# Question 28069: Positive Square-Root Iterates Use Exact Rational Newton Steps [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-r01-0069) 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. R0055-28069 · q-20261015-pulse-20261119-r01-0069 ## 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](https://ocme.evemisslab.com/en/mko/mko-newton-method/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod50-newton-square-root-orbit, newton-square-root-orbit] After derived = [[radicand, initial_iterate, step_budget, step_records, status, final_iterate, final_absolute_residual, general_Newton_proof_claim, finite_scope]] 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 sqrt attempt0 ORIGINAL square := mul(5, 5) After Newton sqrt attempt0 ORIGINAL square = 25 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 sqrt attempt0 ORIGINAL signed residual := sub(25, 20) After Newton sqrt attempt0 ORIGINAL signed residual = 5 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 sqrt attempt0 ORIGINAL A over x := div(20, 5) After Newton sqrt attempt0 ORIGINAL A over x = 4 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 sqrt attempt0 ORIGINAL x plus A over x := sum(5, 4) After Newton sqrt attempt0 ORIGINAL x plus A over x = 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, S004 S006 · calculationBefore Newton sqrt attempt0 ORIGINAL half update := div(9, 2) After Newton sqrt attempt0 ORIGINAL half update = 9/2 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, S005 S007 · calculationBefore Newton sqrt ORIGINAL final square EVEN steps0 := mul(9/2, 9/2) After Newton sqrt ORIGINAL final square EVEN steps0 = 81/4 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 sqrt ORIGINAL final signed residual := sub(81/4, 20) After Newton sqrt ORIGINAL final signed residual = 1/4 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 · impliesBefore source roles = [20, 5, 1] After derived = [[[0, 5, 5, updated, 9/2]], step_budget_exhausted, 9/2, 1/4, No] Conditions: none Positive A/x0 keep(x+A/x)/2 positive and division defined;an exact root stops first.Always compute the final absolute residual,even at steps0.It is|x²-A|,not an unprovided forward error|x-sqrt(A)|,and a finite orbit is not a general convergence certificate. Step identifiers and dependenciesRule: Newton_sqrt_positive_iteration_ROOT_FIRST_residual_NOT_forward_error · Uses: G001, S008 ## 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 5dc376e062c37d1304f15ec3b229743e0865cf6dc57e796579961ffeecf64327 - candidate record SHA-256: 5ffaaa72208f20822070a80a3990063a38d26a12d1eb2cd1570633ee8f1b5ed2 - 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-r01-0069.json)