Question 28027: A Finite Newton Orbit Records Steps without Certifying Convergence

Original question and givens

Coefficients in low-to-high degree order are=[-2, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=3 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

• G001 — Coefficients in low-to-high degree order are=[-2, 0, 1]; all iterations use exact rational arithmetic. Use initial iterate=3 and attempted-step budget=1. Give (coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope).

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-finite-orbit, newton-finite-orbit]

derived = [[coefficients_low_to_high, step_budget, step_records, status, final_iterate, convergence_or_rate_proof_claim, finite_scope]]

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

Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient1 = 0

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

Newton ORIGINAL orbit polynomial ORIGINAL derivative coefficient2 = 2

source roles = [Newton ORIGINAL orbit polynomial, [-2, 0, 1]]

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

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2 := mul(0, 3)

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree2 = 0

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 := sum(0, 1)

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 = 1

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 := mul(1, 3)

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 = 3

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 := sum(3, 0)

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 = 3

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 := mul(3, 3)

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 = 9

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 := sum(9, -2)

Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 = 7

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 := mul(0, 3)

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 = 0

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 := sum(0, 2)

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 = 2

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 := mul(2, 3)

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 = 6

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 := sum(6, 0)

Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 = 6

Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime := div(7, 6)

Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime = 7/6

Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update := sub(3, 7/6)

Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update = 11/6

source roles = [Newton ORIGINAL orbit attempt0, [-2, 0, 1], [0, 2], 3]

derived = [7, 6, updated, 11/6]

source roles = [Newton ORIGINAL orbit, 1, No]

derived = [[[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 11/6]

Reverse checks

• S018 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S017 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S016 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 ORIGINAL tangent zero update with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 ORIGINAL f over fprime with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 fprime ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner multiply degree1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 f ORIGINAL Horner add coefficient2 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Newton ORIGINAL orbit attempt0 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 ORIGINAL orbit polynomial ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Newton ORIGINAL orbit 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

([-2, 0, 1], 1, [[0, 3, 7, 6, updated, 11/6]], step_budget_exhausted, 11/6, No, finite_exact_orbit_is_not_a_convergence_or_quadratic_rate_certificate)

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.