Question 28005: Root-First Stopping, Zero Derivatives and Residual Scaling Stay Distinct

Original question and givens

Use mode=zero_derivative, complete model={"coefficients":[5,0,1],"x":0}. Give (mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary).

• G001 — Use mode=zero_derivative, complete model={"coefficients":[5,0,1],"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

Step-by-step derivation

source roles = [qf-mod50-newton-boundary, newton-boundary]

derived = [[mode, current_iterate, function_value, derivative_value, status, update_qualification_claim, boundary]]

Newton boundary polynomial ORIGINAL derivative coefficient1 := mul(1, 0)

Newton boundary polynomial ORIGINAL derivative coefficient1 = 0

Newton boundary polynomial ORIGINAL derivative coefficient2 := mul(2, 1)

Newton boundary polynomial ORIGINAL derivative coefficient2 = 2

source roles = [Newton boundary polynomial, [5, 0, 1]]

derived = [[5, 0, 1], [0, 2]]

Newton boundary record f ORIGINAL Horner multiply degree2 := mul(0, 0)

Newton boundary record f ORIGINAL Horner multiply degree2 = 0

Newton boundary record f ORIGINAL Horner add coefficient2 := sum(0, 1)

Newton boundary record f ORIGINAL Horner add coefficient2 = 1

Newton boundary record f ORIGINAL Horner multiply degree1 := mul(1, 0)

Newton boundary record f ORIGINAL Horner multiply degree1 = 0

Newton boundary record f ORIGINAL Horner add coefficient1 := sum(0, 0)

Newton boundary record f ORIGINAL Horner add coefficient1 = 0

Newton boundary record f ORIGINAL Horner multiply degree0 := mul(0, 0)

Newton boundary record f ORIGINAL Horner multiply degree0 = 0

Newton boundary record f ORIGINAL Horner add coefficient0 := sum(0, 5)

Newton boundary record f ORIGINAL Horner add coefficient0 = 5

Newton boundary record fprime ORIGINAL Horner multiply degree1 := mul(0, 0)

Newton boundary record fprime ORIGINAL Horner multiply degree1 = 0

Newton boundary record fprime ORIGINAL Horner add coefficient1 := sum(0, 2)

Newton boundary record fprime ORIGINAL Horner add coefficient1 = 2

Newton boundary record fprime ORIGINAL Horner multiply degree0 := mul(2, 0)

Newton boundary record fprime ORIGINAL Horner multiply degree0 = 0

Newton boundary record fprime ORIGINAL Horner add coefficient0 := sum(0, 0)

Newton boundary record fprime ORIGINAL Horner add coefficient0 = 0

source roles = [Newton boundary record, [5, 0, 1], [0, 2], 0]

derived = [5, 0, undefined_zero_derivative, null]

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, 5, 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.