# Question 28009: 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-r01-0084) 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. R0055-28009 · q-20261015-pulse-20261119-r01-0084 ## Original question and givens Use mode=zero_derivative, complete model={"coefficients":[9,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary). - G001 — Use mode=zero_derivative, complete model={"coefficients":[9,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary). ## 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, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary]] 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 boundary polynomial ORIGINAL derivative coefficient1 := mul(1, 0) After Newton boundary polynomial ORIGINAL derivative coefficient1 = 0 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 boundary polynomial ORIGINAL derivative coefficient2 := mul(2, 1) After Newton boundary polynomial ORIGINAL derivative coefficient2 = 2 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 · impliesBefore source roles = [Newton boundary polynomial, [9, 0, 1]] After derived = [[9, 0, 1], [0, 2]] Conditions: none Require raw length2..6 before trimming trailing zeros,and remain nonconstant.Differentiate every monomial to i*c_i*x^(i-1).Source derivative coefficients must pass their1e8 profile even if a root later stops or steps=0. Step identifiers and dependenciesRule: Newton_polynomial_raw_arity_trim_nonconstant_AND_ALL_derivative_coefficients · Uses: G001, S003 S005 · calculationBefore Newton boundary record f ORIGINAL Horner multiply degree2 := mul(0, 0) After Newton boundary record f ORIGINAL Horner multiply degree2 = 0 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 · calculationBefore Newton boundary record f ORIGINAL Horner add coefficient2 := sum(0, 1) After Newton boundary record f ORIGINAL Horner add coefficient2 = 1 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, S005 S007 · calculationBefore Newton boundary record f ORIGINAL Horner multiply degree1 := mul(1, 0) After Newton boundary record f ORIGINAL Horner multiply degree1 = 0 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, S006 S008 · calculationBefore Newton boundary record f ORIGINAL Horner add coefficient1 := sum(0, 0) After Newton boundary record f ORIGINAL Horner add coefficient1 = 0 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, S007 S009 · calculationBefore Newton boundary record f ORIGINAL Horner multiply degree0 := mul(0, 0) After Newton boundary record f ORIGINAL Horner multiply degree0 = 0 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, S008 S010 · calculationBefore Newton boundary record f ORIGINAL Horner add coefficient0 := sum(0, 9) After Newton boundary record f ORIGINAL Horner add coefficient0 = 9 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, S009 S011 · calculationBefore Newton boundary record fprime ORIGINAL Horner multiply degree1 := mul(0, 0) After Newton boundary record fprime ORIGINAL Horner multiply degree1 = 0 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, S010 S012 · calculationBefore Newton boundary record fprime ORIGINAL Horner add coefficient1 := sum(0, 2) After Newton boundary record fprime ORIGINAL Horner add coefficient1 = 2 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, S011 S013 · calculationBefore Newton boundary record fprime ORIGINAL Horner multiply degree0 := mul(2, 0) After Newton boundary record fprime ORIGINAL Horner multiply degree0 = 0 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, S012 S014 · calculationBefore Newton boundary record fprime ORIGINAL Horner add coefficient0 := sum(0, 0) After Newton boundary record fprime ORIGINAL Horner add coefficient0 = 0 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, S013 S015 · impliesBefore source roles = [Newton boundary record, [9, 0, 1], [0, 2], 0] After derived = [9, 0, undefined_zero_derivative, null] Conditions: none Evaluate BOTH f and f_prime completely,then test f=0 FIRST.An exact root stops even at slope0,without0/0.Only a nonroot requires nonzero slope;the tangent-line zero solves z=x-f/f_prime. Step identifiers and dependenciesRule: Newton_EACH_f_AND_fprime_ROOT_FIRST_then_nonzero_derivative_division · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. - S014 · premise_and_result_audit — Recompute Newton boundary record fprime ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Newton boundary record fprime ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Newton boundary record fprime ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Newton boundary record fprime ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Newton boundary record f ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Newton boundary record f ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Newton boundary record f ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Newton boundary record f ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Newton boundary record f ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Newton boundary record f ORIGINAL Horner multiply degree2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct. - S003 · premise_and_result_audit — Recompute Newton boundary polynomial ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Newton boundary polynomial ORIGINAL derivative coefficient1 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 (zero_derivative, 0, 9, 0, undefined_zero_derivative, No, division_by_zero_is_rejected_not_repaired) 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: abc6f30d61f7fb977f26220cae3bac803b0dbed6a8f2998bfa25e20cd73de796 - candidate record SHA-256: 07032af1b7c87fe0a88e3bc8e5b3bf3d3bb35a77948e5b3fad0b54f680f7a4c9 - 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-r01-0084.json)