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: eb30c4b8e888347dc27d4bd3a9142c948867342cf9d9cbc8c3d296974d162dbf - candidate record SHA-256: 2085ae129514fc2640a5577cc477a2e01da4bd3dfac4799a12d7792e9be01912 - semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683 - question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r01-0087.json)