Question 28069: Positive Square-Root Iterates Use Exact Rational Newton Steps

Original question and givens

Use positive radicand A=20, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope).

• G001 — Use positive radicand A=20, positive initial iterate=5, and attempted-step budget=1; f(x)=x^2−A. Give (radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_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-square-root-orbit, newton-square-root-orbit]

derived = [[radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope]]

Newton sqrt attempt0 ORIGINAL square := mul(5, 5)

Newton sqrt attempt0 ORIGINAL square = 25

Newton sqrt attempt0 ORIGINAL signed residual := sub(25, 20)

Newton sqrt attempt0 ORIGINAL signed residual = 5

Newton sqrt attempt0 ORIGINAL A over x := div(20, 5)

Newton sqrt attempt0 ORIGINAL A over x = 4

Newton sqrt attempt0 ORIGINAL x plus A over x := sum(5, 4)

Newton sqrt attempt0 ORIGINAL x plus A over x = 9

Newton sqrt attempt0 ORIGINAL half update := div(9, 2)

Newton sqrt attempt0 ORIGINAL half update = 9/2

Newton sqrt ORIGINAL final square EVEN steps0 := mul(9/2, 9/2)

Newton sqrt ORIGINAL final square EVEN steps0 = 81/4

Newton sqrt ORIGINAL final signed residual := sub(81/4, 20)

Newton sqrt ORIGINAL final signed residual = 1/4

source roles = [20, 5, 1]

derived = [[[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, No]

Reverse checks

• S009 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
• S008 · premise_and_result_audit — Recompute Newton sqrt ORIGINAL final signed residual with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Newton sqrt ORIGINAL final square EVEN steps0 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Newton sqrt attempt0 ORIGINAL half update with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Newton sqrt attempt0 ORIGINAL x plus A over x with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Newton sqrt attempt0 ORIGINAL A over x with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Newton sqrt attempt0 ORIGINAL signed residual with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Newton sqrt attempt0 ORIGINAL square 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

(20, 5, 1, [[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, No, finite_square_root_orbit_does_not_prove_a_general_Newton_theorem)

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.